Class Paraboloid

java.lang.Object

org.episteme.core.mathematics.geometry.surfaces.Paraboloid

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 Paraboloid
extends Object
implements ParametricSurface

Represents a paraboloidal surface.

A paraboloid is a quadric surface with equation z = (x²/a²) + (y²/b²). Parametric form: S(u, v) = (u, v, u²/a² + v²/b²)

Special cases: - a = b: Circular paraboloid (paraboloid of revolution) - a ≠b: Elliptic paraboloid

Since:

1.0

Author:

Silvere Martin-Michiellot, Gemini AI (Google DeepMind)

Constructor Summary

Constructors

Constructor	Description
Paraboloid(PointND vertex, Real scaleA, Real scaleB)	Creates a paraboloid with the given vertex and scale factors.
Paraboloid(Real scaleA, Real scaleB)	Creates a paraboloid with the given scale factors.

Method Summary

Modifier and Type	Method	Description
PointND	at(Real u, Real v)	Evaluates the surface at parameters (u, v).
Real	focalLength()	Returns the focal length for a circular paraboloid (a = b). f = a²/4
Real	gaussianCurvature(Real u, Real v, Real h)	Returns the Gaussian curvature at (u, v).
Real	getScaleA()	Returns the scale factor along x.
Real	getScaleB()	Returns the scale factor along y.
PointND	getVertex()	Returns the vertex point.
Vector<Real>	partialU(Real u, Real v, Real h)	Returns the partial derivative ∂S/∂u.
Vector<Real>	partialV(Real u, Real v, Real h)	Returns the partial derivative ∂S/∂v.

Methods inherited from class Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, 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, normal, surfaceArea, surfaceElement

Methods inherited from interface Relation

getCodomain, getDomain

Constructor Details

Paraboloid

public Paraboloid(Real scaleA,
 Real scaleB)

Creates a paraboloid with the given scale factors. Vertex at origin.

Parameters:

scaleA - the scale factor along x

scaleB - the scale factor along y

Paraboloid

public Paraboloid(PointND vertex,
 Real scaleA,
 Real scaleB)

Creates a paraboloid with the given vertex and scale factors.

Parameters:

vertex - the vertex point

scaleA - the scale factor along x

scaleB - the scale factor along y

Method Details

at

public PointND at(Real u,
 Real v)

Description copied from interface: ParametricSurface

Evaluates the surface at parameters (u, v).

Specified by:

at in interface ParametricSurface

Parameters:

u - the first parameter

v - the second parameter

Returns:

the point on the surface

partialU

public Vector<Real> partialU(Real u,
 Real v,
 Real h)

Description copied from interface: ParametricSurface

Returns the partial derivative ∂S/∂u.

This is the tangent vector in the u-direction.

Specified by:

partialU in interface ParametricSurface

Parameters:

u - the first parameter

v - the second parameter

h - the step size for numerical differentiation

Returns:

the partial derivative vector

partialV

public Vector<Real> partialV(Real u,
 Real v,
 Real h)

Description copied from interface: ParametricSurface

Returns the partial derivative ∂S/∂v.

This is the tangent vector in the v-direction.

Specified by:

partialV in interface ParametricSurface

Parameters:

u - the first parameter

v - the second parameter

h - the step size for numerical differentiation

Returns:

the partial derivative vector

gaussianCurvature

public Real gaussianCurvature(Real u,
 Real v,
 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:

u - the first parameter

v - the second parameter

h - the step size

Returns:

the Gaussian curvature

getScaleA

public Real getScaleA()

Returns the scale factor along x.

Returns:

the scale factor a

getScaleB

public Real getScaleB()

Returns the scale factor along y.

Returns:

the scale factor b

getVertex

public PointND getVertex()

Returns the vertex point.

Returns:

the vertex

focalLength

public Real focalLength()

Returns the focal length for a circular paraboloid (a = b). f = a²/4

Returns:

the focal length

Throws:

IllegalStateException - if not a circular paraboloid