org.bzdev.math.BicubicTriangleInterp

All Implemented Interfaces:: DoubleBinaryOperator, RealValuedFunctTwoOps, RealValuedFunctVAOps, VADomainOps

public class BicubicTriangleInterp extends RealValuedFunctionTwo

Interpolation based on cubic triangular Bernstein-Bézier patches.

This interpolator computes values by using the following expression: $$ \sum_{|\lambda| = 3} \beta_\lambda B^3_\lambda(u,v,w) $$ where u + v + w = 1, λ= (λ₁,λ₂,λ₃) represent three indices, each in the range [0,3], and |λ| is defined as λ₁+λ₂+λ₃. $B^3_\lambda (u,v,w)$ is a Bernstein polynomial of degree 3 over a triangle specified by barycentric coordinates (u, v, w), and is defined by the equation $$ B^3_\lambda (u,v,w) = \frac{3!}{\lambda_1!\lambda_2!\lambda_3!} u^{\lambda_1}v^{\lambda_2}w^{\lambda_3} $$

By convention, we will use (u,v) as independent variables and set w = 1 - (u + v). The control points are located as follows:

                  (0,1)
                   030
                    *
                   / \
                  /   \
                 /     \
                /       \
           021 *---------* 120
              / \       / \
    [u = 0]  /   \     /   \  [w = 0]
   (v axis) /     \   /     \
           /    111\ /       \
      012 *---------*---------* 210
         / \       / \       / \
        /   \     /   \     /   \
       /     \   /     \   /     \
      /       \ /       \ /       \
     *---------*---------*---------*
    003       102       201       300
  (0,0)                          (1,0)
                 [v = 0]
                 (u axis)

The ordered pairs (0,0), (0,1), and (1,0) give the values of (u,v) at the vertices of the triangular region, and the sequences of three numbers denote the λ indices labeling each control point. On the boundaries, either u, v, or w is zero. In many of the methods described below, we use the notation (x, y) instead of (u, w). Some of the constructors allow a range for x and a range for y to be specified. in this case

u = (x - x_min) / (x_max - x_min)
v = (y - y_min) / (y_max - y_min)

The default values are x_min = 0, x_max = 1, y_min = 0, and y_max = 1. For the defaults, u = x and v = y.

Nested Class Summary

Nested classes/interfaces inherited from class org.bzdev.math.RealValuedFunctionVA
RealValuedFunctionVA.Linear
Constructor Summary

Constructors

Constructor

Description

BicubicTriangleInterp(double[] inits)

Constructor using an array.

BicubicTriangleInterp(double xmin, double xmax, double ymin, double ymax, double[] inits)

Constructor using an array and specifying the min/max values this function's domain.

BicubicTriangleInterp(double p003, double p012, double p021, double p030, double p102, double p111, double p120, double p201, double p210, double p300)

Constructor using explicit arguments.

BicubicTriangleInterp(double xmin, double xmax, double ymin, double ymax, double p003, double p012, double p021, double p030, double p102, double p111, double p120, double p201, double p210, double p300)

Constructor using explicit arguments and the minimum and maximum values of this functions domain.
Method Summary

Modifier and Type

Method

Description

double

deriv11At(double x, double y)

Evaluate the partial derivative $ \frac{\partial^2 f}{\partial x_1^2}$ for a function f(x₁,x₂).

double

deriv12At(double x, double y)

Evaluate the partial derivative $\frac{\partial^2 f}{\partial x_1 \partial x_2}$ for a function f(x₁,x₂).

double

deriv1At(double x, double y)

Evaluate the partial derivative $\frac{\partial f}{\partial x_1}$ for a function f(x₁, x₂).

double

deriv21At(double x, double y)

Evaluate the partial derivative $\frac{\partial^2 f}{\partial x_2 \partial x_1}$ for a function f(x₁,x₂).

double

deriv22At(double x, double y)

Evaluate the partial derivative $\frac{\partial^2 f}{\partial x_2^2}$ for a function f(x₁,x₂).

double

deriv2At(double x, double y)

Evaluate the partial derivative $\frac{\partial f}{\partial x_2}$ for a function f(x₁,x₂).

