Shape-related functions and classes. More...

Topics
	Spline
	Splines and curves.

Concepts
concept	geom::BoundedObject
	A shape which can efficiently report its bounding box.

concept	geom::ConvexObject
	A convex shape which implements its convex support function.

concept	geom::RayIntersectableObject
	A shape which can be intersected by a ray.

concept	geom::RegionObject
	A shape which can test for point containment.

concept	geom::SdfObject
	A shape which can report its exact signed distance field.

concept	geom::InteriorMeasurableObject
	A shape which can measure its interior region.

concept	geom::BoundaryMeasurableObject
	A shape which can measure its boundary.

concept	geom::MeasurableObject
	A shape which can measure both its surface area and volume.

concept	geom::ProjectableObject
	A shape which can perform an orthogonal projection to its boundary.

Classes
class	AnyConvex< T, N >
	Helper class which virtualizes the static polymorphism of Convex shapes. More...

class	AnyConvexImpl< Shape >
	Implementation of AnyConvex for a specific Shape. More...

class	Bounded< T, N, Derived >
	Base class describing shapes with finite extents in N dimensions. More...

class	Capsule< T, N >
	An N-dimensional capsule shape. More...

class	Convex< T, N, Derived >
	Base class describing convex shapes in N-dimensional space. More...

class	Cylinder< T, N >
	An N-dimensional cylinder, given by its radius and endpoints. More...

class	Dilated< Shape >
	A wrapper shape which dilates the extents of another Shape. More...

class	Extruded< Shape >
	An axis-aligned extrusion of an arbitrary N-1 dimensional Convex shape. More...

class	Frustum< Shape >
	An N-dimensional frustum (truncated pyramid) with an arbitrary Convex shape as its base, and its (possibly excluded) point at the origin. More...

class	GridIterator< T, N, Order >
	Iterator over the integer points in an N-dimensional grid. More...

class	Hollow< Shape >
	Selects the boundary of a shape. More...

class	KDTree< T, N, Object, NodeData >
	A hierarchical spatial index. More...

class	Plane< T, N >
	A geometric plane or hyperplane. More...

class	Projectable< T, N, Derived >
	Base class describing N-dimensional shapes which implement the ability to project an arbitrary point to the nearest point on their surface. More...

class	RayIntersectable< T, N, Derived >
	Base class describing N-dimensional shapes which can be intersection-tested with a Ray. More...

class	Rect< T, N >
	An N-dimensional axis-aligned interval. More...

class	SdfEvaluable< T, N, Derived >
	Base class describing N-dimensional shapes which implement a signed distance function. More...

class	Similar< Shape >
	A shape transformed by a similarity transform (translation, rotation, and scale). More...

class	Simplex< T, N >
	A simplex in N dimensions (e.g. a tetrahedron, triangle, line, point). More...

class	Sphere< T, N >
	An N-dimensional circle, sphere, or hypersphere with a filled interior. More...

class	SphericalCap< T, N >
	The surface of a sphere within a certain angle from its pole. More...

class	Transformed< Shape >
	A wrapper shape which transforms another arbitrary shape with an AffineTransform. More...

class	Transformed< Rect< T, N > >
	Partial specialization of Transformed for Rects.

Macros
#define	MAX_KDTREE_ARITY 16

Typedefs
template<typename T, index_t N>
using	AffineBox = Transformed<Rect<T,N>>
	Convenience typedef for transformed Rects.

template<typename T, index_t N>
using	Box = Similar<Rect<T,N>>
	Convenience typedef for arbitrarily-oriented Rects.

template<typename T>
using	Circle = Sphere<T,2>
	A 2D circle.

template<typename T>
using	Cone = Frustum<Circle<T>>
	Convenience typedef for a 3D cone with its tip at the origin.

template<typename T, index_t N>
using	SphericalShell = Dilated<Hollow<Sphere<T,N>>>
	Convenience typedef for a spherical shell.

typedef Shape::elem_t	T

template<typename T>
using	Tetrahedron = Simplex<T,3>

template<typename T>
using	TransformedCone = Transformed<Cone<T>>
	Convenience typedef for an oriented cone.

template<typename T>
using	Triangle = Simplex<T,2>

template<typename T, index_t N>
using	ViewFrustum = Transformed< Frustum< Rect<T,N-1> > >
	Convenience typedef for oriented, rectangular, N-dimensional Frustums.

