Style Functions

Constraint Functions

equal

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	First value
y	ℝ	Real Number	Second value

lessThan

Require that the value x is less than the value y with optional padding padding

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	First value
y	ℝ	Real Number	Second value
padding	ℝ	Real Number	Padding

greaterThan

Require that the value x is greater than the value y with optional padding padding

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	First value
y	ℝ	Real Number	Second value
padding	ℝ	Real Number	Padding

lessThanSq

Require that the value x is less than the value y, with steeper penalty

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	First value
y	ℝ	Real Number	Second value

greaterThanSq

Require that the value x is greater than the value y, with steeper penalty

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	First value
y	ℝ	Real Number	Second value

inRange

Require that the value x is in the range defined by [x0, x1].

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	Value
x0	ℝ	Real Number	Lower bound
x1	ℝ	Real Number	Upper bound

contains1D

Require that an interval [l1, r1] contains another interval [l2, r2]. If not possible, returns 0.

Parameters:

Name	Type	Type Description	Description
[l1, r1]	ℝ²	2d Vector of Reals	First interval
[l2, r2]	ℝ²	2d Vector of Reals	Second interval

disjointScalar

Make scalar c disjoint from a range left, right.

Parameters:

Name	Type	Type Description	Description
c	ℝ	Real Number	Scalar
left	ℝ	Real Number	Left bound
right	ℝ	Real Number	Right bound

perpendicular

Require that the vector defined by (q, p) is perpendicular from the vector defined by (r, p).

Parameters:

Name	Type	Type Description	Description
q	ℝⁿ	Vector of Reals	First point
p	ℝⁿ	Vector of Reals	Second point
r	ℝⁿ	Vector of Reals	Third point

collinear

Require that three points be collinear. This does not enforce a specific ordering of points, instead it takes the arrangement of points that is most easily satisfiable.

Parameters:

Name	Type	Type Description	Description
c1	ℝⁿ	Vector of Reals	First point
c2	ℝⁿ	Vector of Reals	Second point
c3	ℝⁿ	Vector of Reals	Third point

collinearOrdered

Require that three points be collinear and enforces the order of these points as provided.

Parameters:

Name	Type	Type Description	Description
c1	ℝⁿ	Vector of Reals	First point
c2	ℝⁿ	Vector of Reals	Second point
c3	ℝⁿ	Vector of Reals	Third point

onCanvas

Require that shape is on the canvas

Parameters:

Name	Type	Type Description	Description
shape	Shape	Any Shape	Shape
canvasWidth	ℝ	Real Number	Width of canvas
canvasHeight	ℝ	Real Number	Height of canvas

overlapping

Require that shape s1 overlaps shape s2 with some overlap overlap. based on the type of the shape, and with an optional overlap between them (e.g. if s1 should be overlapping s2 with margin overlap).

Parameters:

Name	Type	Type Description	Description
s1	Shape	Any Shape	Shape 1
s2	Shape	Any Shape	Shape 2
overlap	ℝ	Real Number	Overlap

overlappingEllipses

Require that ellipse e1 overlaps with ellipse e2 with some overlap overlap.

Parameters:

Name	Type	Type Description	Description
c1	ℝ²	2d Vector of Reals	Center of e1
rx1	ℝ	Real Number	Horizontal radius of e1
ry1	ℝ	Real Number	Vertical radius of e1
c2	ℝ²	2d Vector of Reals	Center of e2
rx2	ℝ	Real Number	Horizontal radius of e2
ry2	ℝ	Real Number	Vertical radius of e2
overlap	ℝ	Real Number	the least amount of overlap

overlappingCircleEllipse

Require that circle c overlaps with ellipse e with some overlap overlap.

Parameters:

Name	Type	Type Description	Description
c1	ℝ²	2d Vector of Reals	Center of c
r1	ℝ	Real Number	Radius of c
c2	ℝ²	2d Vector of Reals	Center of e
rx2	ℝ	Real Number	Horizontal radius of e
ry2	ℝ	Real Number	Vertical radius of e
overlap	ℝ	Real Number	the least amount of overlap

disjoint

Require that a shape s1 is disjoint from shape s2, based on the type of the shape, and with an optional padding between them (e.g. if s1 should be disjoint from s2 with margin padding).

Parameters:

Name	Type	Type Description	Description
s1	Shape	Any Shape	Shape 1
s2	Shape	Any Shape	Shape 2
padding	ℝ	Real Number	Padding

touching

Require that shape s1 is touching shape s2 based on the type of the shape, and with an optional padding between them (e.g. if s1 should be touching s2 with margin padding).

Parameters:

Name	Type	Type Description	Description
s1	Shape	Any Shape	Shape 1
s2	Shape	Any Shape	Shape 2
padding	ℝ	Real Number	Padding

contains

Require that a shape s1 contains another shape s2, based on the type of the shape, and with an optional padding between the sizes of the shapes (e.g. if s1 should contain s2 with margin padding).

Parameters:

Name	Type	Type Description	Description
s1	Shape	Any Shape	Shape 1
s2	Shape	Any Shape	Shape 2
padding	ℝ	Real Number	Padding

containsCircles

Require that a circle c1 contains another circle c2 with optional margin padding.

Parameters:

Name	Type	Type Description	Description
c1	ℝ²	2d Vector of Reals	Center of c1
r1	ℝ	Real Number	Radius of c1
c2	ℝ²	2d Vector of Reals	Center of c2
r2	ℝ	Real Number	Radius of c2
padding	ℝ	Real Number	Margin between the circles

containsPolys

Require that a polygon p1 contains another polygon p2 with optional margin padding.

Parameters:

Name	Type	Type Description	Description
pts1	(ℝ²)ⁿ	List of 2d Real Vectors	List of points for p1
pts2	(ℝ²)ⁿ	List of 2d Real Vectors	List of points for p2
padding	ℝ	Real Number	Margin between the polygons

containsPolyCircle

Require that a polygon p contains circle c with optional margin padding.

Parameters:

Name	Type	Type Description	Description
pts	(ℝ²)ⁿ	List of 2d Real Vectors	List of points for p
c	ℝ²	2d Vector of Reals	Center of c
r	ℝ	Real Number	Radius of c
padding	ℝ	Real Number	Margin between the polygon and the circle

containsPolyPoint

Require that a polygon p contains point pt with optional margin padding.

Parameters:

Name	Type	Type Description	Description
pts	(ℝ²)ⁿ	List of 2d Real Vectors	List of points for polygon p
pt	ℝ²	2d Vector of Reals	Location of point pt
padding	ℝ	Real Number	Margin between the polygon and the point

containsCirclePoint

Require that a circle c contains point pt with optional margin padding.

Parameters:

Name	Type	Type Description	Description
c	ℝ²	2d Vector of Reals	Center of c
r	ℝ	Real Number	Radius of c
pt	ℝ²	2d Vector of Reals	Location of point pt
padding	ℝ	Real Number	Margin between the polygon and the point

containsCirclePoly

Require that a circle c contains polygon p with optional margin padding.

Parameters:

Name	Type	Type Description	Description
c	ℝ²	2d Vector of Reals	Center of c
r	ℝ	Real Number	Radius of c
pts	(ℝ²)ⁿ	List of 2d Real Vectors	List of points for polygon p
padding	ℝ	Real Number	Margin between the polygon and the point

containsCircleRect

Require that a circle c contains rectangle r with optional padding padding.

Parameters:

Name	Type	Type Description	Description
c	ℝ²	2d Vector of Reals	Center of c
r	ℝ	Real Number	Radius of c
rect	(ℝ²)ⁿ	List of 2d Real Vectors	The top-right, top-left, bottom-left, bottom-right points (in that order) of rectangle rect
padding	ℝ	Real Number	Margin between the polygon and the point

containsRectCircle

Require that a rectangle r contains a circle c with optional margin padding.

Parameters:

Name	Type	Type Description	Description
rect	(ℝ²)ⁿ	List of 2d Real Vectors	The top-right, top-left, bottom-left, bottom-right points (in that order) of rectangle rect
c	ℝ²	2d Vector of Reals	Center of c
r	ℝ	Real Number	Radius of c
padding	ℝ	Real Number	Margin between the polygon and the point

containsRects

Requires that rect1 contains rect2 with some optional margin padding.

Parameters:

Name	Type	Type Description	Description
rect1	(ℝ²)ⁿ	List of 2d Real Vectors	The top-right, top-left, bottom-left, bottom-right points (in that order) of rectangle rect1
rect2	(ℝ²)ⁿ	List of 2d Real Vectors	The top-right, top-left, bottom-left, bottom-right points (in that order) points of rectangle rect2
padding	ℝ	Real Number	Margin between the polygon and the point

distributeHorizontally

Requires that the bounding boxes for shapes within a shapeList are evenly spaced horizontally with a set padding.

Parameters:

Name	Type	Type Description	Description	Default Value
shapes	Shape[]	List of shapes	List containing the top-left, top-right, bottom-right, bottom-left points (in that order) of the axis-aligned bounding box of a shape
padding	ℝ	Real Number	Margin between bounding boxes of shapes	0
leftToRight	true | false	Boolean Value	true if wanting to distribute left-to-right, false if wanting to distribute right-to-left	true

distributeVertically

Requires that the bounding boxes for shapes within a shapeList are evenly spaced vertically with a set padding.

Parameters:

Name	Type	Type Description	Description	Default Value
shapes	Shape[]	List of shapes	List containing the top-left, top-right, bottom-right, bottom-left points (in that order) of the axis-aligned bounding box of a shape
padding	ℝ	Real Number	Margin between bounding boxes of shapes	0
topToBottom	true | false	Boolean Value	true if wanting to distribute top-to-bottom, false if wanting to distribute bottom-to-top	true

disjointIntervals

Make two intervals disjoint. They must be 1D intervals (line-like shapes) sharing a y-coordinate.

Parameters:

Name	Type	Type Description	Description
s1	LineShape	Line Shape	Line 1
s2	LineShape	Line Shape	Line 2

isLocallyConvex

The shape should be locally convex (all angles between consecutive edges would have the same sign)

Parameters:

Name	Type	Type Description	Description
points	(ℝⁿ)ᵐ	List of Real Vectors	points of polygonal chain
closed	true | false	Boolean Value	whether the polygonic chain is closed

isConvex

The enclosed area should be convex

Parameters:

Name	Type	Type Description	Description
points	(ℝⁿ)ᵐ	List of Real Vectors	points of polygonal chain
closed	true | false	Boolean Value	whether the polygonic chain is closed

isEquilateral

All edges should have the same length

Parameters:

Name	Type	Type Description	Description
points	(ℝⁿ)ᵐ	List of Real Vectors	points of polygonal chain
closed	true | false	Boolean Value	whether the polygonic chain is closed

isEquiangular

All angles between consecutive edges should be equal

Parameters:

Name	Type	Type Description	Description
points	(ℝⁿ)ᵐ	List of Real Vectors	points of polygonal chain
closed	true | false	Boolean Value	whether the polygonic chain is closed

Objective Functions

minimal

Encourage the input value to be close to negative infinity

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	Value

maximal

Encourage the input value to be close to infinity

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	Value

equal

Encourage the inputs to have the same value: (x - y)^2

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	First value
y	ℝ	Real Number	Second value

nearVec

Encourage two vectors v1 and v2 to be near each other with distance offset.

Parameters:

Name	Type	Type Description	Description
v1	ℝⁿ	Vector of Reals	a vector
v2	ℝⁿ	Vector of Reals	a vector
offset	ℝ	Real Number	distance between two vectors

greaterThan

Encourage x to be greater than or equal to y: max(0,y - x)^2

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	First value
y	ℝ	Real Number	Second value

lessThan

Encourage x to be less than or equal to y: max(0,x - y)^2

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	First value
y	ℝ	Real Number	Second value

repelPt

Repel point a from another scalar b with weight weight.

Parameters:

Name	Type	Type Description	Description
weight	ℝ	Real Number	Weight
a	ℝⁿ	Vector of Reals	First point
b	ℝⁿ	Vector of Reals	Second point

repelScalar

Repel scalar c from another scalar d.

