Sphere Class Reference

Spheres. More...

#include <sphere.h>

Inheritance diagram for Sphere:

Public Member Functions
	Sphere ()
	Empty.
	Sphere (const double &)
	Creates a sphere centered at origin with specified radius.
	Sphere (const Vector &, const double &=0.0)
	Creates a sphere given center and radius.
	Sphere (const Vector &, const Vector &)
	Creates a sphere given two points.
	Sphere (const Vector &, const Vector &, const Vector &)
	Creates a sphere given three vertices.
	Sphere (const Vector &, const Vector &, const Vector &, const Vector &)
	Create a sphere circumsizing four vertices.
	Sphere (const QVector< Vector > &)
	Compute the minimal bounding sphere of a set of points.
	~Sphere ()
	Empty.
Vector	Center () const
	Gets the center of a sphere.
double	Radius () const
	Gets the radius of a sphere.
Box	GetBox () const
	Compute the bounding box of a sphere.
bool	Intersect (const Ray &) const
	Check the intersection between a sphere and a ray.
int	Intersect (const Ray &, double &, double &) const
	Compute the intersection between a sphere and a ray.
int	Intersect (const Ray &, double &, double &, Vector &, Vector &) const
	This function computes the intersections between a sphere and a ray.
bool	Intersect (const Box &) const
	Box-sphere intersection test.
bool	Intersect (const Sphere &, Circle &) const
	Check if two spheres intersect and compute the circle of intersection.
bool	Intersect (const Sphere &) const
	Check if two spheres intersect.
bool	SurfaceIntersect (const Sphere &) const
	Check if two spheres intersect, only considering their surface.
bool	Intersect (const Ray &, double &) const
	This function computes the first intersection between a sphere and a ray.
bool	Inside (const Vector &) const
	Check if a point is inside or outside the sphere.
double	Volume () const
	Compute the volume of the sphere.
double	Volume (const Sphere &) const
	Compute the volume of the intersection of two spheres.
double	Area () const
	Compute the surface area of a sphere.
Vector	Normal (const Vector &) const
	Computes the normal vector between a point and the sphere.
double	R (const Vector &) const
	Compute the squared distance between a point and the sphere.
double	Signed (const Vector &) const
	Compute the signed distance between a point and the sphere.
double	R (const Sphere &) const
	Compute the signed distance between two spheres.
double	R (const Vector &, const Vector &) const
	Compute the great-circle or orthodromic distance.
void	Rotate (const Matrix &)
	Rotates a sphere.
void	Translate (const Vector &)
	Translates a sphere.
void	Scale (const double &)
	Uniformly scales a sphere.
Sphere	Translated (const Vector &) const
	Translate a sphere.
Sphere	Scaled (const Vector &) const
	Scales a sphere by a given vector.
Sphere	Rotated (const Matrix &) const
	Rotates a sphere.
Sphere	Transformed (const Frame &) const
	Transforms a sphere.
Sphere	InverseTransformed (const Frame &) const
	Inverse transforms a sphere.
void	Extend (const Vector &)
	Extend the sphere so that the argument point should be embedded in the new sphere.
void	Extend (const double &)
	Extend the sphere, i.e. increase the radius of the sphere.
Sphere	Extended (const double &) const
	Extend the sphere, i.e. increase the radius of the sphere.
Vector	RandomSurface (Random &=Random::R239) const
	Generate a random point on the sphere.
Vector	RandomInside (Random &=Random::R239) const
	Generate a random vector inside the sphere.
Vector	Fibonacci (int, int) const
QVector< Vector >	Poisson (const double &, int, Random &=Random::R239) const
	Create a Poisson Disc sampling of the surface of the sphere.
Vector2	Euler (const Vector &) const
	Compute the spherical (Euler) coordinates of a point relative to the sphere.

Static Public Member Functions
static Vector	RandomNormal (Random &=Random::R239)
	Generate a random unit vector orthonormal to the sphere.
static double	Area (const double &)
	Compute the surface area of a sphere.
static double	Volume (const double &)
	Compute the volume of a sphere.
static Vector2	EquiRectangular (int, int, int, int)
	Compute the Euler coordinates of a point defined in a rectangle map.
static bool	Intersection (const Sphere &, const Sphere &, const Sphere &, Vector &, Vector &)
	Compute the intersection of three spheres.
static bool	Intersection (const Sphere &, const Sphere &, const Vector &, Vector &, Vector &)
	Compute the intersection of two spheres and a plane.