Enumerations
enum	ArrayOrder { ARRAYORDER_FIRST_DIM_CONSECUTIVE , ARRAYORDER_LAST_DIM_CONSECUTIVE }
	Array traversal order, specified in terms of which axes to increment first. More...

enum class	KDAxisChoice { AXIS_LONGEST , AXIS_HIGHEST_VARIANCE , AXIS_CYCLICAL }
	Strategy for choosing an axis along which to split a group of objects in a spatial index. More...

enum class	KDInsertionChoice { INSERT_SMALLEST_VOLUME_INCREASE , INSERT_SHORTEST_DISTANCE }
	Strategy for insertion of an object into an existing tree. More...

enum class	KDPivotChoice { PIVOT_MEAN , PIVOT_MEDIAN }
	Strategy for choosing the pivot value when splitting a group of objects in a spatial index. More...

Functions
	AnyConvexImpl (const Shape &&s)

	AnyConvexImpl (const Shape &s)

virtual Rect< T, N >	_impl_bounds () const =0

virtual Rect< T, N >	_impl_bounds () const

virtual point_t	_impl_convex_support (Vec< T, N > p) const =0

virtual point_t	_impl_convex_support (Vec< T, N > p) const

template<typename Shape>
AnyConvexImpl< Shape >	as_any_convex (const Shape &s)
	Wrap `s` in a virtual class which implements the Convex concept.

Rect< T, N >	bounds () const

Rect< T, N >	bounds () const
	Produces an axis-aligned box completely enclosing this shape.

Rect< T, N >	bounds () const
	Produces an axis-aligned box completely enclosing this shape.

Vec< T, N >	clip (Vec< T, N > p) const
	Nearest point on the interior of the shape.

bool	contains (Vec< T, N > p) const
	Shape-point overlap test.

point_t	convex_support (point_t d) const

point_t	convex_support (point_t d) const
	Geometric convex support function.

template<typename Shape>
Dilated< Shape >	dilate (const Dilated< Shape > &s, typename Shape::elem_t dilation)
	Dilate the shape `s` by the amount `dilation`.

template<typename Shape>
Dilated< Shape >	dilate (const Shape &s, typename Shape::elem_t dilation)
	Dilate the shape `s` by the amount `dilation`.

template<typename T, index_t N>
Sphere< T, N >	dilate (const Sphere< T, N > &s, T dilation)
	Dilate the Sphere `s` by the amount `dilation`.

template<typename T, index_t N>
Plane< T, N >	dilate (Plane< T, N > p, T dilation)
	Dilate the Plane `p` by the amount `dilation`.

template<typename Shape>
Extruded< Shape >	extrude (const Shape &s, typename Shape::elem_t h0, typename Shape::elem_t h1)
	Convenience function to extrude the shape `s` between heights `h0` and `h1` by wrapping `s` in the `Extruded` template.

template<typename Shape>
Frustum< Shape >	frustum (const Shape &s, typename Shape::elem_t h0, typename Shape::elem_t h1)
	Convenience function to raise the shape `s` into a frustum between heights `h0` and `h1`, by wrapping `s` in the `Frustum` template.

Rect< T, 1 >	intersect (const Ray< T, N > &r) const
	Ray/shape intersection.

template<typename Shape>
bool	intersects (const Convex< T, N, Shape > &other) const
	Convex shape overlap test.

template<typename T, index_t N>
bool	intersects (const AnyConvex< T, N > &shape_a, const AnyConvex< T, N > &shape_b)

Vec< T, N >	normal (Vec< T, N > p) const
	Unit-length outward-facing direction.

template<typename T, index_t N>
Capsule< T, N >	operator* (const Isometry< T, N > &xf, const Capsule< T, N > &c)
	Transform a capsule by an isometry.

template<typename T, index_t N>
Cylinder< T, N >	operator* (const Isometry< T, N > &xf, const Cylinder< T, N > &c)
	Transform a cylinder by an isometry.

template<typename T, index_t N, typename Shape>
Dilated< Shape >	operator* (const Isometry< T, N > &xf, const Dilated< Shape > &s)
	Transform a dilated shape by an isometry.