Parameters:

Name	Type	Type Description	Description
c	ℝ	Real Number	First scalar
d	ℝ	Real Number	Second scalar

inDirection

Encourage the point p to be in the direction direction with respect to point pRef. The direction vector does not need to be normalized. The offset parameter is the shortest allowed distance between the points.

Parameters:

Name	Type	Type Description	Description
p	ℝ²	2d Vector of Reals
pRef	ℝ²	2d Vector of Reals
direction	ℝ²	2d Vector of Reals
offset	ℝ	Real Number

below

Encourage the center of bottom to be below the center of top.

Parameters:

Name	Type	Type Description	Description	Default Value
bottom	Shape	Any Shape	shape on the bottom
top	Shape	Any Shape	shape on the top
offset	ℝ	Real Number	distance between the two centers	100

above

Encourage the center of top to be above the center of bottom.

Parameters:

Name	Type	Type Description	Description	Default Value
top	Shape	Any Shape	shape on the top
bottom	Shape	Any Shape	shape on the bottom
offset	ℝ	Real Number	distance between the two centers	100

leftwards

Encourage the center of left to be leftwards to the center of right.

Parameters:

Name	Type	Type Description	Description	Default Value
left	Shape	Any Shape	shape on the left
right	Shape	Any Shape	shape on the right
offset	ℝ	Real Number	distance between the two centers	100

rightwards

Encourage the center of right to be rightwards to the center of left.

Parameters:

Name	Type	Type Description	Description	Default Value
right	Shape	Any Shape	shape on the right
left	Shape	Any Shape	shape on the left
offset	ℝ	Real Number	distance between the two centers	100

sameCenter

Encourage shape s1 to have the same center position as shape s2.

Parameters:

Name	Type	Type Description	Description
s1	Shape	Any Shape	a shape
s2	Shape	Any Shape	a shape

notTooClose

Try to repel shapes s1 and s2 with some weight.

Parameters:

Name	Type	Type Description	Description	Default Value
s1	Shape	Any Shape	a shape
s2	Shape	Any Shape	a shape
weight	ℝ	Real Number	weight of repel	10

near

Try to place shape s1 near shape s2 (putting their centers at the same place).

Parameters:

Name	Type	Type Description	Description	Default Value
s1	Shape	Any Shape	a shape
s2	Shape	Any Shape	a shape
offset	ℝ	Real Number	offset	10

nearPt

Try to place shape s1 near a location (x, y).

Parameters:

Name	Type	Type Description	Description
s1	Shape	Any Shape	a shape
x	ℝ	Real Number	x
y	ℝ	Real Number	y

nonDegenerateAngle

Try to place shape s1 near a location (x, y).

Parameters:

Name	Type	Type Description	Description	Default Value
s0	Shape	Any Shape	a shape
s1	Shape	Any Shape	a shape
s2	Shape	Any Shape	a shape
strength	ℝ	Real Number	strength	20
range	ℝ	Real Number	range	10

centerLabelAbove

Parameters:

Name	Type	Type Description	Description
s1	LineShape	Line Shape
s2	EquationShape | ImageShape | RectangleShape | TextShape	any of: Equation Shape, or Image Shape, or Rectangle Shape, or Text Shape
w	ℝ	Real Number

centerLabel

Try to center a label s2 with respect to some shape s1.

Parameters:

Name	Type	Type Description	Description	Default Value
s1	LineShape | EquationShape | ImageShape | RectangleShape | TextShape	any of: Line Shape, or any of: Equation Shape, or Image Shape, or Rectangle Shape, or Text Shape
s2	EquationShape | ImageShape | RectangleShape | TextShape	any of: Equation Shape, or Image Shape, or Rectangle Shape, or Text Shape
w	ℝ	Real Number
padding	ℝ	Real Number		10

pointLineDist

Try to make distance between a point and a segment s1 equal to padding.

Parameters:

Name	Type	Type Description	Description
point	ℝ²	2d Vector of Reals
s1	LineShape	Line Shape
padding	ℝ	Real Number

isRegular

Try to make the shape regular

Parameters:

Name	Type	Type Description	Description
points	(ℝⁿ)ᵐ	List of Real Vectors	points of polygonal chain
closed	true | false	Boolean Value	whether the polygonic chain is closed

isEquilateral

Try to make the shape equilateral

Parameters:

Name	Type	Type Description	Description
points	(ℝⁿ)ᵐ	List of Real Vectors	points of polygonal chain
closed	true | false	Boolean Value	whether the polygonic chain is closed

isEquiangular

Try to make the shape equiangular

Parameters:

Name	Type	Type Description	Description
points	(ℝⁿ)ᵐ	List of Real Vectors	points of polygonal chain
closed	true | false	Boolean Value	whether the polygonic chain is closed

Computation Functions

makePath

See https://github.com/penrose/penrose/issues/716

Parameters:

Name	Type	Type Description	Description
start	ℝ²	2d Vector of Reals	Start point of the path
end	ℝ²	2d Vector of Reals	End point of the path
curveHeight	ℝ	Real Number	Height of the curve
padding	ℝ	Real Number	Padding between the curve and the labels

Returns: PathCmd (Path Command)

get

Return ith element of list `xs, assuming lists only hold floats.

Parameters:

Name	Type	Type Description	Description
xs	ℝⁿ	Vector of Reals	List of floats
i	ℕ	Natural Number	Index of the element to return

Returns: ℝ (Real Number)

rgba

Return a paint color of elements r, g, b, a (red, green, blue, opacity).

Parameters:

Name	Type	Type Description	Description
r	[0,1]	Real Number in Unit Interval	Red
g	[0,1]	Real Number in Unit Interval	Green
b	[0,1]	Real Number in Unit Interval	Blue
a	[0,1]	Real Number in Unit Interval	Opacity

Returns: Color (Color)

selectColor

Parameters:

Name	Type	Type Description	Description
color1	Color	Color	First color
color2	Color	Color	Second color
level	ℝ	Real Number	Level

Returns: Color (Color)

hsva

Return a paint color of elements h, s, v, a (hue, saturation, value, opacity).

Parameters:

Name	Type	Type Description	Description
h	ℝ	Real Number	Hue in [0, 360)
s	ℝ	Real Number	Saturation in [0, 100]
v	ℝ	Real Number	Value in [0, 100]
a	[0,1]	Real Number in Unit Interval	Opacity

Returns: Color (Color)

none

Return a paint of none (no paint)

Returns: Color (Color)

oneBasedElement

Index a point list using 1-based indexing.

Parameters:

Name	Type	Type Description	Description
points	(ℝⁿ)ᵐ	List of Real Vectors	list of points
i	ℤ⁺	Positive Integer	1-based index

Returns: ℝ² (2d Vector of Reals)

acosh

Return acosh(x).

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	x

Returns: ℝ (Real Number)

acos

Return acos(x).

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	x

Returns: ℝ (Real Number)

asin

Return asin(x).

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	x

Returns: ℝ (Real Number)

asinh

Return asinh(x).

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	x

Returns: ℝ (Real Number)

mod

Return mod(a, n).

Parameters:

Name	Type	Type Description	Description
a	ℝ	Real Number	a
n	ℝ	Real Number	n

Returns: ℝ (Real Number)

atan

Return atan(x).

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	x

Returns: ℝ (Real Number)

atan2

Return atan2(x, y).

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	x
y	ℝ	Real Number	y

Returns: ℝ (Real Number)

atanh

Return atanh(x).

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	x

Returns: ℝ (Real Number)

cbrt

Return cbrt(x).

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	x

Returns: ℝ (Real Number)

ceil

Return ceil(x).

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	x

Returns: ℝ (Real Number)

cos

Return cos(x).

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	x

Returns: ℝ (Real Number)

cosh

Return cosh(x).

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	x

Returns: ℝ (Real Number)

exp

Return exp(x).

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	x

Returns: ℝ (Real Number)

expm1

Return expm1(x).

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	x

Returns: ℝ (Real Number)

floor

Return floor(x).

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	x

Returns: ℝ (Real Number)

log

Return log(x).

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	x

Returns: ℝ (Real Number)

log2

Return log2(x).

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	x

Returns: ℝ (Real Number)

log10

Return log10(x).

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	x

Returns: ℝ (Real Number)

log1p

Return log1p(x).

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	x

Returns: ℝ (Real Number)

pow

Return pow(x, y).

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	x
y	ℝ	Real Number	y

Returns: ℝ (Real Number)

round

Return round(x).

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	x

Returns: ℝ (Real Number)

sign

Return sign(x).

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	x

Returns: ℝ (Real Number)

sin

Return sin(x).

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	x

Returns: ℝ (Real Number)

sinh

Return sinh(x).

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	x

Returns: ℝ (Real Number)

tan

Return tan(x).

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	x

Returns: ℝ (Real Number)

tanh

Return tanh(x).

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	x

Returns: ℝ (Real Number)

trunc

Return trunc(x).

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	x

Returns: ℝ (Real Number)

sum

Return the sum of elements in a vector.

Parameters:

Name	Type	Type Description	Description
xs	ℝⁿ	Vector of Reals	elements

Returns: ℝ (Real Number)

sumVectors

Return the sum of vectors in a list of vectors.

Parameters:

Name	Type	Type Description	Description
vecs	(ℝⁿ)ᵐ	List of Real Vectors	vectors

Returns: ℝⁿ (Vector of Reals)

maxList

Return the maximum of the elements in a vector.

Parameters:

Name	Type	Type Description	Description
xs	ℝⁿ	Vector of Reals	elements

Returns: ℝ (Real Number)

minList

Return the minimum of the elements in a vector.

Parameters:

Name	Type	Type Description	Description
xs	ℝⁿ	Vector of Reals	elements

Returns: ℝ (Real Number)

count

Return the number of the elements in a vector.

Parameters:

Name	Type	Type Description	Description
xs	ℝⁿ	Vector of Reals	elements

Returns: ℝ (Real Number)

dot

Return the dot product of v and w.

Parameters:

Name	Type	Type Description	Description
v	ℝⁿ	Vector of Reals	Vector v
w	ℝⁿ	Vector of Reals	Vector w

Returns: ℝ (Real Number)

identity

identity(n) returns the $n \times n$ identity matrix

$I = \left[ \begin{array}{cccc} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{array} \right].$

Parameters:

Name	Type	Type Description	Description
n	ℤ⁺	Positive Integer	dimension

Returns: (ℝⁿ)ᵐ (List of Real Vectors)

diagonal

diagonal(v) takes a vector $v$ of length $n$, and returns the $n \times n$ diagonal matrix

$D = \left[ \begin{array}{cccc} v_1 & 0 & \cdots & 0 \\ 0 & v_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & v_n \end{array} \right].$

Parameters:

Name	Type	Type Description	Description
v	ℝⁿ	Vector of Reals	vector of diagonal entries

Returns: (ℝⁿ)ᵐ (List of Real Vectors)

trace

trace(A) takes a square matrix $A$, and returns the trace $\text{tr}(A)$, equal to the sum of its diagonal entries.

Parameters:

Name	Type	Type Description	Description
A	(ℝⁿ)ᵐ	List of Real Vectors	a square matrix

Returns: ℝ (Real Number)

determinant

determinant(A) takes a $2 \times 2$, $3 \times 3$, or $4 \times 4$ matrix $A$, and returns its determinant $\text{det}(A)$.

Parameters:

Name	Type	Type Description	Description
A	(ℝⁿ)ᵐ	List of Real Vectors	a square matrix

Returns: ℝ (Real Number)

inverse

inverse(A) takes a $2 \times 2$, $3 \times 3$, or $4 \times 4$ matrix $A$, and returns its inverse $A^{-1}$. If the matrix is not invertible, the result may be numerically invalid (with INF or NaN entries).

Parameters:

Name	Type	Type Description	Description
A	(ℝⁿ)ᵐ	List of Real Vectors	a 2x2, 3x3, or 4x4 matrix

Returns: (ℝⁿ)ᵐ (List of Real Vectors)

outerProduct

outerProduct(u,v) takes two vectors $u$, $v$ of equal length $n$, and returns the $n \times n$ outer product matrix $A$, with entries $A_{ij} = u_i v_j$.

