Class Sphere

java.lang.Object

org.episteme.core.mathematics.geometry.surfaces.Sphere

All Implemented Interfaces:

Function<Vector<Real>, Vector<Real>>, ContinuousFunction<Vector<Real>, Vector<Real>>, DifferentiableFunction<Vector<Real>, Vector<Real>>, Function<Vector<Real>, Vector<Real>>, Relation<Vector<Real>, Vector<Real>>, ParametricSurface

public class Sphere
extends Object
implements ParametricSurface

Represents a sphere as a parametric surface.

Spherical coordinates parametrization: S(θ,φ) = center + (r*sin(θ)*cos(φ), r*sin(θ)*sin(φ), r*cos(θ)) where θ ∈ [0,π] is the polar angle and φ∈ [0,2π] is the azimuthal angle.

Since:

1.0

Author:

Silvere Martin-Michiellot, Gemini AI (Google DeepMind)

Constructor Summary

Constructors

Constructor	Description
Sphere(PointND center, Real radius)	Creates a sphere.

Method Summary

Modifier and Type	Method	Description
PointND	at(Real theta, Real phi)	Evaluates the surface at parameters (u, v).
boolean	contains(PointND point, Real tolerance)	Checks if a point is on the surface of the sphere.
Real	gaussianCurvature(Real theta, Real phi, Real h)	Returns the Gaussian curvature at (u, v).
PointND	getCenter()	Returns the center of the sphere.
Real	getRadius()	Returns the radius of the sphere.
Vector<Real>	normal(Real theta, Real phi, Real h)	Returns the normal vector at (u, v).
Real	surfaceArea()	Returns the surface area of the sphere.
String	toString()
Real	volume()	Returns the volume of the sphere.

Methods inherited from class Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, wait, wait, wait

Methods inherited from interface ContinuousFunction

isContinuous

Methods inherited from interface DifferentiableFunction

isDifferentiable

Methods inherited from interface Function

compose

Methods inherited from interface Function

andThen, apply, compose, contains, evaluate, getBackend, setBackend

Methods inherited from interface ParametricSurface

differentiate, evaluate, firstFundamentalForm, getOutputDimension, partialU, partialV, surfaceArea, surfaceElement

Methods inherited from interface Relation

getCodomain, getDomain

Constructor Details

Sphere

public Sphere(PointND center,
 Real radius)

Creates a sphere.

Parameters:

center - the center point (must be 3D)

radius - the radius

Method Details

at

public PointND at(Real theta,
 Real phi)

Description copied from interface: ParametricSurface

Evaluates the surface at parameters (u, v).

Specified by:

at in interface ParametricSurface

Parameters:

theta - the first parameter

phi - the second parameter

Returns:

the point on the surface

normal

public Vector<Real> normal(Real theta,
 Real phi,
 Real h)

Description copied from interface: ParametricSurface

Returns the normal vector at (u, v).

The normal is computed as: N = ∂S/∂u × ∂S/∂v

Specified by:

normal in interface ParametricSurface

Parameters:

theta - the first parameter

phi - the second parameter

h - the step size for numerical differentiation

Returns:

the unit normal vector

gaussianCurvature

public Real gaussianCurvature(Real theta,
 Real phi,
 Real h)

Description copied from interface: ParametricSurface

Returns the Gaussian curvature at (u, v).

K = (LN - M²) / (EG - F²) where L, M, N are second fundamental form coefficients and E, F, G are first fundamental form coefficients.

Specified by:

gaussianCurvature in interface ParametricSurface

Parameters:

theta - the first parameter

phi - the second parameter

h - the step size

Returns:

the Gaussian curvature

surfaceArea

public Real surfaceArea()

Returns the surface area of the sphere.

Returns:

4πr²

volume

public Real volume()

Returns the volume of the sphere.

Returns:

(4/3)πr³

getCenter

public PointND getCenter()

Returns the center of the sphere.

Returns:

the center point

getRadius

public Real getRadius()

Returns the radius of the sphere.

Returns:

the radius

contains

public boolean contains(PointND point,
 Real tolerance)

Checks if a point is on the surface of the sphere.

Parameters:

point - the point to check

tolerance - the tolerance

Returns:

true if on surface

toString

public String toString()

Overrides:

toString in class Object