Static Public Attributes
static const double	epsilon = 1.0e-6
	\( \varepsilon \) for intersection tests.
static const Sphere	Null
	Empty sphere.
static const Sphere	Infinity
	Infinite sphere.
static const Sphere	Unit
	Unit sphere.

Protected Attributes
Vector	c = Vector::Null
	Center.
double	r = 0.0
	Radius.

Friends
Sphere	operator+ (const Sphere &a, const Sphere &b)
	Computes the Minkowski sum of two spheres.
Sphere	operator* (const double &t, const Sphere &sphere)
	Scales a sphere by a scalar factor. The homothetic sphere has a positive radius whatever the scaling factor.
Sphere	operator* (const Sphere &sphere, const double &t)
	Scales a sphere by a scalar factor. The homothetic sphere has a positive radius whatever the scaling factor.
std::ostream &	operator<< (std::ostream &s, const Sphere &sphere)
	Overloaded.

Detailed Description

Spheres.

Spheres are defined by their center and radius. The squared radius is not stored in the data-structure, and is computed on the fly whenever needed.

This class implements several high level functions to compute the minimal bounding sphere of a set of points. It also implements the Minkowski sum of two spheres.

The sphere-ray intersection functions assume that the ray is normalized, i.e. has a unit direction vector.

Constructor & Destructor Documentation

Sphere() [1/6]

Sphere::Sphere ( const double & r )

explicit

Creates a sphere centered at origin with specified radius.

Parameters

r

Radius.

Sphere() [2/6]

Sphere::Sphere ( const Vector & c, const double & r = 0.0 )

explicit

Creates a sphere given center and radius.

Parameters

c	Center.
r	Radius.

Sphere() [3/6]

Sphere::Sphere ( const Vector & a, const Vector & b )

explicit

Creates a sphere given two points.

Basically set the center to the middle of the segment and set the radius to the half length of the segment.

Parameters

a,b	The two vertices.

Sphere() [4/6]

Sphere::Sphere ( const Vector & x, const Vector & y, const Vector & z )

explicit

Creates a sphere given three vertices.

Set the center to the center of the triangle, and compute the corresponding radius. Note that the argument vertices should not be aligned.

Parameters

x,y,z

The three vertices.

Sphere() [5/6]

Sphere::Sphere ( const Vector & p0, const Vector & p1, const Vector & p2, const Vector & p3 )

explicit

Create a sphere circumsizing four vertices.

This is the same as a sphere bounding a tetrahedron.

See also

Tetrahedra::GetSphere()

Parameters

p0,p1,p2,p3

Argument vertices

Sphere() [6/6]

Sphere::Sphere ( const QVector< Vector > & p )

explicit

Compute the minimal bounding sphere of a set of points.

Error around 1%~2%. Based on Bo Tian, Bouncing bubble: a fast algorithm for minimal enclosing pall problem, 2012.

The sphere of an empty set of points is the null sphere, centered at the origin, with a null radius.

Parameters

p	Set of points.

Member Function Documentation

Area()

double Sphere::Area ( const double & r )

inlinestatic

Compute the surface area of a sphere.

Parameters

r

Radius.

EquiRectangular()

Vector2 Sphere::EquiRectangular ( int i, int j, int x, int y )

static

Compute the Euler coordinates of a point defined in a rectangle map.

Parameters

i,j	Point coordinates.
x,y	Size of the map.

Returns

A vector containing the theta and phi angles.

Euler()

Vector2 Sphere::Euler

(

const Vector &

p

)

const

Compute the spherical (Euler) coordinates of a point relative to the sphere.

The point p is first projected onto the unit sphere centered at c by computing the unit direction vector \(\mathbf{u} = \mathrm{Normalized}(p - c)\). The two angles are then:

• \(\theta \in [0,\pi]\) : polar (colatitude) angle from the Z axis, \(\theta = \arccos(u_z)\).
• \(\phi \in (-\pi,\pi]\) : azimuthal angle in the XY plane, \(\phi = \mathrm{atan2}(u_x, u_y)\).

Parameters

p	Point in world space (need not lie on the sphere surface).

Returns

Vector2(theta, phi) with angles in radians.

Extend() [1/2]

void Sphere::Extend

(

const double &

e

)

Extend the sphere, i.e. increase the radius of the sphere.