Parameters:

Name	Type	Type Description	Description
v	ℝⁿ	Vector of Reals	Vector v
w	ℝⁿ	Vector of Reals	Vector w

Returns: (ℝⁿ)ᵐ (List of Real Vectors)

crossProductMatrix

crossProductMatrix(v) takes a 3-vector $v$, and returns a $3 \times 3$ skew symmetric matrix $A^T = -A$ such that $Au = v \times u$ for any vector $u$.

Parameters:

Name	Type	Type Description	Description
v	ℝⁿ	Vector of Reals	Vector v

Returns: (ℝⁿ)ᵐ (List of Real Vectors)

matrix

matrix(a,b,c,d,e,f) specifies a transformation matrix

$\left[ \begin{array}{ccc} a & c & e \\ b & d & f \\ 0 & 0 & 1 \end{array} \right].$

This function mirrors the SVG/CSS matrix transform function.

Parameters:

Name	Type	Type Description	Description
a	ℝ	Real Number	top left entry
b	ℝ	Real Number	middle left entry
c	ℝ	Real Number	top center entry
d	ℝ	Real Number	middle center entry
e	ℝ	Real Number	top right entry
f	ℝ	Real Number	middle right entry

Returns: (ℝⁿ)ᵐ (List of Real Vectors)

matrix3d

matrix(a1, a2, a3, a4, b1, b2, b3, b4, c1, c2, c3, c4, d1, d2, d3, d4) specifies a transformation matrix

$\left[ \begin{array}{cccc} a1 & a2 & a3 & a4 \\ b1 & b2 & b3 & b4 \\ c1 & c2 & c3 & c4 \\ d1 & d2 & d3 & d4 \end{array} \right].$

This function mirrors the CSS matrix3d transform function.

Parameters:

Name	Type	Type Description	Description
a1	ℝ	Real Number	1st column of 1st row
b1	ℝ	Real Number	1st column of 2nd row
c1	ℝ	Real Number	1st column of 3rd row
d1	ℝ	Real Number	1st column of 4th row
a2	ℝ	Real Number	2nd column of 1st row
b2	ℝ	Real Number	2nd column of 2nd row
c2	ℝ	Real Number	2nd column of 3rd row
d2	ℝ	Real Number	2nd column of 4th row
a3	ℝ	Real Number	3rd column of 1st row
b3	ℝ	Real Number	3rd column of 2nd row
c3	ℝ	Real Number	3rd column of 3rd row
d3	ℝ	Real Number	3rd column of 4th row
a4	ℝ	Real Number	4th column of 1st row
b4	ℝ	Real Number	4th column of 2nd row
c4	ℝ	Real Number	4th column of 3rd row
d4	ℝ	Real Number	4th column of 4th row

Returns: (ℝⁿ)ᵐ (List of Real Vectors)

rotate

rotate(theta, [x], [y]) returns a counter-clockwise 2D rotation by an angle $\theta$, optionally around the point $(x,y)$. If no point is specified, the rotation is around the origin. Since this transformation is in general an affine transformation, it is encoded in homogeneous coordinates via the $3 \times 3$ matrix

$\left[ \begin{array}{rrr} \cos(\theta) & -\sin(\theta) & 0 \\ \sin(\theta) & \cos(\theta) & 0 \\ 0 & 0 & 1 \end{array} \right].$

For the $2 \times 2$ linear version, see rotate2d().)

Parameters:

Name	Type	Type Description	Description
theta	ℝ	Real Number	angle of rotation (in radians)
x	ℝ	Real Number	center of rotation (x coordinate)
y	ℝ	Real Number	center of rotation (y coordinate)

Returns: (ℝⁿ)ᵐ (List of Real Vectors)

rotate2d

rotate2d(theta) returns a 2D rotation around the origin by a given angle $\theta$, encoded via the $2 \times 2$ matrix

$\left[ \begin{array}{rr} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{array} \right].$

This matrix cannot directly be composed with 2D affine transformations; for the $3 \times 3$ affine version, see rotate().

Parameters:

Name	Type	Type Description	Description
theta	ℝ	Real Number	angle of rotation (in radians)

Returns: (ℝⁿ)ᵐ (List of Real Vectors)

rotate3d

rotate3d(theta, v) returns a 3D rotation by an angle $\theta$ around a unit axis $v$. The matrix is constructed via Rodrigues' rotation formula

$I + \sin(\theta)\hat{v} + (1-\cos(\theta))\hat{v}^2,$

where $I$ is the $3 \times 3$ identity matrix, and $\hat{v}$ is the cross product matrix associated with $v$ (see crossProductMatrix()). This matrix cannot directly be composed with 3D affine transformations; for the $4 \times 4$ affine version, see rotate3dh().

Parameters:

Name	Type	Type Description	Description
theta	ℝ	Real Number	angle of rotation (in radians)
v	ℝⁿ	Vector of Reals	axis of rotation (unit vector)

Returns: (ℝⁿ)ᵐ (List of Real Vectors)

rotate3dh

rotate3dh(theta, v) returns a 3D rotation by a given angle $\theta$ around a unit axis $v$. This transformation is encoded as a $4 \times 4$ matrix in homogeneous coordinates, so that it can be composed with 3D affine transformations. For the $3 \times 3$ linear version, see rotate3d().

Parameters:

Name	Type	Type Description	Description
theta	ℝ	Real Number	angle of rotation (in radians)
v	ℝⁿ	Vector of Reals	axis of rotation (unit vector)

Returns: (ℝⁿ)ᵐ (List of Real Vectors)

scale

scale(sx, sy) returns a nonuniform scaling by factors $s_x$, $s_y$ along $x$, $y$ axes, encoded via the matrix

$\left[ \begin{array}{ccc} s_x & 0 & 0 \\ 0 & s_y & 0 \\ 0 & 0 & 1 \end{array} \right].$

This transformation is encoded as a $3 \times 3$ matrix in homogeneous coordinates, so that it can be composed with 2D affine transformations. For the $2 \times 2$ linear version, see scale2d().

Parameters:

Name	Type	Type Description	Description
sx	ℝ	Real Number	horizontal scale factor
sy	ℝ	Real Number	vertical scale factor

Returns: (ℝⁿ)ᵐ (List of Real Vectors)

scale2d

scale2d(sx,sy) returns a $2 \times 2$ matrix representing nonuniform scaling by factors $s_x$, $s_y$ along $x$, $y$ axes, encoded via the matrix

$\left[ \begin{array}{cc} s_x & 0 \\ 0 & s_y \end{array} \right].$

This transformation is encoded as a $2 \times 2$ matrix that cannot directly be composed with 2D affine transformations. For the $3 \times 3$ affine version, see rotate().

Parameters:

Name	Type	Type Description	Description
sx	ℝ	Real Number	horizontal scale factor
sy	ℝ	Real Number	vertical scale factor

Returns: (ℝⁿ)ᵐ (List of Real Vectors)

scale3d

scale3d(sx, sy, sz) returns a nonuniform scaling by factors $s_x$, $s_y$, $s_z$ along $x$, $y$, $z$ axes, via the matrix

$\left[ \begin{array}{ccc} s_x & 0 & 0 \\ 0 & s_y & 0 \\ 0 & 0 & s_z \end{array} \right].$

This transformation is encoded as a $3 \times 3$ matrix that cannot directly be composed with 3D affine transformations. For the $4 \times 4$ affine version, see scale3dh().

Parameters:

Name	Type	Type Description	Description
sx	ℝ	Real Number	x scale factor
sy	ℝ	Real Number	y scale factor
sz	ℝ	Real Number	z scale factor

Returns: (ℝⁿ)ᵐ (List of Real Vectors)

scale3dh

scale3dh(sx, sy, sz) returns a $4 \times 4$ matrix representing nonuniform scaling by factors $s_x$, $s_y$, $s_z$ along $x$, $y$, $z$ axes, via the matrix

$\left[ \begin{array}{cccc} s_x & 0 & 0 & 0 \\ 0 & s_y & 0 & 0 \\ 0 & 0 & s_z & 0 \\ 0 & 0 & 0 & 1 \end{array} \right].$

This transformation is encoded as a $4 \times 4$ matrix in homogeneous coordinates, so that it can be composed with 3D affine transformations. For the $3 \times 3$ linear version, see scale3D().

Parameters:

Name	Type	Type Description	Description
sx	ℝ	Real Number	x scale factor
sy	ℝ	Real Number	y scale factor
sz	ℝ	Real Number	z scale factor

Returns: (ℝⁿ)ᵐ (List of Real Vectors)

skew

skew(a_x, a_y) takes angles $a_x$ and $a_y$, and returns a 2D skew transformation encoded by the matrix

$\left[ \begin{array}{ccc} 1 & \tan(a_x) & 0 \\ \tan(a_y) & 1 & 0 \\ 0 & 0 & 1 \end{array} \right].$

If $a_y$ is not defined, its default value is 0, resulting in a purely horizontal skewing. This transformation is encoded as a $3 \times 3$ matrix in homogeneous coordinates, so that it can be composed with affine transformations. For the linear version, see skew2d().

Parameters:

Name	Type	Type Description	Description
ax	ℝ	Real Number	horizontal angle
ay	ℝ	Real Number	vertical angle

Returns: (ℝⁿ)ᵐ (List of Real Vectors)

skew2d

skew2d(a_x, a_y) takes angles $a_x$ and $a_y$, and returns a 2D skew transformation encoded via the matrix

$\left[ \begin{array}{cc} 1 & \tan(a_x) \\ \tan(a_y) & 1 \end{array} \right].$

If $a_y$ is not defined, its default value is 0, resulting in a purely horizontal skewing. This transformation is encoded as a $2 \times 2$ matrix that cannot directly be composed with 2D affine transformations. For the $3 \times 3$ affine version, see skew().

Parameters:

Name	Type	Type Description	Description
ax	ℝ	Real Number	horizontal angle
ay	ℝ	Real Number	vertical angle

Returns: (ℝⁿ)ᵐ (List of Real Vectors)

shear

shear(u,v) takes two $n$-dimensional vectors $u$ and $v$, and returns a transformation that shears any given point $x$ in the direction $u$ according to its extent along the direction $v$, i.e., that performs the transformation

$x \mapsto x + \langle v,x \rangle u.$

This transformation is encoded as an $(n+1) \times (n+1)$ matrix in homogeneous coordinates, so that it can be composed with affine transformations. For the linear version, see shear2d() or shear3d().

Parameters:

Name	Type	Type Description	Description
u	ℝⁿ	Vector of Reals	offset direction
v	ℝⁿ	Vector of Reals	shear axis

Returns: (ℝⁿ)ᵐ (List of Real Vectors)

shear2d

shear2d(u,v) takes two 2-dimensional vectors $u$ and $v$, and returns a transformation that shears any given point $x$ in the direction $u$ according to its extent along the direction $v$, i.e., that performs the transformation

$x \mapsto x + \langle v,x \rangle u.$

This transformation is encoded as a $2 \times 2$ matrix that cannot directly be composed with 2-dimensional affine transformations. For the affine version, see shear().

Parameters:

Name	Type	Type Description	Description
u	ℝⁿ	Vector of Reals	offset direction
v	ℝⁿ	Vector of Reals	shear axis

Returns: (ℝⁿ)ᵐ (List of Real Vectors)

shear3d

shear3d(u,v) takes two 3-dimensional vectors $u$ and $v$, returns a transformation that shears any given point $x$ in the direction $u$ according to its extent along the direction $v$, i.e., that performs the transformation

$x \mapsto x + \langle v,x \rangle u.$

This transformation is encoded as a $3 \times 3$ matrix that cannot directly be composed with 3-dimensional affine transformations. For the affine version, see shear().