double

getDomainMax1()

Get the maximum value of the first argument in the domain of the function.

double

getDomainMax2()

Get the maximum value of the second argument in the domain of the function.

double

getDomainMin1()

Get the minimum value of the first argument in the domain of the function.

double

getDomainMin2()

Get the minimum value of the second argument in the domain of the function.

boolean

isInDomain(double x, double y)

Determine if a point (x, y) is within the domain of a real-valued function of two arguments.

double

uFromXY(double x, double y)

Get the normalized barycentric coordinate u given the coordinates (x, y).

double

valueAt(double x, double y)

Call the function.

double

vFromXY(double x, double y)

Get the normalized barycentric coordinate v given the coordinates (x, y).

double

wFromXY(double x, double y)

Get the normalized barycentric coordinate w given the coordinates (x, y).

double

xForZeroW(double y)

Get the value of first argument to the function represented by this interpolator corresponding to a particular value of the second argument when w is zero.

double

xFromUV(double u, double v)

Get the X coordinate given the barycentric coordinates u and v.

double

xFromVW(double v, double w)

Get the X coordinate given the barycentric coordinates v and w.

double

xFromWU(double w, double u)

Get the X coordinate given the barycentric coordinates w and u.

double

yForZeroW(double x)

Get the value of second argument to the function represented by this interpolator corresponding to a particular value of the first argument when w is zero.

double

yFromUV(double u, double v)

Get the Y coordinate given the barycentric coordinates u and v.

double

yFromVW(double v, double w)

Get the Y coordinate given the barycentric coordinates v and w.

double

yFromWU(double w, double u)

Get the Y coordinate given the barycentric coordinates w and u.

Methods inherited from class org.bzdev.math.RealValuedFunctionTwo
deriv, deriv1, deriv11, deriv12, deriv2, deriv21, deriv22, derivAt, domainMax1Closed, domainMax2Closed, domainMaxClosed, domainMin1Closed, domainMin2Closed, domainMinClosed, getDomainMax, getDomainMin, isInDomain, secondDeriv, secondDerivAt, valueAt

Methods inherited from class org.bzdev.math.RealValuedFunctionVA
jacobian, jacobian, maxArgLength, minArgLength

Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Methods inherited from interface org.bzdev.math.RealValuedFunctTwoOps
applyAsDouble, maxArgLength, minArgLength

Constructor Details
- BicubicTriangleInterp
  
  public BicubicTriangleInterp(double[] inits)
  
  Constructor using an array. The array contains the control points whose indices are 003, 012, 021, 030, 102, 111, 120, 201, 210, 300, listed in that order.
  
  Parameters:
  
  inits - the control points
- BicubicTriangleInterp
  
  public BicubicTriangleInterp(double xmin, double xmax, double ymin, double ymax, double[] inits)
  
  Constructor using an array and specifying the min/max values this function's domain. The array contains the control points whose indices are 003, 012, 021, 030, 102, 111, 120, 201, 210, 300, listed in that order.
  
  Parameters:
  
  xmin - the bound on the interpolation region, corresponding to u = 0, for the first argument of f
  
  xmax - the bound on the interpolation region, corresponding to u = 1, for the first argument of f
  
  ymin - the bound on the interpolation region, corresponding to v = 0, for the second argument of f
  
  ymax - the bound on the interpolation region, corresponding to v = 1, for the second argument of f
  
  inits - the control points
- BicubicTriangleInterp
  
  public BicubicTriangleInterp(double p003, double p012, double p021, double p030, double p102, double p111, double p120, double p201, double p210, double p300)
  
  Constructor using explicit arguments. The arguments provide the control points.
  
  Parameters:
  
  p003 - the control point indexed by λ=(0,0,3)
  
  p012 - the control point indexed by λ=(0,1,2)
  
  p021 - the control point indexed by λ=(0,2,1)
  
  p030 - the control point indexed by λ=(0,3,0)
  
  p102 - the control point indexed by λ=(1,0,2)
  
  p111 - the control point indexed by λ=(1,1,1)
  
  p120 - the control point indexed by λ=(1,2,0)
  
  p201 - the control point indexed by λ=(2,0,1)
  
  p210 - the control point indexed by λ=(2,1,0)
  
  p300 - the control point indexed by λ=(3,0,0)
- BicubicTriangleInterp
  
  public BicubicTriangleInterp(double xmin, double xmax, double ymin, double ymax, double p003, double p012, double p021, double p030, double p102, double p111, double p120, double p201, double p210, double p300)
  
  Constructor using explicit arguments and the minimum and maximum values of this functions domain. The arguments provide the control points.
  