Contrary to Scale, this function does not change the center.

Parameters

e	Radius change.

Extend() [2/2]

void Sphere::Extend

(

const Vector &

p

)

Extend the sphere so that the argument point should be embedded in the new sphere.

This is the same as computing the smallest enclosing sphere of a sphere and a point.

Parameters

p

Point.

See also

Circle2::Extend(const Vector2&)

Extended()

Sphere Sphere::Extended

(

const double &

e

)

const

Extend the sphere, i.e. increase the radius of the sphere.

See also

Extend

Parameters

e	Radius change.

Fibonacci()

Vector Sphere::Fibonacci	(	int	i,
		int	n ) const

Compute the i-th point of the Fibonacci distribution of points on the sphere.

In other words, phyllotaxic samples for nearly optimal distribution over a sphere.

Parameters

i	Point index.
n	Number of samples on the sphere.

Inside()

bool Sphere::Inside

(

const Vector &

p

)

const

Check if a point is inside or outside the sphere.

Parameters

p	The point.

Intersect() [1/7]

bool Sphere::Intersect

(

const Box &

box

)

const

Box-sphere intersection test.

The algorithm calculates the square of the distance from the box to the sphere by analyzing the orientation of the sphere relative to the box in a single loop. If the box is not axis aligned, transform the center of the sphere to the box's local coordinate frame.

After J. Arvo, A simple method for box-sphere intersection testing, In A. Glassner, Graphics Gems: 335-339, Academic Press 1990.

See also

Box::R()

Intersect() [2/7]

bool Sphere::Intersect

(

const Ray &

ray

)

const

Check the intersection between a sphere and a ray.

Note that intersections are sorted.

This function assumes that the ray is normalized, i.e. has a unit direction vector.

Parameters

ray	The (normalized) ray.

See also

Sphere::Intersect(const Ray&,double&,double&) const

Intersect() [3/7]

bool Sphere::Intersect	(	const Ray &	ray,
		double &	t ) const

This function computes the first intersection between a sphere and a ray.

This function assumes that the ray is normalized, i.e., has a unit direction vector.

Parameters

ray	The (normalized) ray.
t	Returned intersection depth.

Intersect() [4/7]

int Sphere::Intersect	(	const Ray &	ray,
		double &	ta,
		double &	tb ) const

Compute the intersection between a sphere and a ray.

Note that intersections are sorted.

This function assumes that the ray is normalized, i.e. has a unit direction vector.

Parameters

ray	The (normalized) ray.
ta,tb	Intersection depths.

Intersect() [5/7]

int Sphere::Intersect	(	const Ray &	ray,
		double &	ta,
		double &	tb,
		Vector &	na,
		Vector &	nb ) const

This function computes the intersections between a sphere and a ray.

The intersections are sorted.

This function assumes that the ray is normalized, i.e. has a unit direction vector.

Parameters

ray	The (normalized) ray.
ta,tb	Intersection depths.
na,nb	Normals at intersection points.

Intersect() [6/7]

bool Sphere::Intersect

(

const Sphere &

sphere

)

const

Check if two spheres intersect.

Parameters

sphere

Sphere.

Intersect() [7/7]

bool Sphere::Intersect	(	const Sphere &	sphere,
		Circle &	circle ) const

Check if two spheres intersect and compute the circle of intersection.

The spheres intersect if the distance between the centers is between the absolute value of the difference of the radii and the sum of the radii. There is then a circle of intersection, in the plane orthogonal to the line of the centers. The plane may be outside of the segment joining the centers, if the radii are very different.

The function returns false if the spheres are concentric, as there is no circle of intersection (there are infinitely many common points if the radii are equal).

Parameters

sphere	Sphere.
circle	Returned Circle, whose axis is the normalized vector from the center of the other sphere to the center of this sphere.

Intersection() [1/2]

bool Sphere::Intersection ( const Sphere & sa, const Sphere & sb, const Sphere & sc, Vector & u, Vector & v )

static

Compute the intersection of three spheres.

Parameters

sa,sb,sc	Spheres.
u,v	Returned points.

Intersection() [2/2]

bool Sphere::Intersection ( const Sphere & sa, const Sphere & sb, const Vector & n, Vector & u, Vector & v )

static

Compute the intersection of two spheres and a plane.

Parameters

sa,sb	Spheres.
n	Unit normal of the plane containing both centers, should be orthogonal to the direction between the centers.
u,v	Returned points.