Parameters:

Name	Type	Type Description	Description
u	ℝⁿ	Vector of Reals	offset direction
v	ℝⁿ	Vector of Reals	shear axis

Returns: (ℝⁿ)ᵐ (List of Real Vectors)

translate

translate(x,y) returns a translation by the given offset $(x,y)$. If $y$ is not specified, it is assumed to be $0$. Since translation is affine rather than linear, it is encoded as a $3 \times 3$ matrix in homogeneous coordinates, namely

$T = \left[ \begin{array}{ccc} 1 & 0 & x \\ 0 & 1 & y \\ 0 & 0 & 1 \end{array} \right].$

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	horizontal offset
y	ℝ	Real Number	vertical offset

Returns: (ℝⁿ)ᵐ (List of Real Vectors)

translate3dh

translate3dh(x,y,z) returns a translation by $(x,y,z)$. Since translation is affine rather than linear, it is encoded as a $4 \times 4$ matrix in homogeneous coordinates, namely

$T = \left[ \begin{array}{cccc} 1 & 0 & 0 & x \\ 0 & 1 & 0 & y \\ 0 & 0 & 0 & z \\ 0 & 0 & 0 & 1 \end{array} \right].$

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	x offset
y	ℝ	Real Number	y offset
z	ℝ	Real Number	z offset

Returns: (ℝⁿ)ᵐ (List of Real Vectors)

lookAt

lookAt(eye, center, up) returns a $4 \times 4$ viewing matrix derived from an eye point $e$, a reference point $c$ indicating the center of the scene, and an up vector $u$. The matrix maps the reference point to the negative z axis and the eye point to the origin. When a typical projection matrix is used, the center of the scene therefore maps to the center of the viewport. Similarly, the direction described by the up vector projected onto the viewing plane is mapped to the positive y axis so that it points upward in the viewport. The up vector must not be parallel to the line of sight from the eye point to the reference point. The matrix is given explicitly by $V = MT$ where $T$ is a translation by $-e$ (a la translate3dh), and $M$ is given by

$M = \left[ \begin{array}{rrrc} s_1 & s_2 & s_3 & 0 \\ v_1 & v_2 & v_3 & 0 \\ -f_1 & -f_2 & -f_3 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right],$

where $f := (c-e)/|c-e|$ is the unit vector from the eye to the center, $s = f \times u$ is a vector pointing to the side, and $v = s \times f$ is a vector in roughly the same direction as $u$ that completes an orthonormal basis with $s$ and $f$ (hence, $M$ is a rotation matrix).

Parameters:

Name	Type	Type Description	Description
eye	ℝⁿ	Vector of Reals	position of the eye point
center	ℝⁿ	Vector of Reals	position of the reference point
up	ℝⁿ	Vector of Reals	unit vector in the upward direction

Returns: (ℝⁿ)ᵐ (List of Real Vectors)

perspective

perspective(fovy, aspect, [zNear], [zFar]) returns a $4 \times 4$ perspective projection matrix, given a vertical field of view $\theta_y$, an aspect ratio $a$, and optional near/far $z$-values, $z_0$ and $z_1$. The aspect ratio should match the aspect ratio of the associated viewport. For example, an aspect ratio of 2 means the viewer's angle of view is twice as wide in $x$ as it is in $y$. Coordinates within $[z_0,z_1]$ will get mapped to depth values in the range $[0,1]$. Letting $f := \cot(\theta_y/2)$, the perspective matrix is given by

$P = \left[ \begin{array}{cccc} f/a & 0 & 0 & 0 \\ 0 & f & 0 & 0 \\ 0 & 0 & \frac{z_0+z_1}{z_0-z_1} & \frac{2 z_0 z_1}{z_0 - z_1} \\ 0 & 0 & -1 & 0 \end{array} \right].$

Parameters:

Name	Type	Type Description	Description	Default Value
fovy	ℝ	Real Number	field of view angle, in degrees, in the y direction
aspect	ℝ	Real Number	aspect ratio that determines the field of view in the x direction, equal to the ratio of x (width) to y (height)
zNear	ℝ	Real Number	distance from the viewer to the near clipping plane (always positive), with a default value of 0.1	0.1
zFar	ℝ	Real Number	distance from the viewer to the far clipping plane (always positive), with a default value of 100.0	100

Returns: (ℝⁿ)ᵐ (List of Real Vectors)

ortho

ortho(left, right, bottom, top, [zNear], [zFar]) returns a $4 \times 4$ transformation that produces a parallel projection, given four horizontal/vertical clipping planes, and an optional near/far clipping plane. Letting $l$, $r$, $b$, and $t$ denote the left/right/bottom/top clipping planes, respectively, the projection matrix is given by

$P = \left[ \begin{array}{cccc} \frac{2}{r-l} & 0 & 0 & -u_x \\ 0 & \frac{2}{t-b} & 0 & -u_y \\ 0 & 0 & \frac{-2}{z_1-z_0} & -u_z \\ 0 & 0 & 0 & 1 \\ \end{array} \right],$

where

$\begin{array}{rcl} u_x &=& (r + l) / (r - l), \\ u_y &=& (t + b) / (t - b), \\ u_z &=& (z_1 + z_0) / (z_1 - z_0). \end{array}$

Parameters:

Name	Type	Type Description	Description	Default Value
Left	ℝ	Real Number	coordinate of the left vertical clipping plane
Right	ℝ	Real Number	coordinate of the right vertical clipping plane
Bottom	ℝ	Real Number	coordinate of the bottom horizontal clipping plane
Top	ℝ	Real Number	coordinate of the top horizontal clipping plane
zNear	ℝ	Real Number	distance to the nearer depth clipping plane (negative if the plane is behind the viewer), with a default value of 0.1	0.1
zFar	ℝ	Real Number	distance to the farther depth clipping plane (negative if the plane is behind the viewer), with a default value of 100	100

Returns: (ℝⁿ)ᵐ (List of Real Vectors)

project

project(p, model, proj, view) transforms the specified 3D object coordinates $p$ into 2D window coordinates $q$ using a given $4 \times 4$ model transformation $M$, $4 \times 4$ projection transformation $P$, and viewport $V = [ \begin{array}{cccc} x & y & w & h \end{array} ]$, where $x$, $y$ give the lower-left corner of the canvas, and $w$, $h$ are its width and height. To also recover the depth, see projectDepth(). The projection is performed by first computing $r := PM\hat{p}$, where $\hat{p} := (p,1)$ are the homogeneous coordinates for $p$. The first two components of $q := r/r_w$ (where $r_w$ is the final component of $r$) gives coordinates in the range $[-1,1] \times [-1,1]$, which are then stretched to the range $[x,x+w] \times [y,y+h]$.

Parameters:

Name	Type	Type Description	Description
p	ℝⁿ	Vector of Reals	3D object coordinates (x,y,z)
model	(ℝⁿ)ᵐ	List of Real Vectors	4x4 modelview matrix
proj	(ℝⁿ)ᵐ	List of Real Vectors	4x4 projection matrix
view	ℝⁿ	Vector of Reals	viewport (x, y, width, height)

Returns: ℝⁿ (Vector of Reals)

projectDepth

projectDepth(p, model, proj, view) transforms the specified 3D object coordinates $p$ into 2D window coordinates $q$ using a given $4 \times 4$ model transformation $M$, $4 \times 4$ projection transformation $P$, and viewport $V = [ \begin{array}{cccc} x & y & w & h \end{array} ]$, where $x$, $y$ give the lower-left corner of the canvas, and $w$, $h$ are its width and height. It returns the projected coordinates, as well as the depth relative to the view. See project() for further information.

Parameters:

Name	Type	Type Description	Description
p	ℝⁿ	Vector of Reals	3D object coordinates (x,y,z)
model	(ℝⁿ)ᵐ	List of Real Vectors	4x4 modelview matrix
proj	(ℝⁿ)ᵐ	List of Real Vectors	4x4 projection matrix
view	ℝⁿ	Vector of Reals	viewport (x, y, width, height)

Returns: ℝⁿ (Vector of Reals)

projectList

projectList(P, model, proj, view) transforms a list of 3D object coordinates $P = p_1, \ldots, p_k$ into window 2D coordinates using a given $4 \times 4$ model transformation $M$, $4 \times 4$ projection transformation $P$, and viewport $V = [ \begin{array}{cccc} x & y & w & h \end{array} ]$, where $x$, $y$ give the lower-left corner of the canvas, and $w$, $h$ are its width and height. Returns the list of projected 2D coordinates $q_1, \ldots, q_k$. See project() for further information.

Parameters:

Name	Type	Type Description	Description
p	(ℝⁿ)ᵐ	List of Real Vectors	list of 3D object coordinates (x,y,z)
model	(ℝⁿ)ᵐ	List of Real Vectors	4x4 modelview matrix
proj	(ℝⁿ)ᵐ	List of Real Vectors	4x4 projection matrix
view	ℝⁿ	Vector of Reals	viewport (x, y, width, height)

Returns: (ℝ²)ⁿ (List of 2d Real Vectors)

matrixMultiplyList

matrixMultiplyList(A, V) multiplies each vector $v_1, \ldots, v_k$ in the list $V$ by the given $n \times n$ matrix $A$, returning the list of products. List elements must all be vectors of length $n$.

Parameters:

Name	Type	Type Description	Description
A	(ℝⁿ)ᵐ	List of Real Vectors	nxn matrix
V	(ℝⁿ)ᵐ	List of Real Vectors	list of n-dimensional vectors

Returns: (ℝⁿ)ᵐ (List of Real Vectors)

fromHomogeneous

fromHomogeneous(q) takes a vector $q$ of length $n+1$, encoding a point in $n$-dimensional homogeneous coordinates, and returns a vector $p$ of length $n$, encoding the same point in Cartesian coordinates, i.e,. $p = (q_1, \ldots, q_{k-1})/q_k$.

Parameters:

Name	Type	Type Description	Description
q	ℝⁿ	Vector of Reals	homogeneous coordinates

Returns: ℝⁿ (Vector of Reals)

fromHomogeneousList

fromHomogeneousList(Q) takes a list $Q = q_1, \ldots, q_k$ of vectors of length $n+1$, encoding points in $n$-dimensional homogeneous coordinates, and returns a list $P = p_1, \ldots, p_k$ of vectors of length $n$, encoding the same points in Cartesian coordinates.

Parameters:

Name	Type	Type Description	Description
Q	(ℝⁿ)ᵐ	List of Real Vectors	list of points in homogeneous coordinates

Returns: (ℝⁿ)ᵐ (List of Real Vectors)

toHomogeneous

toHomogeneous(p) takes a vector $p$ of length $n$, encoding a point in $n$-dimensional Cartesian coordinates, and returns a vector $q$ of length $n+1$, encoding the same point in homogeneous coordinates, i.e., $q = (p,1)$.

Parameters:

Name	Type	Type Description	Description
p	ℝⁿ	Vector of Reals	Cartesian coordinates

Returns: ℝⁿ (Vector of Reals)

toHomogeneousList

toHomogeneousList(P) takes a list $P = p_1, \ldots, p_k$ of vectors of length $n$, encoding points in $n$-dimensional Cartesian coordinates, and returns a list $Q = q_1, \ldots, q_k$ of vectors of length $n+1$, encoding the same points in homogeneous coordinates.

Parameters:

Name	Type	Type Description	Description
P	(ℝⁿ)ᵐ	List of Real Vectors	list of points in Cartesian coordinates

Returns: (ℝⁿ)ᵐ (List of Real Vectors)

toHomogeneousMatrix

toHomogeneousMatrix(A) takes a square $n \times n$ matrix $A$ representing a spatial transformation in $A$ dimensions, and returns an $(n+1) \times (n+1)$ matrix representing the same transformation in homogeneous coordinates.

Parameters:

Name	Type	Type Description	Description
A	(ℝⁿ)ᵐ	List of Real Vectors	matrix encoding linear transformation

Returns: (ℝⁿ)ᵐ (List of Real Vectors)

length

Return the length of the Line shape.

Parameters:

Name	Type	Type Description	Description
l	LineShape	Line Shape	A line

Returns: ℝ (Real Number)

normalize

Return the normalized version of vector v.

Parameters:

Name	Type	Type Description	Description
v	ℝⁿ	Vector of Reals	Vector v

Returns: ℝⁿ (Vector of Reals)

pathFromPoints

Given a list of points pts, returns a PathData that can be used as input to the Path shape's pathData attribute to be drawn on the screen.

Parameters:

Name	Type	Type Description	Description
pathType	"open" | "closed"	Path Type	Path Type
pts	(ℝ²)ⁿ	List of 2d Real Vectors	List of points

Returns: PathCmd (Path Command)

quadraticCurveFromPoints

Given a list of points pts, returns a PathData that can be used as input to the Path shape's pathData attribute to be drawn on the screen.