template<typename T, index_t N>
Simplex< T, N >	operator* (const Isometry< T, N > &xf, const Simplex< T, N > &s)
	Transform a simplex by an isometry.

template<typename T, index_t N>
Sphere< T, N >	operator* (const Isometry< T, N > &xf, const Sphere< T, N > &s)
	Transform a sphere by an isometry.

template<typename Shape>
Similar< Shape >	operator* (const Similarity< typename Shape::elem_t, Shape::N > &xf, const Shape &shape)
	Transform the shape `shape` by wrapping it with a Similarity transform.

template<typename Shape>
Similar< Shape >	operator* (const Similarity< typename Shape::elem_t, Shape::N > &xf, const Similar< Shape > &s)
	Transform the shape `s` by `xf`.

template<typename T, index_t N>
Capsule< T, N >	operator* (const Similarity< T, N > &xf, const Capsule< T, N > &c)
	Transform a capsule by a similarity transform.

template<typename T, index_t N>
Cylinder< T, N >	operator* (const Similarity< T, N > &xf, const Cylinder< T, N > &c)
	Transform a cylinder by a similarity transform.

template<typename T, index_t N, typename Shape>
Dilated< Shape >	operator* (const Similarity< T, N > &xf, const Dilated< Shape > &s)
	Transform a dilated shape by a similarity transform.

template<typename T, index_t N>
Simplex< T, N >	operator* (const Similarity< T, N > &xf, const Simplex< T, N > &s)
	Transform a simplex by a similarity transform.

template<typename T, index_t N>
Sphere< T, N >	operator* (const Similarity< T, N > &xf, const Sphere< T, N > &s)
	Transform a sphere by a similarity transform.

template<typename Shape>
Transformed< Shape >	operator* (const AffineTransform< typename Shape::elem_t, Shape::N > &xf, const Shape &s)
	Transform the shape `s` by wrapping it in an `Transformed` class.

template<typename Shape>
Transformed< Shape >	operator* (const AffineTransform< typename Shape::elem_t, Shape::N > &xf, const Transformed< Shape > &s)
	Transform the transformed shape `s` by `xf`.

template<typename Shape, typename T, index_t N>
Transformed< Shape >	operator* (const Isometry< T, N > &xf, const Transformed< Shape > &s)
	Transform the shape `s` by an isometry `xf`.

template<typename Shape, typename T, index_t N>
Transformed< Shape >	operator* (const Similarity< T, N > &xf, const Transformed< Shape > &s)
	Transform the shape `s` by a similarity transform `xf`.

template<typename Shape>
Similar< Shape > &	*operator=** (Similar< Shape > &s, const Similarity< typename Shape::elem_t, Shape::N > &xf)
	In-place transform the shape `s` by `xf`.

template<typename Shape>
Transformed< Shape > &	*operator=** (Transformed< Shape > &s, const AffineTransform< typename Shape::elem_t, Shape::N > &xf)
	In-place transform the transformed shape `s` by `xf`.

template<typename T, index_t N>
Capsule< T, N >	operator/ (const Capsule< T, N > &c, const Isometry< T, N > &xf)
	Inverse-transform a capsule by an isometry.

template<typename T, index_t N>
Cylinder< T, N >	operator/ (const Cylinder< T, N > &c, const Isometry< T, N > &xf)
	Inverse-transform a cylinder by an isometry.

template<typename T, index_t N, typename Shape>
Dilated< Shape >	operator/ (const Dilated< Shape > &s, const Isometry< T, N > &xf)
	Inverse-transform a dilated shape by an isometry.

template<typename T, index_t N>
Simplex< T, N >	operator/ (const Simplex< T, N > &s, const Isometry< T, N > &xf)
	Inverse-transform a simplex by an isometry.

template<typename T, index_t N>
Sphere< T, N >	operator/ (const Sphere< T, N > &s, const Isometry< T, N > &xf)
	Inverse-transform a sphere by an isometry.

template<typename Shape>
Similar< Shape >	operator/ (const Shape &s, const Similarity< typename Shape::elem_t, Shape::N > &xf)
	Transform the shape `s` by the inverse of `xf`.

template<typename Shape>
Similar< Shape >	operator/ (const Similar< Shape > &s, const Similarity< typename Shape::elem_t, Shape::N > &xf)
	Transform the shape `s` by the inverse of `xf`.