  Parameters:
  
  xmin - the lower bound of the interpolation region for the first argument of f
  
  xmax - the upper bound of the interpolation region for the first argument of f
  
  ymin - the lower bound of the interpolation region for the second argument of f
  
  ymax - the upper bound of the interpolation region for the second argument of f
  
  p003 - the control point indexed by λ=(0,0,3)
  
  p012 - the control point indexed by λ=(0,1,2)
  
  p021 - the control point indexed by λ=(0,2,1)
  
  p030 - the control point indexed by λ=(0,3,0)
  
  p102 - the control point indexed by λ=(1,0,2)
  
  p111 - the control point indexed by λ=(1,1,1)
  
  p120 - the control point indexed by λ=(1,2,0)
  
  p201 - the control point indexed by λ=(2,0,1)
  
  p210 - the control point indexed by λ=(2,1,0)
  
  p300 - the control point indexed by λ=(3,0,0)
Method Details
- getDomainMin1
  
  public double getDomainMin1()
  
  Description copied from class: RealValuedFunctionTwo
  
  Get the minimum value of the first argument in the domain of the function.
  
  Overrides:
  
  getDomainMin1 in class RealValuedFunctionTwo
  
  Returns:
  
  the minimum value
- getDomainMax1
  
  public double getDomainMax1()
  
  Description copied from class: RealValuedFunctionTwo
  
  Get the maximum value of the first argument in the domain of the function.
  
  Overrides:
  
  getDomainMax1 in class RealValuedFunctionTwo
  
  Returns:
  
  the maximum value
- getDomainMin2
  
  public double getDomainMin2()
  
  Description copied from class: RealValuedFunctionTwo
  
  Get the minimum value of the second argument in the domain of the function.
  
  Overrides:
  
  getDomainMin2 in class RealValuedFunctionTwo
  
  Returns:
  
  the minimum value
- getDomainMax2
  
  public double getDomainMax2()
  
  Description copied from class: RealValuedFunctionTwo
  
  Get the maximum value of the second argument in the domain of the function.
  
  Overrides:
  
  getDomainMax2 in class RealValuedFunctionTwo
  
  Returns:
  
  the maximum value
- isInDomain
  
  public boolean isInDomain(double x, double y) throws UnsupportedOperationException
  
  Description copied from class: RealValuedFunctionTwo
  
  Determine if a point (x, y) is within the domain of a real-valued function of two arguments.
  The default behavior of this method assumes the domain is a rectangular region and uses the methods RealValuedFunctionTwo.getDomainMin1(), RealValuedFunctionTwo.getDomainMin2(), RealValuedFunctionTwo.getDomainMax1(), RealValuedFunctionTwo.getDomainMax2() RealValuedFunctionTwo.domainMin1Closed(), RealValuedFunctionTwo.domainMin2Closed(), RealValuedFunctionTwo.domainMax1Closed(), and RealValuedFunctionTwo.domainMax2Closed() to determine if the arguments represent a point in the functions domain. If the domain is not rectangular with each side either in or not in the domain, then this method must be overridden. If it is not possible with a reasonable amount of computation to determine that a point is in the domain, an UnsupportedOperationException may be thrown. If this exception is thrown, it should be thrown regardless of the arguments.
  
  Overrides:
  
  isInDomain in class RealValuedFunctionTwo
  
  Parameters:
  
  x - the 1st coordinate
  
  y - the 2nd coordinate
  
  Returns:
  
  true if the point (x, y) is in this function's domain; false otherwise
  
  Throws:
  
  UnsupportedOperationException - domain membership could not be determined.
- uFromXY
  
  public double uFromXY(double x, double y)
  
  Get the normalized barycentric coordinate u given the coordinates (x, y). The barycentric coordinates are denoted as (u, v, w) with the constraint that u + v + w = 1 and with each coordinate in the range [0, 1].
  