Parameters:

Name	Type	Type Description	Description
pathType	"open" | "closed"	Path Type	Path Type
pts	(ℝ²)ⁿ	List of 2d Real Vectors	List of points

Returns: PathCmd (Path Command)

interpolateQuadraticFromPoints

Draw a curve interpolating three given points. (Note that this is different from specifying the three control points of a quadratic Bézier curve, since a Bézier does not interpolate the middle control point.)

Parameters:

Name	Type	Type Description	Description
pathType	"open" | "closed"	Path Type	Path Type
p0	ℝ²	2d Vector of Reals	First point
p1	ℝ²	2d Vector of Reals	Second point
p2	ℝ²	2d Vector of Reals	Third point

Returns: PathCmd (Path Command)

cubicCurveFromPoints

Given a list of points pts, returns a PathData that can be used as input to the Path shape's pathData attribute to be drawn on the screen.

Parameters:

Name	Type	Type Description	Description
pathType	"open" | "closed"	Path Type	Path type
pts	(ℝ²)ⁿ	List of 2d Real Vectors	List of points

Returns: PathCmd (Path Command)

connectPaths

Given a list of PathDatas, returns a PathData representing the union of these paths with lines connecting the start and end points.

Parameters:

Name	Type	Type Description	Description
pathType	"open" | "closed"	Path Type	Path type
pathDataList	PathData[]	List of paths	List of path data

Returns: PathCmd (Path Command)

concatenatePaths

Given a list of PathDatas, return the union of these paths.

Parameters:

Name	Type	Type Description	Description
pathDataList	PathData[]	List of paths	List of path data

Returns: PathCmd (Path Command)

joinPaths

Given a list of PathDatas, join them into one SVG path. For correct results, the end points and start pointsof each path must already coincide.

Parameters:

Name	Type	Type Description	Description
pathDataList	PathData[]	List of paths	List of path data

Returns: PathCmd (Path Command)

Penrose

Return path data describing an "impossible polygon."

Parameters:

Name	Type	Type Description	Description	Default Value
center	ℝⁿ	Vector of Reals	(x,y) translation	[ 0, 0 ]
radius	ℝ	Real Number	radius of outer polygon (must be positive)	50
holeSize	ℝ	Real Number	radius of inner polygon as a fraction of the outer radius (in range (0,1])	0.35
angle	ℝ	Real Number	angle of rotation	0
nSides	ℤ⁺	Positive Integer	number of sides (integer ≥ 3)	5
chirality	String	String	either "cw" for clockwise, or "ccw" for counterclockwise	ccw

Returns: PathCmd (Path Command)

firstPoint

Returns the first point in a list.

Parameters:

Name	Type	Type Description	Description
points	(ℝⁿ)ᵐ	List of Real Vectors	list of points

Returns: ℝ² (2d Vector of Reals)

lastPoint

Returns the last point in a list.

Parameters:

Name	Type	Type Description	Description
points	(ℝⁿ)ᵐ	List of Real Vectors	list of points

Returns: ℝ² (2d Vector of Reals)

averagePoint

Returns the average (mean) of all points in a list.

Parameters:

Name	Type	Type Description	Description
points	(ℝⁿ)ᵐ	List of Real Vectors	list of points

Returns: ℝⁿ (Vector of Reals)

interpolatingSpline

Returns path data for a curve that smoothly interpolates the given points. Interpolation is performed via a Catmull-Rom spline.

Parameters:

Name	Type	Type Description	Description	Default Value
pathType	"open" | "closed"	Path Type	either "open" or "closed."
points	(ℝⁿ)ᵐ	List of Real Vectors	points to be interpolated
tension	ℝ	Real Number	smoothness of curve (0=piecewise linear, .25=default)	0.25

Returns: PathCmd (Path Command)

diffusionProcess

Return n points sampled from a diffusion process starting at X0, with covariance matrix A and constant drift omega. This path approximately integrates the stochastic differential equation dX_t = omega dt + A dW_t, where W_t is a Wiener process.

Parameters:

Name	Type	Type Description	Description
n	ℤ⁺	Positive Integer	number of points
X0	ℝ²	2d Vector of Reals	starting location
A	(ℝⁿ)ᵐ	List of Real Vectors	covariance matrix
omega	ℝ²	2d Vector of Reals	drift direction

Returns: (ℝ²)ⁿ (List of 2d Real Vectors)

unitMark

Return two points parallel to line s1 using its normal line s2.

Parameters:

Name	Type	Type Description	Description
s1	LineShape	Line Shape
s2	LineShape	Line Shape
padding	ℝ	Real Number

Returns: (ℝ²)ⁿ (List of 2d Real Vectors)

unitMark2

Return two points to "cap off" the line made in unitMark.

Parameters:

Name	Type	Type Description	Description
[start, end]	(ℝ²)ⁿ	List of 2d Real Vectors
t	String	String
size	ℝ	Real Number

Returns: (ℝ²)ⁿ (List of 2d Real Vectors)

arc

Return series of elements that can render an arc SVG. See: https://css-tricks.com/svg-path-syntax-illustrated-guide/ for the "A" spec. Returns elements that can be passed to Path shape spec to render an SVG arc.

Parameters:

Name	Type	Type Description	Description
pathType	"open" | "closed"	Path Type	The path type: either "open" or "closed." whether the SVG should automatically draw a line between the final point and the start point
start	ℝ²	2d Vector of Reals	coordinate to start drawing the arc
end	ℝ²	2d Vector of Reals	coordinate to finish drawing the arc
[width, height]	ℝ²	2d Vector of Reals	width and height of the ellipse to draw the arc along
rotation	ℝ	Real Number	angle in degrees to rotate ellipse about its center
largeArc	ℝ	Real Number	0 to draw shorter of 2 arcs, 1 to draw longer
arcSweep	ℝ	Real Number	0 to rotate CCW, 1 to rotate CW

Returns: PathCmd (Path Command)

circularArc

Return path data that describes a circular arc. The arc is equivalent to the parametric curve center + r*(cos(t),sin(t)) for t in the range [theta0,theta1]. More general arcs (e.g., along an ellipse) can be drawn using arc().

Parameters:

Name	Type	Type Description	Description
pathType	"open" | "closed"	Path Type	The path type: either "open" or "closed." whether the SVG should automatically draw a line between the final point and the start point
center	ℝ²	2d Vector of Reals	circle center
r	ℝ	Real Number	circle radius
theta0	ℝ	Real Number	start angle in radians
theta1	ℝ	Real Number	end angle in radians

Returns: PathCmd (Path Command)

repeatedArcs

Generate multiple concentric arcs. Useful for denoting equal angles.

Parameters:

Name	Type	Type Description	Description
innerStart	ℝ²	2d Vector of Reals	coordinate to start drawing the inner arc
innerEnd	ℝ²	2d Vector of Reals	coordinate to end the inner arc
outerStart	ℝ²	2d Vector of Reals	coordinate to start drawing the outer arc
outerEnd	ℝ²	2d Vector of Reals	coordinate to end the outer arc
innerRadius	ℝ²	2d Vector of Reals	radii of the ellipse to draw the inner arc along (width, height)
repeat	ℤ⁺	Positive Integer	number of times to repeat the arc
spacing	ℝ	Real Number	spacing between arcs
arcSweep	ℝ	Real Number	arc length to sweep

Returns: PathCmd (Path Command)

wedge

Return series of elements that render a "wedge", which is the same as the arc above except that it's connected to the circle center and filled. Returns elements that can be passed to Path shape spec to render an SVG arc.

Parameters:

Name	Type	Type Description	Description
center	ℝ²	2d Vector of Reals	center of the circle on which the arc sits
start	ℝ²	2d Vector of Reals	coordinate to start drawing the arc
end	ℝ²	2d Vector of Reals	coordinate to finish drawing the arc
radius	ℝ²	2d Vector of Reals	width and height of the ellipse to draw the arc along (i.e. [width, height])
rotation	ℝ	Real Number	angle in degrees to rotate ellipse about its center
largeArc	ℝ	Real Number	0 to draw shorter of 2 arcs, 1 to draw longer
arcSweep	ℝ	Real Number	0 to rotate CCW, 1 to rotate CW

Returns: PathCmd (Path Command)

ptOnLine

Find the point that is located at dist r along a line between p1 and p2. Returns vector representation of the point of intersection.

Parameters:

Name	Type	Type Description	Description
p1	ℝⁿ	Vector of Reals	start point of line segment
p2	ℝⁿ	Vector of Reals	endpoint of line segment
r	ℝ	Real Number	distance from p1 to travel along the line

Returns: ℝⁿ (Vector of Reals)

arcSweepFlag

Return 0 if direction of rotation is CCW, 1 if direction of rotation is CW.

Parameters:

Name	Type	Type Description	Description
[x1, y1]	ℝ²	2d Vector of Reals	x, y coordinates of the circle/ellipse that the arc is drawn on
start	ℝ²	2d Vector of Reals	start point of the arc
end	ℝ²	2d Vector of Reals	end point of the arc

Returns: ℝ (Real Number)

angleBetween

Return the unsigned angle between vectors u, v, in radians. Assumes that both u and v have nonzero magnitude. The returned value will be in the range [0,pi].

Parameters:

Name	Type	Type Description	Description
u	ℝⁿ	Vector of Reals	A vector
v	ℝⁿ	Vector of Reals	A vector

Returns: ℝ (Real Number)

angleFrom

Return the signed angle from vector u to vector v, in radians. Assumes that both u and v are 2D vectors and have nonzero magnitude. The returned value will be in the range [-pi,pi].

Parameters:

Name	Type	Type Description	Description
u	ℝⁿ	Vector of Reals	A vector
v	ℝⁿ	Vector of Reals	A vector

Returns: ℝ (Real Number)

cross2D

Return the 2D cross product of u and v, equal to the determinant of the 2x2 matrix [u v]

Parameters:

Name	Type	Type Description	Description
u	ℝ²	2d Vector of Reals	A vector
v	ℝ²	2d Vector of Reals	A vector

Returns: ℝ (Real Number)

cross

Return the 3D cross product of 3D vectors u and v.

Parameters:

Name	Type	Type Description	Description
u	ℝ³	3d Vector of Reals	A vector
v	ℝ³	3d Vector of Reals	A vector

Returns: ℝ³ (3d Vector of Reals)

lineLineIntersection

Return the intersection of a line passing through a0 and a1 with a line passing through b0 and b1.

Parameters:

Name	Type	Type Description	Description
a0	ℝ²	2d Vector of Reals	First point of first line
a1	ℝ²	2d Vector of Reals	Second point of first line
b0	ℝ²	2d Vector of Reals	First point of second line
b1	ℝ²	2d Vector of Reals	Second point of second line

Returns: ℝ² (2d Vector of Reals)

midpoint

Return a point located at the midpoint between pts start and end

Parameters:

Name	Type	Type Description	Description
start	ℝⁿ	Vector of Reals	First point
end	ℝⁿ	Vector of Reals	Second point

Returns: ℝⁿ (Vector of Reals)

midpointOffset

Return a point located at the midpoint of a line s1 but offset by padding in its normal direction (for labeling).

Parameters:

Name	Type	Type Description	Description
s1	LineShape	Line Shape	A line
padding	ℝ	Real Number	Padding between midpoint and label

Returns: ℝ² (2d Vector of Reals)

chevron

Return a list of points for a chevron shape comprised of two line segments intersecting at a right angle at the midpoint of s1, which can then be passed to pathFromPoints to draw the chevron.

Parameters:

Name	Type	Type Description	Description
s1	LineShape	Line Shape	A line
padding	ℝ	Real Number	Length of each line segment

Returns: (ℝ²)ⁿ (List of 2d Real Vectors)

innerPointOffset

Return a point located at padding of a line s1 offset by padding in its normal direction (for making right angle markers).