InverseTransformed()

Sphere Sphere::InverseTransformed

(

const Frame &

t

)

const

Inverse transforms a sphere.

Parameters

t	Transformation.

Normal()

Vector Sphere::Normal

(

const Vector &

p

)

const

Computes the normal vector between a point and the sphere.

Simply project point onto the sphere, and return the corresponding Euclidean distance vector.

Parameters

p

Point.

Poisson()

QVector< Vector > Sphere::Poisson	(	const double &	r,
		int	n,
		Random &	random = Random::R239 ) const

Create a Poisson Disc sampling of the surface of the sphere.

Simple dart throwing algorithm.

Parameters

r	Radius of the discs.
n	Maximum number of darts thrown on the sphere.
random	Random number generator.

R() [1/3]

double Sphere::R

(

const Sphere &

s

)

const

Compute the signed distance between two spheres.

If the spheres intersect, the result is negative.

Parameters

s	The other sphere.

R() [2/3]

double Sphere::R

(

const Vector &

p

)

const

Compute the squared distance between a point and the sphere.

Parameters

p	The point.

R() [3/3]

double Sphere::R	(	const Vector &	a,
		const Vector &	b ) const

Compute the great-circle or orthodromic distance.

It is the geodesic distance between two points on the surface of a sphere.

This function is computationnally intensive, requiring several square root and inverse trigonometric function evaluations.

Parameters

a,b	Points on the sphere.

RandomInside()

Vector Sphere::RandomInside

(

Random &

random = Random::R239

)

const

Generate a random vector inside the sphere.

Parameters

random

Random number generator.

RandomNormal()

Vector Sphere::RandomNormal ( Random & random = Random::R239 )

static

Generate a random unit vector orthonormal to the sphere.

Parameters

random

Random number generator.

RandomSurface()

Vector Sphere::RandomSurface

(

Random &

random = Random::R239

)

const

Generate a random point on the sphere.

Parameters

random

Random number generator.

Rotate()

void Sphere::Rotate

(

const Matrix &

r

)

Rotates a sphere.

Parameters

r	Rotation matrix.

Rotated()

Sphere Sphere::Rotated

(

const Matrix &

r

)

const

Rotates a sphere.

Parameters

r	Rotation matrix.

Scale()

void Sphere::Scale

(

const double &

s

)

Uniformly scales a sphere.

Parameters

s	Scaling factor.

Scaled()

Sphere Sphere::Scaled

(

const Vector &

s

)

const

Scales a sphere by a given vector.

Since the sphere may become an ellipsoid, compute the enclosing sphere.

Parameters

s	Scaling vector.

Signed()

double Sphere::Signed

(

const Vector &

p

)

const

Compute the signed distance between a point and the sphere.

Parameters

p	The point.

SurfaceIntersect()

bool Sphere::SurfaceIntersect

(

const Sphere &

sphere

)

const

Check if two spheres intersect, only considering their surface.

Parameters

sphere

Sphere.

Transformed()

Sphere Sphere::Transformed

(

const Frame &

t

)

const

Transforms a sphere.

Parameters

t	Transformation.

Translate()

void Sphere::Translate

(

const Vector &

t

)

Translates a sphere.

Parameters

t	Translation vector.

Translated()

Sphere Sphere::Translated

(

const Vector &

t

)

const

Translate a sphere.

Parameters

t	Translation vector.

Volume() [1/2]

double Sphere::Volume ( const double & r )

inlinestatic

Compute the volume of a sphere.

Parameters

r

Radius.

Volume() [2/2]

double Sphere::Volume

(

const Sphere &

sphere

)

const

Compute the volume of the intersection of two spheres.

If the spheres do not intersect, the function returns 0; if the smallest is included in the largest, the function returns the volume of the smallest, otherwise the volume of the lens is computed.

For two spheres of radii \( R \) and \( r \) whose centers are at distance \( d \), with \( |R-r|<d<R+r \), the volume of the intersection is:

\[V=\frac{\pi(R+r-d)^2\left(d^2+2dr-3r^2+2dR+6rR-3R^2\right)}{12d}. \]

See also

Circle::Area(const Circle2&)

Parameters

sphere

The other sphere.

Friends And Related Symbol Documentation

operator<<

std::ostream & operator<< ( std::ostream & s, const Sphere & sphere )

friend

Overloaded.

Parameters

s	Stream.
sphere	The sphere.