template<typename T, index_t N>
Capsule< T, N >	operator/ (const Capsule< T, N > &c, const Similarity< T, N > &xf)
	Inverse-transform a capsule by a similarity transform.

template<typename T, index_t N>
Cylinder< T, N >	operator/ (const Cylinder< T, N > &c, const Similarity< T, N > &xf)
	Inverse-transform a cylinder by a similarity transform.

template<typename T, index_t N, typename Shape>
Dilated< Shape >	operator/ (const Dilated< Shape > &s, const Similarity< T, N > &xf)
	Inverse-transform a dilated shape by a similarity transform.

template<typename T, index_t N>
Simplex< T, N >	operator/ (const Simplex< T, N > &s, const Similarity< T, N > &xf)
	Inverse-transform a simplex by a similarity transform.

template<typename T, index_t N>
Sphere< T, N >	operator/ (const Sphere< T, N > &s, const Similarity< T, N > &xf)
	Inverse-transform a sphere by a similarity transform.

template<typename Shape>
Transformed< Shape >	operator/ (const Shape &s, const AffineTransform< typename Shape::elem_t, Shape::N > &xf)
	Transform the shape `s` by the inverse of `xf`.

template<typename Shape>
Transformed< Shape >	operator/ (const Transformed< Shape > &s, const AffineTransform< typename Shape::elem_t, Shape::N > &xf)
	Transform the transformed shape `s` by the inverse of `xf`.

template<typename Shape, typename T, index_t N>
Transformed< Shape >	operator/ (const Transformed< Shape > &s, const Isometry< T, N > &xf)
	Inverse transform the shape `s` by an isometry `xf`.

template<typename Shape, typename T, index_t N>
Transformed< Shape >	operator/ (const Transformed< Shape > &s, const Similarity< T, N > &xf)
	Inverse transform the shape `s` by a similarity transform `xf`.

template<typename Shape>
Similar< Shape > &	operator/= (Similar< Shape > &s, const Similarity< typename Shape::elem_t, Shape::N > &xf)
	In-place transform the shape `s` by the inverse of `xf`.

template<typename Shape>
Transformed< Shape > &	operator/= (Transformed< Shape > &s, const AffineTransform< typename Shape::elem_t, Shape::N > &xf)
	In-place transform the transformed shape `s` by the inverse of `xf`.

Vec< T, N >	project (Vec< T, N > p) const
	Nearest point on the surface of the shape.

template<typename T, index_t N>
Dilated< Rect< T, N > >	roundrect (const Rect< T, N > &rect, T radius)
	Produce a rounded rectangle with corner radius `radius`.

T	sdf (Vec< T, N > p) const

T	sdf (Vec< T, N > p) const
	Signed distance function.

template<typename T, index_t N>
Dilated< Hollow< Sphere< T, N > > >	shell (const Sphere< T, N > &s, T thickness)
	Create a spherical shell with wall thickness `thickness` around the Sphere `s`.

template<typename T, index_t N>
Dilated< Hollow< Sphere< T, N > > >	shell (Vec< T, N > center, T r0, T r1)
	Create a spherical shell around `center`, with inner and outer radii `r0` and `r1`.

template<typename T, index_t N>
bool	trace_simplex (const Vec< T, N > verts[N], const Ray< T, N > &ray, Vec< T, N-1 > uv, T s)

Variables
static constexpr index_t	N = Shape::N

Shape	shape

template<typename Shape, Transform< typename Shape::elem_t, Shape::N > Xf>
Hollow< Shape >	operator* (const Xf &xf, const Hollow< Shape > &s)
	Transform a hollow shape.

template<typename Shape, Transform< typename Shape::elem_t, Shape::N > Xf>
Hollow< Shape >	operator/ (const Hollow< Shape > &s, const Xf &xf)
	Inverse transform a hollow shape.

template<typename Shape, Transform< typename Shape::elem_t, Shape::N > Xf>
Hollow< Shape > &	*operator=** (Hollow< Shape > &s, const Xf &xf)
	In-place transform a hollow shape.

template<typename Shape, Transform< typename Shape::elem_t, Shape::N > Xf>
Hollow< Shape > &	operator/= (Hollow< Shape > &s, const Xf &xf)
	In-place inverse transform a hollow shape.