Parameters:

Name	Type	Type Description	Description
pt1	ℝ²	2d Vector of Reals	First point
pt2	ℝ²	2d Vector of Reals	Second point
pt3	ℝ²	2d Vector of Reals	Third point
padding	ℝ	Real Number	Offset from line to returned point

Returns: ℝ² (2d Vector of Reals)

ticksOnLine

Create equally spaced tick marks centered at the midpoint of a line

Parameters:

Name	Type	Type Description	Description
pt1	ℝ²	2d Vector of Reals	starting point of a line
pt2	ℝ²	2d Vector of Reals	ending point of a line
spacing	ℝ	Real Number	space in px between each tick
numTicks	ℤ⁺	Positive Integer	number of tick marks to create
tickLength	ℝ	Real Number	1/2 length of each tick

Returns: PathCmd (Path Command)

orientedSquare

Given two orthogonal segments that intersect at intersection, and a size len return a path comprised of three points that describe a perpendicular mark at the angle where the segments intersect.

Parameters:

Name	Type	Type Description	Description
s1	LineShape	Line Shape	First line segment
s2	LineShape	Line Shape	Second line segment
intersection	ℝ²	2d Vector of Reals	Point of intersection
len	ℝ	Real Number	Side length of square marker

Returns: PathCmd (Path Command)

triangle

Given three lines l1, l2, l3 that already form a triangle, return a path that describes the triangle (which can then be filled, etc.).

Parameters:

Name	Type	Type Description	Description
l1	LineShape	Line Shape	First line
l2	LineShape	Line Shape	Second line
l3	LineShape	Line Shape	Third line

Returns: PathCmd (Path Command)

average2

Return the average of floats x and y.

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	x
y	ℝ	Real Number	y

Returns: ℝ (Real Number)

average

Return the average of the floats in the list xs.

Parameters:

Name	Type	Type Description	Description
xs	ℝⁿ	Vector of Reals	xs

Returns: ℝ (Real Number)

unit

Return the normalized version of vector v.

Parameters:

Name	Type	Type Description	Description
v	ℝⁿ	Vector of Reals	v

Returns: ℝⁿ (Vector of Reals)

random

Uniformly sample a random value in the range from minVal to maxVal.

Parameters:

Name	Type	Type Description	Description
minVal	ℝ	Real Number	minimum value
maxVal	ℝ	Real Number	maximum value

Returns: ℝ (Real Number)

unitRandom

Uniformly sample a random value in the range [0,1).

Returns: ℝ (Real Number)

diskRandom

Sample the uniform distribution on the unit disk.

Returns: ℝⁿ (Vector of Reals)

circleRandom

Sample the uniform distribution on the unit circle.

Returns: ℝⁿ (Vector of Reals)

sphereRandom

Sample the uniform distribution on the unit sphere.

Returns: ℝⁿ (Vector of Reals)

normalRandom

Sample a normal distribution with mean 0 and standard deviation 1.

Returns: ℝ (Real Number)

triangleRandom

Sample a point from the uniform distribution over a triangle with vertices a, b, and c.

Parameters:

Name	Type	Type Description	Description
a	ℝ²	2d Vector of Reals	First vertex
b	ℝ²	2d Vector of Reals	Second vertex
c	ℝ²	2d Vector of Reals	Third vertex

Returns: ℝⁿ (Vector of Reals)

sampleColor

Sample a random color once, with opacity alpha and color type colorType ("rgb" or "hsv").

Parameters:

Name	Type	Type Description	Description
alpha	[0,1]	Real Number in Unit Interval	Opacity
colorType	"rgb" | "hsv"	Color Type	Color model

Returns: Color (Color)

randomIndex

Uniformly sample a random integer value in the range from minIndex to maxIndex.

Parameters:

Name	Type	Type Description	Description
minIndex	ℝ	Real Number	minimum index
maxIndex	ℝ	Real Number	maximum index

Returns: ℤ⁺ (Positive Integer)

setOpacity

Set the opacity of a color color to frac.

Parameters:

Name	Type	Type Description	Description
color	Color	Color	Color
frac	[0,1]	Real Number in Unit Interval	Opacity

Returns: Color (Color)

mul

Multiply a matrix m and a vector v (where v is implicitly treated as a column vector).

Parameters:

Name	Type	Type Description	Description
m	(ℝⁿ)ᵐ	List of Real Vectors	A matrix
v	ℝⁿ	Vector of Reals	A vector

Returns: ℝⁿ (Vector of Reals)

barycenter

Return the barycenter of the triangle with vertices a, b, c.

Parameters:

Name	Type	Type Description	Description
a	ℝ²	2d Vector of Reals	First vertex
b	ℝ²	2d Vector of Reals	Second vertex
c	ℝ²	2d Vector of Reals	Third vertex

Returns: ℝ² (2d Vector of Reals)

circumcenter

Return the circumcenter of the triangle with vertices p, q, r.

Parameters:

Name	Type	Type Description	Description
p	ℝ²	2d Vector of Reals	First vertex
q	ℝ²	2d Vector of Reals	Second vertex
r	ℝ²	2d Vector of Reals	Third vertex

Returns: ℝ² (2d Vector of Reals)

circumradius

Return the circumradius of the triangle with vertices p, q, r.

Parameters:

Name	Type	Type Description	Description
p	ℝ²	2d Vector of Reals	First vertex
q	ℝ²	2d Vector of Reals	Second vertex
r	ℝ²	2d Vector of Reals	Third vertex

Returns: ℝ (Real Number)

incenter

Return the incenter of the triangle with vertices p, q, r.

Parameters:

Name	Type	Type Description	Description
p	ℝ²	2d Vector of Reals	First vertex
q	ℝ²	2d Vector of Reals	Second vertex
r	ℝ²	2d Vector of Reals	Third vertex

Returns: ℝ² (2d Vector of Reals)

inradius

Return the inradius of the triangle with vertices p, q, r.

Parameters:

Name	Type	Type Description	Description
p	ℝ²	2d Vector of Reals	First vertex
q	ℝ²	2d Vector of Reals	Second vertex
r	ℝ²	2d Vector of Reals	Third vertex

Returns: ℝ (Real Number)

sqr

Return the square of the number x.

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	x

Returns: ℝ (Real Number)

sqrt

Return the square root of number x. (Note: if x < 0 you may get NaNs)

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	x

Returns: ℝ (Real Number)

max

Return the max of the numbers x, y.

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	x
y	ℝ	Real Number	y

Returns: ℝ (Real Number)

min

Return the min of the numbers x, y.

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	x
y	ℝ	Real Number	y

Returns: ℝ (Real Number)

abs

Return the absolute value of the number x.

Parameters:

Name	Type	Type Description	Description
x	ℝ	Real Number	x

Returns: ℝ (Real Number)

toRadians

Convert the angle theta from degrees to radians.

Parameters:

Name	Type	Type Description	Description
theta	ℝ	Real Number	theta

Returns: ℝ (Real Number)

toDegrees

Convert the angle theta from radians to degrees.

Parameters:

Name	Type	Type Description	Description
theta	ℝ	Real Number	theta

Returns: ℝ (Real Number)

norm

Return the Euclidean norm of the vector v.

Parameters:

Name	Type	Type Description	Description
v	ℝⁿ	Vector of Reals	A vector

Returns: ℝ (Real Number)

normsq

Return the Euclidean norm squared of the vector v.

Parameters:

Name	Type	Type Description	Description
v	ℝⁿ	Vector of Reals	A vector

Returns: ℝ (Real Number)

vdist

Return the Euclidean distance between the vectors v and w.

Parameters:

Name	Type	Type Description	Description
v	ℝⁿ	Vector of Reals	A vector
w	ℝⁿ	Vector of Reals	A vector

Returns: ℝ (Real Number)

vmul

Returns the scalar-vector product.

Parameters:

Name	Type	Type Description	Description
s	ℝ	Real Number	A scalar
v	ℝⁿ	Vector of Reals	A vector

Returns: ℝⁿ (Vector of Reals)

vdistsq

Return the Euclidean distance squared between the vectors v and w.

Parameters:

Name	Type	Type Description	Description
v	ℝⁿ	Vector of Reals	A vector
w	ℝⁿ	Vector of Reals	A vector

Returns: ℝ (Real Number)

angleOf

Return the angle made by the vector v with the positive x-axis.

Parameters:

Name	Type	Type Description	Description
v	ℝⁿ	Vector of Reals	A vector

Returns: ℝ (Real Number)

MathE

Base e of the natural logarithm.

Returns: ℝ (Real Number)

MathPI

Ratio of the circumference of a circle to its diameter.

Returns: ℝ (Real Number)

rot90

Rotate a 2D vector v by 90 degrees counterclockwise.

Parameters:

Name	Type	Type Description	Description
v	ℝ²	2d Vector of Reals	A vector

Returns: ℝ² (2d Vector of Reals)

rotateBy

Rotate a 2D vector v by theta degrees counterclockwise.

Parameters:

Name	Type	Type Description	Description
v	ℝ²	2d Vector of Reals	A vector
theta	ℝ	Real Number	degrees to rotate counterclockwise

Returns: ℝ² (2d Vector of Reals)

signedDistance

Return the signed distance between a shape and a point

Parameters:

Name	Type	Type Description	Description
s	EquationShape | ImageShape | RectangleShape | TextShape | CircleShape | PolygonShape | LineShape | PolylineShape	any of: any of: Equation Shape, or Image Shape, or Rectangle Shape, or Text Shape, or Circle Shape, or Polygon Shape, or Line Shape, or Polyline Shape	A shape
p	ℝ²	2d Vector of Reals	A point

Returns: ℝ (Real Number)

signedDistanceRect

Returns the distance between a rect and a point

Parameters:

Name	Type	Type Description	Description
rect	(ℝ²)ⁿ	List of 2d Real Vectors
pt	ℝ²	2d Vector of Reals

Returns: ℝ (Real Number)

signedDistanceCircle

Returns the distance between a circle and a point

Parameters:

Name	Type	Type Description	Description
c	ℝ²	2d Vector of Reals	center of circle
r	ℝ	Real Number	radius of circle
pt	ℝ²	2d Vector of Reals	the point

Returns: ℝ (Real Number)

signedDistancePolygon

Returns the distance between a polygon and a point

Parameters:

Name	Type	Type Description	Description
pts	(ℝ²)ⁿ	List of 2d Real Vectors	points of the polygon
pt	ℝ²	2d Vector of Reals	the point

Returns: ℝ (Real Number)

signedDistanceEllipse

Returns the distance between an ellipse and a point

Parameters:

Name	Type	Type Description	Description
c	ℝ²	2d Vector of Reals	center of ellipse
rx	ℝ	Real Number	horizontal radius of ellipse
ry	ℝ	Real Number	vertical radius of ellipse
pt	ℝ²	2d Vector of Reals	the point

Returns: ℝ (Real Number)

signedDistanceLine

Returns the distance between a line and a point

Parameters:

Name	Type	Type Description	Description
start	ℝ²	2d Vector of Reals	start of line
end	ℝ²	2d Vector of Reals	end of line
pt	ℝ²	2d Vector of Reals	the point

Returns: ℝ (Real Number)

signedDistancePolyline

Returns the distance between a line and a polyline

Parameters:

Name	Type	Type Description	Description
pts	(ℝ²)ⁿ	List of 2d Real Vectors	points of the polyline
pt	ℝ²	2d Vector of Reals	the point

Returns: ℝ (Real Number)

signedDistanceGroup

Returns the signed distance between a group of shapes and a point

Parameters:

Name	Type	Type Description	Description
shapes	Shape[]	List of shapes
pt	ℝ²	2d Vector of Reals

Returns: ℝ (Real Number)

unitVector

Construct a unit vector u in the direction of the given angle theta (in radians).

Parameters:

Name	Type	Type Description	Description
theta	ℝ	Real Number	direction

Returns: ℝ² (2d Vector of Reals)

rayIntersect

Given a point p and vector v, find the first point where the ray r(t)=p+tv intersects the given shape S. If there are no intersections, returns p.

Parameters:

Name	Type	Type Description	Description
S	EquationShape | ImageShape | RectangleShape | TextShape | CircleShape | PolygonShape | LineShape | PolylineShape | EllipseShape | GroupShape	any of: any of: Equation Shape, or Image Shape, or Rectangle Shape, or Text Shape, or Circle Shape, or Polygon Shape, or Line Shape, or Polyline Shape, or Ellipse Shape, or Group Shape	A shape
p	ℝ²	2d Vector of Reals	A point
v	ℝ²	2d Vector of Reals	A vector

Returns: ℝ² (2d Vector of Reals)

rayIntersectDistance

Given a point p and vector v, returns the distance to the first point where the ray r(t)=p+tv intersects the shape S. If there are no intersections, returns Infinity.

