Primitives

Primitive{Dim,T}

We say that a geometry is a primitive when it can be expressed as a single entity with no parts (a.k.a. atomic). For example, a sphere is a primitive described in terms of a mathematical expression involving a metric and a radius. See https://en.wikipedia.org/wiki/Geometric_primitive.

Point(x₁, x₂, ..., xₙ)
Point((x₁, x₂, ..., xₙ))
Point{Dim,T}(x₁, x₂, ..., xₙ)
Point{Dim,T}((x₁, x₂, ..., xₙ))

A point in Dim-dimensional space with coordinates of type T.

The coordinates of the point are given with respect to the canonical Euclidean basis, and Integer coordinates are converted to Float64.

Examples

# 2D points
A = Point(0.0, 1.0) # double precision as expected
B = Point(0f0, 1f0) # single precision as expected
C = Point(0, 0) # Integer is converted to Float64 by design
D = Point2(0, 1) # explicitly ask for double precision
E = Point2f(0, 1) # explicitly ask for single precision

# 3D points
F = Point(1.0, 2.0, 3.0) # double precision as expected
G = Point(1f0, 2f0, 3f0) # single precision as expected
H = Point(1, 2, 3) # Integer is converted to Float64 by design
I = Point3(1, 2, 3) # explicitly ask for double precision
J = Point3f(1, 2, 3) # explicitly ask for single precision

Notes

• Type aliases are Point1, Point2, Point3, Point1f, Point2f, Point3f
• Integer coordinates are not supported because most geometric processing algorithms assume a continuous space. The conversion to Float64 avoids InexactError and other unexpected results.

coordinates(point)

Return the coordinates of the point with respect to the canonical Euclidean basis.

-(A::Point, B::Point)

Return the Vec associated with the direction from point B to point A.

+(A::Point, v::Vec)
+(v::Vec, A::Point)

Return the point at the end of the vector v placed at a reference (or start) point A.

-(A::Point, v::Vec)
-(v::Vec, A::Point)

Return the point at the end of the vector -v placed at a reference (or start) point A.

Ray(p, v)

A ray originating at point p, pointed in direction v. It can be called as r(t) with t > 0 to cast it at p + t * v.

Line(a, b)

A line passing through points a and b.

See also Segment.

BezierCurve(points)

A recursive Bézier curve with control points points. See https://en.wikipedia.org/wiki/Bézier_curve. A point on the curve b can be evaluated by calling b(t) with t between 0 and 1. The evaluation method defaults to DeCasteljau's algorithm for accurate evaluation. Horner's method, faster with a large number of points but less precise, can be used via b(t, Horner()).

Examples

BezierCurve(Point2[(0.,0.),(1.,-1.)])

Plane(p, u, v)

A plane embedded in R³ passing through point p, defined by non-parallel vectors u and v.

Plane(p, n)

Alternatively specify point p and a given normal vector n to the plane.

Box(min, max)

An axis-aligned box with min and max corners. See https://en.wikipedia.org/wiki/Hyperrectangle.

Examples

Box(Point(0, 0, 0), Point(1, 1, 1))
Box((0, 0), (1, 1))

Ball(center, radius)

A ball with center and radius.

See also Sphere.

Sphere(center, radius)

A sphere with center and radius.

See also Ball.

Ellipsoid(radii, center=(0, 0, 0), rotation=I)

A 3D ellipsoid with given radii, center and rotation.

Disk(plane, radius)

A disk embedded in 3-dimensional space on a given plane with given radius.

See also Circle.

Circle(plane, radius)

A circle embedded in 3-dimensional space on a given plane with given radius.

See also Disk.

Cylinder(bottom, top, radius)

A solid circular cylinder embedded in R³ with given radius, delimited by bottom and top planes.

Cylinder(start, finish, radius)

Alternatively, construct a right circular cylinder with given radius along the segment with start and finish end points.

Cylinder(start, finish)

Or construct a right circular cylinder with unit radius along the segment with start and finish end points.

Cylinder(radius)

Finally, construct a right vertical circular cylinder with given radius.

See https://en.wikipedia.org/wiki/Cylinder.

CylinderSurface(bottom, top, radius)

A circular cylinder surface embedded in R³ with given radius, delimited by bottom and top planes.

CylinderSurface(start, finish, radius)

Alternatively, construct a right circular cylinder surface with given radius along the segment with start and finish end points.

CylinderSurface(start, finish)

Or construct a right circular cylinder surface with unit radius along the segment with start and finish end points.

CylinderSurface(radius)

Finally, construct a right vertical circular cylinder surface with given radius.

See https://en.wikipedia.org/wiki/Cylinder.

Cone(base, apex)

A cone with base disk and apex. See https://en.wikipedia.org/wiki/Cone.

See also ConeSurface.

ConeSurface(base, apex)

A cone surface with base disk and apex. See https://en.wikipedia.org/wiki/Cone.

See also Cone.

Frustum(bot, top)

A frustum (truncated cone) with bot and top disks. See https://en.wikipedia.org/wiki/Frustum.

See also FrustumSurface.

FrustumSurface(bot, top)

A frustum (truncated cone) surface with bot and top disks. See https://en.wikipedia.org/wiki/Frustum.

See also Frustum.

Torus(center, normal, major, minor)

A torus centered at center with axis of revolution directed by normal and with radii major and minor.

ParaboloidSurface(apex, radius, focallength)

A paraboloid surface embedded in R³ and extending up to a distance radius from its focal axis, which is aligned along the z direction and passes through apex (the apex of the paraboloid). The equation of the paraboloid is the following:

\[f(x, y) = \frac{(x - x_0)^2 + (y - y_0)^2}{4f} + z_0\qquad\text{for } x^2 + y^2 < r^2,\]

where $(x_0, y_0, z_0)$ is the apex of the parabola, $f$ is the focal length, and $r$ is the clip radius.

ParaboloidSurface(apex, radius)

This creates a paraboloid surface with focal length equal to 1.

ParaboloidSurface(apex)

This creates a paraboloid surface with focal length equal to 1 and a rim with unit radius.

ParaboloidSurface()

Same as above, but here the apex is at Apex(0, 0, 0).

See also https://en.wikipedia.org/wiki/Paraboloid.