Detailed Description

Shape-related functions and classes.

Typedef Documentation

◆ Circle

template<typename T, index_t N>

template<typename T>

using Circle = Sphere<T,2>

A 2D circle.

Template Parameters

T	Coordinate type

◆ Cone

template<typename Shape>

template<typename T>

using Cone = Frustum<Circle<T>>

Convenience typedef for a 3D cone with its tip at the origin.

Formed from a frustum with a circular base.

Template Parameters

T	Coordinate type

◆ TransformedCone

template<typename Shape>

template<typename T>

using TransformedCone = Transformed<Cone<T>>

Convenience typedef for an oriented cone.

Template Parameters

T	Coordinate type

Enumeration Type Documentation

◆ ArrayOrder

enum ArrayOrder

Array traversal order, specified in terms of which axes to increment first.

Enumerator

ARRAYORDER_FIRST_DIM_CONSECUTIVE

Array traversal by incrementing the first coordinate in the innermost loop.

For matrices, whose coordinates are ordered (row, col), this represens column-major (i.e. "fortran") order. If coordinates are (x, y, ...), then this is row-major order.

ARRAYORDER_LAST_DIM_CONSECUTIVE

Array traversal by incrementing the last coordinate in the innermost loop.

For matrices, whose coordinates are ordered (row, col), this represents row-major (i.e. "C") order. If coordinates are (x, y, ...), then this is column-major order.

◆ KDAxisChoice

enum class KDAxisChoice

strong

Strategy for choosing an axis along which to split a group of objects in a spatial index.

Enumerator
AXIS_LONGEST	Divide a node along its longest axis.
AXIS_HIGHEST_VARIANCE	Divide a node along the axis with the highest position variance.
AXIS_CYCLICAL	Divide each node along an axis that cycles with tree depth, starting with the x (first) axis at the root node.

◆ KDInsertionChoice

enum class KDInsertionChoice

strong

Strategy for insertion of an object into an existing tree.

Enumerator
INSERT_SMALLEST_VOLUME_INCREASE	Insert objects into the node that will experience the smallest volume increase as a result of the addition.
INSERT_SHORTEST_DISTANCE	Insert objects into the node which is closest.

◆ KDPivotChoice

enum class KDPivotChoice

strong

Strategy for choosing the pivot value when splitting a group of objects in a spatial index.

Enumerator
PIVOT_MEAN	Split a node at the population average.
PIVOT_MEDIAN	Split a node at the population median.

Function Documentation

◆ _impl_bounds()

template<typename Shape>

virtual Rect< T, N > _impl_bounds ( ) const

inlineprotectedvirtual

Implements AnyConvex< Shape::elem_t, Shape::N >.

◆ _impl_convex_support()

template<typename Shape>

virtual point_t _impl_convex_support ( Vec< T, N > p ) const

inlineprotectedvirtual

Implements AnyConvex< Shape::elem_t, Shape::N >.

◆ bounds()

template<typename T, index_t N, typename Derived>

Rect< T, N > bounds ( ) const

inline

Produces an axis-aligned box completely enclosing this shape.

The default implementation calls convex_support() along each of the principal axes to find the extents.

◆ clip()

template<typename T, index_t N, typename Derived>

Vec< T, N > clip ( Vec< T, N > p ) const

inline

Nearest point on the interior of the shape.

If p is on the interior of the shape, return p unaltered; otherwise orthogonally project p to the shape's surface.

◆ contains()

template<typename T, index_t N, typename Derived>

bool contains ( Vec< T, N > p ) const

inline

Shape-point overlap test.

Return true if the point p is on the surface or interior of the shape, false otherwise.

◆ convex_support()

template<typename T, index_t N, typename Derived>

point_t convex_support ( point_t d ) const

inline

Geometric convex support function.

Returns the point on the surface of this convex shape that is furthest along direction d (i.e., has the highest dot product with d).

All shapes which implement this function automatically support geometrical intersection tests with any other Convex object.

Parameters

d	Direction along which to find a support plane.

Returns: A point on the surface of this convex shape.

◆ dilate() [1/2]

template<typename T, index_t N>

Sphere< T, N > dilate	(	const Sphere< T, N > &	s,
		T	dilation )

Dilate the Sphere s by the amount dilation.