Parameters:

Name	Type	Type Description	Description
S	EquationShape | ImageShape | RectangleShape | TextShape | CircleShape | PolygonShape | LineShape | PolylineShape | EllipseShape | GroupShape	any of: any of: Equation Shape, or Image Shape, or Rectangle Shape, or Text Shape, or Circle Shape, or Polygon Shape, or Line Shape, or Polyline Shape, or Ellipse Shape, or Group Shape	A shape
p	ℝ²	2d Vector of Reals	A point
v	ℝ²	2d Vector of Reals	A vector

Returns: ℝ (Real Number)

rayIntersectCircle

Parameters:

Name	Type	Type Description	Description
c	ℝ²	2d Vector of Reals	center of circle
r	ℝ	Real Number	radius of circle
p	ℝ²	2d Vector of Reals	A point
v	ℝ²	2d Vector of Reals	A vector

Returns: ℝ² (2d Vector of Reals)

rayIntersectCircleDistance

Parameters:

Name	Type	Type Description	Description
c	ℝ²	2d Vector of Reals	center of circle
r	ℝ	Real Number	radius of circle
p	ℝ²	2d Vector of Reals	A point
v	ℝ²	2d Vector of Reals	A vector

Returns: ℝ (Real Number)

rayIntersectEllipse

Parameters:

Name	Type	Type Description	Description
c	ℝ²	2d Vector of Reals
rx	ℝ	Real Number
ry	ℝ	Real Number
p	ℝ²	2d Vector of Reals	A point
v	ℝ²	2d Vector of Reals	A vector

Returns: ℝ² (2d Vector of Reals)

rayIntersectEllipseDistance

Parameters:

Name	Type	Type Description	Description
c	ℝ²	2d Vector of Reals
rx	ℝ	Real Number
ry	ℝ	Real Number
p	ℝ²	2d Vector of Reals	A point
v	ℝ²	2d Vector of Reals	A vector

Returns: ℝ (Real Number)

rayIntersectLine

Parameters:

Name	Type	Type Description	Description
start	ℝ²	2d Vector of Reals
end	ℝ²	2d Vector of Reals
p	ℝ²	2d Vector of Reals	A point
v	ℝ²	2d Vector of Reals	A vector

Returns: ℝ² (2d Vector of Reals)

rayIntersectLineDistance

Parameters:

Name	Type	Type Description	Description
start	ℝ²	2d Vector of Reals
end	ℝ²	2d Vector of Reals
p	ℝ²	2d Vector of Reals	A point
v	ℝ²	2d Vector of Reals	A vector

Returns: ℝ (Real Number)

rayIntersectRect

Parameters:

Name	Type	Type Description	Description
rect	(ℝ²)ⁿ	List of 2d Real Vectors	The top-right, top-left, bottom-left, bottom-right points (in that order) of the rectangle
p	ℝ²	2d Vector of Reals	A point
v	ℝ²	2d Vector of Reals	A vector

Returns: ℝ² (2d Vector of Reals)

rayIntersectRectDistance

Parameters:

Name	Type	Type Description	Description
rect	(ℝ²)ⁿ	List of 2d Real Vectors	The top-right, top-left, bottom-left, bottom-right points (in that order) of the rectangle
p	ℝ²	2d Vector of Reals	A point
v	ℝ²	2d Vector of Reals	A vector

Returns: ℝ (Real Number)

rayIntersectPoly

Parameters:

Name	Type	Type Description	Description
pts	(ℝ²)ⁿ	List of 2d Real Vectors
closed	true | false	Boolean Value
p	ℝ²	2d Vector of Reals	A point
v	ℝ²	2d Vector of Reals	A vector

Returns: ℝ² (2d Vector of Reals)

rayIntersectPolyDistance

Parameters:

Name	Type	Type Description	Description
pts	(ℝ²)ⁿ	List of 2d Real Vectors
closed	true | false	Boolean Value
p	ℝ²	2d Vector of Reals	A point
v	ℝ²	2d Vector of Reals	A vector

Returns: ℝ (Real Number)

rayIntersectGroup

Parameters:

Name	Type	Type Description	Description
shapes	Shape[]	List of shapes
p	ℝ²	2d Vector of Reals	A point
v	ℝ²	2d Vector of Reals	A vector

Returns: ℝ² (2d Vector of Reals)

rayIntersectGroupDistance

Parameters:

Name	Type	Type Description	Description
shapes	Shape[]	List of shapes
p	ℝ²	2d Vector of Reals	A point
v	ℝ²	2d Vector of Reals	A vector

Returns: ℝ (Real Number)

rayIntersectNormal

Given a point p and vector v, find the unit normal at the first point where the ray r(t)=p+tv intersects the given shape S. If there are no intersections, returns (0,0).

Parameters:

Name	Type	Type Description	Description
S	EquationShape | ImageShape | RectangleShape | TextShape | CircleShape | PolygonShape | LineShape | PolylineShape | EllipseShape | GroupShape	any of: any of: Equation Shape, or Image Shape, or Rectangle Shape, or Text Shape, or Circle Shape, or Polygon Shape, or Line Shape, or Polyline Shape, or Ellipse Shape, or Group Shape	A shape
p	ℝ²	2d Vector of Reals	A point
v	ℝ²	2d Vector of Reals	A vector

Returns: ℝ² (2d Vector of Reals)

rayIntersectNormalCircle

Parameters:

Name	Type	Type Description	Description
c	ℝ²	2d Vector of Reals	center of circle
r	ℝ	Real Number	radius of circle
p	ℝ²	2d Vector of Reals	A point
v	ℝ²	2d Vector of Reals	A vector

Returns: ℝ² (2d Vector of Reals)

rayIntersectNormalEllipse

Parameters:

Name	Type	Type Description	Description
c	ℝ²	2d Vector of Reals
rx	ℝ	Real Number
ry	ℝ	Real Number
p	ℝ²	2d Vector of Reals	A point
v	ℝ²	2d Vector of Reals	A vector

Returns: ℝ² (2d Vector of Reals)

rayIntersectNormalLine

Parameters:

Name	Type	Type Description	Description
start	ℝ²	2d Vector of Reals
end	ℝ²	2d Vector of Reals
p	ℝ²	2d Vector of Reals	A point
v	ℝ²	2d Vector of Reals	A vector

Returns: ℝ² (2d Vector of Reals)

rayIntersectNormalRect

Parameters:

Name	Type	Type Description	Description
rect	(ℝ²)ⁿ	List of 2d Real Vectors	The top-right, top-left, bottom-left, bottom-right points (in that order) of the rectangle
p	ℝ²	2d Vector of Reals	A point
v	ℝ²	2d Vector of Reals	A vector

Returns: ℝ² (2d Vector of Reals)

rayIntersectNormalPoly

Parameters:

Name	Type	Type Description	Description
pts	(ℝ²)ⁿ	List of 2d Real Vectors
closed	true | false	Boolean Value
p	ℝ²	2d Vector of Reals	A point
v	ℝ²	2d Vector of Reals	A vector

Returns: ℝ² (2d Vector of Reals)

rayIntersectNormalGroup

Parameters:

Name	Type	Type Description	Description
shapes	Shape[]	List of shapes
p	ℝ²	2d Vector of Reals	A point
v	ℝ²	2d Vector of Reals	A vector

Returns: ℝ² (2d Vector of Reals)

closestPoint

Returns a point on the shape s closest to a query point p. If this point is not unique, an arbitrary choice is made.

Parameters:

Name	Type	Type Description	Description
s	EquationShape | ImageShape | RectangleShape | TextShape | CircleShape | PolygonShape | LineShape | PolylineShape | EllipseShape | GroupShape	any of: any of: Equation Shape, or Image Shape, or Rectangle Shape, or Text Shape, or Circle Shape, or Polygon Shape, or Line Shape, or Polyline Shape, or Ellipse Shape, or Group Shape	A shape
p	ℝ²	2d Vector of Reals	A vector

Returns: ℝ² (2d Vector of Reals)

closestPointCircle

Parameters:

Name	Type	Type Description	Description
c	ℝ²	2d Vector of Reals	center of circle
r	ℝ	Real Number	radius of circle
pt	ℝ²	2d Vector of Reals	the point

Returns: ℝ² (2d Vector of Reals)

closestPointRect

Parameters:

Name	Type	Type Description	Description
rect	(ℝ²)ⁿ	List of 2d Real Vectors	The top-right, top-left, bottom-left, bottom-right points (in that order) of the rectangle
pt	ℝ²	2d Vector of Reals	the point

Returns: ℝ² (2d Vector of Reals)

closestPointLine

Parameters:

Name	Type	Type Description	Description
start	ℝ²	2d Vector of Reals	start point of line
end	ℝ²	2d Vector of Reals	end point of line
pt	ℝ²	2d Vector of Reals	the point

Returns: ℝ² (2d Vector of Reals)

closestPointEllipse

Parameters:

Name	Type	Type Description	Description
c	ℝ²	2d Vector of Reals	center of ellipse
rx	ℝ	Real Number	horizontal radius of ellipse
ry	ℝ	Real Number	vertical radius of ellipse
pt	ℝ²	2d Vector of Reals	the point

Returns: ℝ² (2d Vector of Reals)

closestPointPoly

Parameters:

Name	Type	Type Description	Description
pts	(ℝ²)ⁿ	List of 2d Real Vectors	points of the polygon
closed	true | false	Boolean Value	whether or not the polygon is closed
pt	ℝ²	2d Vector of Reals	the point

Returns: ℝ² (2d Vector of Reals)

closestPointGroup

Parameters:

Name	Type	Type Description	Description
shapes	Shape[]	List of shapes	shapes of the group
pt	ℝ²	2d Vector of Reals	the point

Returns: ℝ² (2d Vector of Reals)

closestSilhouettePoint

Returns a point on the visibility silhouette of shape s closest to a query point p. If this point is not unique, an arbitrary choice is made. If no such point exists, the query point p is returned.

Parameters:

Name	Type	Type Description	Description
s	EquationShape | ImageShape | RectangleShape | TextShape | CircleShape | PolygonShape | LineShape | PolylineShape | EllipseShape | GroupShape	any of: any of: Equation Shape, or Image Shape, or Rectangle Shape, or Text Shape, or Circle Shape, or Polygon Shape, or Line Shape, or Polyline Shape, or Ellipse Shape, or Group Shape	A shape
p	ℝ²	2d Vector of Reals	A point

Returns: ℝ² (2d Vector of Reals)

closestSilhouettePointCircle

Parameters:

Name	Type	Type Description	Description
c	ℝ²	2d Vector of Reals	center of circle
r	ℝ	Real Number	radius of circle
p	ℝ²	2d Vector of Reals	A point

Returns: ℝ² (2d Vector of Reals)

closestSilhouettePointEllipse

Parameters:

Name	Type	Type Description	Description
c	ℝ²	2d Vector of Reals
rx	ℝ	Real Number
ry	ℝ	Real Number
p	ℝ²	2d Vector of Reals	A point

Returns: ℝ² (2d Vector of Reals)

closestSilhouettePointLine

Parameters:

Name	Type	Type Description	Description
start	ℝ²	2d Vector of Reals
end	ℝ²	2d Vector of Reals
p	ℝ²	2d Vector of Reals	A point

Returns: ℝ² (2d Vector of Reals)

closestSilhouettePointRect

Parameters:

Name	Type	Type Description	Description
rect	(ℝ²)ⁿ	List of 2d Real Vectors	The top-right, top-left, bottom-left, bottom-right points (in that order) of the rectangle
p	ℝ²	2d Vector of Reals	A point

Returns: ℝ² (2d Vector of Reals)

closestSilhouettePointPolyline

Parameters:

Name	Type	Type Description	Description
points	(ℝ²)ⁿ	List of 2d Real Vectors
p	ℝ²	2d Vector of Reals	A point

Returns: ℝ² (2d Vector of Reals)

closestSilhouettePointPolygon

Parameters:

Name	Type	Type Description	Description
points	(ℝ²)ⁿ	List of 2d Real Vectors
p	ℝ²	2d Vector of Reals	A point

Returns: ℝ² (2d Vector of Reals)

closestSilhouettePointGroup

Parameters:

Name	Type	Type Description	Description
shapes	Shape[]	List of shapes
p	ℝ²	2d Vector of Reals	A point

Returns: ℝ² (2d Vector of Reals)

closestSilhouetteDistance

Returns the distance to the closest point on the visibility silhouette of shape s relative to query point p. If no such point exists, returns Infinity.