(This just adds the amount dilation to the Sphere's radius).

◆ dilate() [2/2]

template<typename T, index_t N>

Plane< T, N > dilate	(	Plane< T, N >	p,
		T	dilation )

Dilate the Plane p by the amount dilation.

(This just moves the plane dilation units along its normal).

◆ intersect()

template<typename T, index_t N, typename Derived>

Rect< T, 1 > intersect ( const Ray< T, N > & r ) const

inline

Ray/shape intersection.

Return the possibly-empty range of ray parameters s such that the Ray overlaps the shape at r.origin + s * r.direction.

For non-convex objects with multiple regions of overlap, this method should return the interval containing the smallest positive value.

◆ intersects()

template<typename T, index_t N, typename Derived>

template<typename Shape>

bool intersects ( const Convex< T, N, Shape > & other ) const

inline

Convex shape overlap test.

Returns: True if and only if this convex shape overlaps other; false otherwise.

◆ normal()

template<typename T, index_t N, typename Derived>

Vec< T, N > normal ( Vec< T, N > p ) const

inline

Unit-length outward-facing direction.

For any point, return the direction which points directly away from the nearest point on the shape, away from its interior.

This should be the same as the gradient of the sdf().

◆ project()

template<typename T, index_t N, typename Derived>

Vec< T, N > project ( Vec< T, N > p ) const

inline

Nearest point on the surface of the shape.

Compute the point on the boundary of the shape which is closest to p.

◆ roundrect()

template<typename T, index_t N>

Dilated< Rect< T, N > > roundrect	(	const Rect< T, N > &	rect,
		T	radius )

inline

Produce a rounded rectangle with corner radius radius.

The rounded rectangle will have the extents of rect, but with corners rounded by radius. The radius will be clamped to the maximum possible value that will not cause the corners to overlap.

◆ sdf()

template<typename T, index_t N, typename Derived>

T sdf ( Vec< T, N > p ) const

inline

Signed distance function.

Compute a signed distance to the nearest surface point on the shape. Points on the exterior have positive value; points on the interior have negative value; surface points have sdf value 0.

◆ shell() [1/2]

template<typename T, index_t N>

Dilated< Hollow< Sphere< T, N > > > shell	(	const Sphere< T, N > &	s,
		T	thickness )

Create a spherical shell with wall thickness thickness around the Sphere s.

The wall will be centered on the Sphere's surface.

◆ shell() [2/2]

template<typename T, index_t N>

Dilated< Hollow< Sphere< T, N > > > shell	(	Vec< T, N >	center,
		T	r0,
		T	r1 )

Create a spherical shell around center, with inner and outer radii r0 and r1.

It is not required that r0 < r1; the shell will always be constructed with positive thickness.

◆ trace_simplex()

template<typename T, index_t N>

bool trace_simplex	(	const Vec< T, N >	verts[N],
		const Ray< T, N > &	ray,
		Vec< T, N-1 > *	uv,
		T *	s )

inline

Ray trace an N-1 dimensional simplex (e.g. triangle in 3D, tetrahedron in 4D, a line in 2D etc.), returning the surface parameters of the hit point, given in coordinates of the basis spanned by the edges radiating from verts[0].

Parameters

verts	An array of the simplex's N vertices.
ray	The ray to intersect with the simplex.
uv	Buffer for the surface coordinates of the hitpoint. (return variable; may be null).
s	The ray parameter of the hit point. (return variable)

Returns: true if the ray hit the simplex.

Topics

Concepts

Classes

Macros

Typedefs

Enumerations

Functions

Variables

Detailed Description

Typedef Documentation

◆ Circle

◆ Cone

◆ TransformedCone

Enumeration Type Documentation

◆ ArrayOrder

◆ KDAxisChoice

◆ KDInsertionChoice

◆ KDPivotChoice

Function Documentation

◆ _impl_bounds()

◆ _impl_convex_support()

◆ bounds()

◆ clip()

◆ contains()

◆ convex_support()

◆ dilate() [1/2]

◆ dilate() [2/2]

◆ intersect()

◆ intersects()

◆ normal()

◆ project()

◆ roundrect()

◆ sdf()

◆ shell() [1/2]

◆ shell() [2/2]

◆ trace_simplex()