Parameters:

Name	Type	Type Description	Description
s	EquationShape | ImageShape | RectangleShape | TextShape | CircleShape | PolygonShape | LineShape | PolylineShape | EllipseShape | GroupShape	any of: any of: Equation Shape, or Image Shape, or Rectangle Shape, or Text Shape, or Circle Shape, or Polygon Shape, or Line Shape, or Polyline Shape, or Ellipse Shape, or Group Shape	A shape
p	ℝ²	2d Vector of Reals	A point

Returns: ℝ (Real Number)

rectLineDist

Return the distance between a rectangle (defined using the bottom-left and top-right points) and a line (defined using start and end points)

Parameters:

Name	Type	Type Description	Description
bottomLeft	ℝ²	2d Vector of Reals	bottom-left point of rectangle
topRight	ℝ²	2d Vector of Reals	top-right point of rectangle
start	ℝ²	2d Vector of Reals	start point of line
end	ℝ²	2d Vector of Reals	end point of line

Returns: ℝ (Real Number)

shapeDistance

Return the distance between two shapes.

Parameters:

Name	Type	Type Description	Description
s1	Shape	Any Shape	a shape
s2	Shape	Any Shape	a shape

Returns: ℝ (Real Number)

shapeDistanceCircles

Return the distance between two circles.

Parameters:

Name	Type	Type Description	Description
c1	ℝ²	2d Vector of Reals	center of first circle
r1	ℝ	Real Number	radius of first circle
c2	ℝ²	2d Vector of Reals	center of second circle
r2	ℝ	Real Number	radius of second circle

Returns: ℝ (Real Number)

shapeDistanceRects

Return the distance between two rectangles.

Parameters:

Name	Type	Type Description	Description
rect1	(ℝ²)ⁿ	List of 2d Real Vectors	The top-right, top-left, bottom-left, bottom-right points (in that order) of the first rectangle.
rect2	(ℝ²)ⁿ	List of 2d Real Vectors	The top-right, top-left, bottom-left, bottom-right points (in that order) of the second rectangle.

Returns: ℝ (Real Number)

shapeDistanceRectLine

Returns the distance between a rectangle and a line.

Parameters:

Name	Type	Type Description	Description
rect	(ℝ²)ⁿ	List of 2d Real Vectors	The top-right, top-left, bottom-left, bottom-right points (in that order) of the rectangle.
start	ℝ²	2d Vector of Reals	The start point of the line
end	ℝ²	2d Vector of Reals	The end point of the line

Returns: ℝ (Real Number)

shapeDistanceRectlikePolyline

Returns the distance between a rectangle and a polyline.

Parameters:

Name	Type	Type Description	Description
rect	(ℝ²)ⁿ	List of 2d Real Vectors	The top-right, top-left, bottom-left, bottom-right points (in that order) of the rectangle.
points	(ℝⁿ)ᵐ	List of Real Vectors	points of polyline

Returns: ℝ (Real Number)

shapeDistancePolys

Returns the distance between two polygons.

Parameters:

Name	Type	Type Description	Description
pts1	(ℝ²)ⁿ	List of 2d Real Vectors	The list of points for the first polygon
pts2	(ℝ²)ⁿ	List of 2d Real Vectors	The list of points for the second polygon

Returns: ℝ (Real Number)

shapeDistanceRectCircle

Returns the distance between a rectangle and a circle.

Parameters:

Name	Type	Type Description	Description
rect	(ℝ²)ⁿ	List of 2d Real Vectors	The top-right, top-left, bottom-left, bottom-right points (in that order) of the rectangle.
c	ℝ²	2d Vector of Reals	center of the circle
r	ℝ	Real Number	radius of the circle

Returns: ℝ (Real Number)

shapeDistancePolyEllipse

Returns the distance between a polygon and an ellipse.

Parameters:

Name	Type	Type Description	Description
pts	(ℝ²)ⁿ	List of 2d Real Vectors	The list of points for the polygon
c	ℝ²	2d Vector of Reals	center of the ellipse
rx	ℝ	Real Number	horizontal radius of ellipse
ry	ℝ	Real Number	vertical radius of ellipse

Returns: ℝ (Real Number)

shapeDistanceCircleLine

Returns the distance between a circle and a line.

Parameters:

Name	Type	Type Description	Description
c	ℝ²	2d Vector of Reals	center of the circle
r	ℝ	Real Number	radius of the circle
start	ℝ²	2d Vector of Reals	start point of line
end	ℝ²	2d Vector of Reals	end point of line

Returns: ℝ (Real Number)

shapeDistanceLines

Returns the distance between two lines.

Parameters:

Name	Type	Type Description	Description
start1	ℝ²	2d Vector of Reals	start point of line
end1	ℝ²	2d Vector of Reals	end point of line
start2	ℝ²	2d Vector of Reals	start point of line
end2	ℝ²	2d Vector of Reals	end point of line

Returns: ℝ (Real Number)

signedArea

Returns the signed area enclosed by a polygonal chain given its nodes

Parameters:

Name	Type	Type Description	Description
points	(ℝⁿ)ᵐ	List of Real Vectors	points of polygonal chain
closed	true | false	Boolean Value	whether the polygonic chain is closed

Returns: ℝ (Real Number)

turningNumber

Returns the turning number of polygonal chain given its nodes

Parameters:

Name	Type	Type Description	Description
points	(ℝⁿ)ᵐ	List of Real Vectors	points of polygonal chain
closed	true | false	Boolean Value	whether the polygonic chain is closed

Returns: ℝ (Real Number)

perimeter

Returns the total length of polygonal chain given its nodes

Parameters:

Name	Type	Type Description	Description
points	(ℝⁿ)ᵐ	List of Real Vectors	points of polygonal chain
closed	true | false	Boolean Value	whether the polygonic chain is closed

Returns: ℝ (Real Number)

isoperimetricRatio

Returns the isoperimetric ratio (perimeter squared divided by enclosed area)

Parameters:

Name	Type	Type Description	Description
points	(ℝⁿ)ᵐ	List of Real Vectors	points of curve
closed	true | false	Boolean Value	whether the curve is closed

Returns: ℝ (Real Number)

elasticEnergy

Returns integral of curvature squared along the curve

Parameters:

Name	Type	Type Description	Description
points	(ℝⁿ)ᵐ	List of Real Vectors	points of curve
closed	true | false	Boolean Value	whether curve is closed

Returns: ℝ (Real Number)

totalCurvature

Returns integral of curvature along the curve

Parameters:

Name	Type	Type Description	Description
points	(ℝⁿ)ᵐ	List of Real Vectors	points of curve
closed	true | false	Boolean Value	whether curve is closed
signed	true | false	Boolean Value	whether curvature is signed

Returns: ℝ (Real Number)

lengthK

Returns the sum of all line segment lengths raised to k

Parameters:

Name	Type	Type Description	Description
points	(ℝⁿ)ᵐ	List of Real Vectors	points of curve
closed	true | false	Boolean Value	whether curve is closed
k	ℝ	Real Number	exponent for line segments

Returns: ℝ (Real Number)

maxCurvature

Returns the maximum value of curvature along the curve

Parameters:

Name	Type	Type Description	Description
points	(ℝⁿ)ᵐ	List of Real Vectors	points of curve
closed	true | false	Boolean Value	whether curve is closed

Returns: ℝ (Real Number)

pElasticEnergy

Returns integral of curvature raised to p along the curve

Parameters:

Name	Type	Type Description	Description
points	(ℝⁿ)ᵐ	List of Real Vectors	points of curve
closed	true | false	Boolean Value	whether curve is closed
p	ℝ	Real Number	exponent for curvature

Returns: ℝ (Real Number)

inflectionEnergy

Returns integral of curvature derivative raised to p along the curve

Parameters:

Name	Type	Type Description	Description
points	(ℝⁿ)ᵐ	List of Real Vectors	points of curve
closed	true | false	Boolean Value	whether curve is closed
p	ℝ	Real Number	exponent for curvature derivative

Returns: ℝ (Real Number)

tangentVectors

Returns list of n tangent vectors given a list of n points.

Parameters:

Name	Type	Type Description	Description
points	(ℝⁿ)ᵐ	List of Real Vectors	points of curve
closed	true | false	Boolean Value	whether curve is closed

Returns: (ℝⁿ)ᵐ (List of Real Vectors)

normalVectors

Returns list of n normal vectors given a list of n points.

Parameters:

Name	Type	Type Description	Description
points	(ℝⁿ)ᵐ	List of Real Vectors	points of curve
closed	true | false	Boolean Value	whether curve is closed

Returns: (ℝⁿ)ᵐ (List of Real Vectors)

binormalVectors

Returns list of n binormal vectors given a list of n points.

Parameters:

Name	Type	Type Description	Description
points	(ℝⁿ)ᵐ	List of Real Vectors	points of curve
closed	true | false	Boolean Value	whether curve is closed

Returns: (ℝⁿ)ᵐ (List of Real Vectors)

evoluteCurve

Returns evolute curve from a list of n points.

Parameters:

Name	Type	Type Description	Description
points	(ℝⁿ)ᵐ	List of Real Vectors	points of curve
closed	true | false	Boolean Value	whether curve is closed

Returns: (ℝⁿ)ᵐ (List of Real Vectors)

offsetCurve

Returns an offset version of the input curve.

Parameters:

Name	Type	Type Description	Description
points	(ℝⁿ)ᵐ	List of Real Vectors	points of curve
closed	true | false	Boolean Value	whether curve is closed
magnitude	ℝ	Real Number	magnitude of the offset

Returns: (ℝⁿ)ᵐ (List of Real Vectors)

curvatures

Returns list of n curvature values given a list of n points.

Parameters:

Name	Type	Type Description	Description
points	(ℝⁿ)ᵐ	List of Real Vectors	points of curve
closed	true | false	Boolean Value	whether curve is closed

Returns: ℝⁿ (Vector of Reals)

centerOfMass

Returns center of mass for a 2D point cloud

Parameters:

Name	Type	Type Description	Description
points	(ℝ²)ⁿ	List of 2d Real Vectors	points of curve

Returns: ℝ² (2d Vector of Reals)

noClip

Describes no shape clipping

Returns: ClipData (Shape clip data)

clip

Describes clipping to a shape

Parameters:

Name	Type	Type Description	Description
shape	Shape	Any Shape

Returns: ClipData (Shape clip data)

bboxPts

Returns the top-left, top-right, bottom-right, bottom-left points (in that order) of the axis-aligned bounding box of a shape

Parameters:

Name	Type	Type Description	Description
s	Shape	Any Shape	a shape

Returns: (ℝ²)ⁿ (List of 2d Real Vectors)

tsneEnergy

Returns T-SNE energy

Parameters:

Name	Type	Type Description	Description
points	(ℝⁿ)ᵐ	List of Real Vectors	high dimensional points
projectedPoints	(ℝⁿ)ᵐ	List of Real Vectors	projected, low dimensional points

Returns: ℝ (Real Number)

rectPts

Returns the top-left, top-right, bottom-right, bottom-left points of a rect-like shape. This takes into account rotation.

Parameters:

Name	Type	Type Description	Description
s	EquationShape | ImageShape | RectangleShape | TextShape	any of: Equation Shape, or Image Shape, or Rectangle Shape, or Text Shape

Returns: (ℝ²)ⁿ (List of 2d Real Vectors)

TeXify

Returns the TeX-fied version of a string where subscripts are handled in a way suitable for equation rendering. For example, "hello_world" becomes "{hello}_{world}".

Parameters:

Name	Type	Type Description	Description
str	String	String

Returns: String (String)

repeat

Returns a vector of n elements of K

Parameters:

Name	Type	Type Description	Description
n	ℕ	Natural Number
k	ℝ	Real Number

Returns: ℝⁿ (Vector of Reals)