  Parameters:
  
  x - the first argument for this interpolator's function
  
  y - the second argument for this interpolator's function
  
  Returns:
  
  the value of u
- vFromXY
  
  public double vFromXY(double x, double y)
  
  Get the normalized barycentric coordinate v given the coordinates (x, y). The barycentric coordinates are denoted as (u, v, w) with the constraint that u + v + w = 1 and with each coordinate in the range [0, 1].
  
  Parameters:
  
  x - the first argument for this interpolator's function
  
  y - the second argument for this interpolator's function
  
  Returns:
  
  the value of v
- wFromXY
  
  public double wFromXY(double x, double y)
  
  Get the normalized barycentric coordinate w given the coordinates (x, y). The barycentric coordinates are denoted as (u, v, w) with the constraint that u + v + w = 1 and with each coordinate in the range [0, 1].
  
  Parameters:
  
  x - the first argument for this interpolator's function
  
  y - the second argument for this interpolator's function
  
  Returns:
  
  the value of w
- xFromUV
  
  public double xFromUV(double u, double v)
  
  Get the X coordinate given the barycentric coordinates u and v. The barycentric coordinates are denoted as (u, v, w) with the constraint that u + v + w = 1 and with each coordinate in the range [0, 1].
  
  Parameters:
  
  u - the barycentric coordinate u
  
  v - the barycentric coordinate v
  
  Returns:
  
  the corresponding X coordinate
- xFromWU
  
  public double xFromWU(double w, double u)
  
  Get the X coordinate given the barycentric coordinates w and u. The barycentric coordinates are denoted as (u, v, w) with the constraint that u + v + w = 1 and with each coordinate in the range [0, 1].
  
  Parameters:
  
  w - the barycentric coordinate w
  
  u - the barycentric coordinate u
  
  Returns:
  
  the corresponding X coordinate
- xFromVW
  
  public double xFromVW(double v, double w)
  
  Get the X coordinate given the barycentric coordinates v and w. The barycentric coordinates are denoted as (u, v, w) with the constraint that u + v + w = 1 and with each coordinate in the range [0, 1].
  
  Parameters:
  
  v - the barycentric coordinate v
  
  w - the barycentric coordinate w
  
  Returns:
  
  the corresponding X coordinate
- yFromUV
  
  public double yFromUV(double u, double v)
  
  Get the Y coordinate given the barycentric coordinates u and v. The barycentric coordinates are denoted as (u, v, w) with the constraint that u + v + w = 1 and with each coordinate in the range [0, 1].
  
  Parameters:
  
  u - the barycentric coordinate u
  
  v - the barycentric coordinate v
  
  Returns:
  
  the corresponding Y coordinate
- yFromWU
  
  public double yFromWU(double w, double u)
  
  Get the Y coordinate given the barycentric coordinates w and u. The barycentric coordinates are denoted as (u, v, w) with the constraint that u + v + w = 1 and with each coordinate in the range [0, 1].
  
  Parameters:
  
  w - the barycentric coordinate w
  
  u - the barycentric coordinate u
  
  Returns:
  
  the corresponding Y coordinate
- yFromVW
  
  public double yFromVW(double v, double w)
  
  Get the Y coordinate given the barycentric coordinates v and w. The barycentric coordinates are denoted as (u, v, w) with the constraint that u + v + w = 1 and with each coordinate in the range [0, 1].
  
  Parameters:
  
  v - the barycentric coordinate v
  
  w - the barycentric coordinate w
  
  Returns:
  
  the corresponding Y coordinate
- yForZeroW
  
  public double yForZeroW(double x)
  
  Get the value of second argument to the function represented by this interpolator corresponding to a particular value of the first argument when w is zero.
  
  Parameters:
  
  x - the value of the first argument
  
  Returns:
  
  the value of the second argument
- xForZeroW
  
  public double xForZeroW(double y)
  
  Get the value of first argument to the function represented by this interpolator corresponding to a particular value of the second argument when w is zero.
  
  Parameters:
  
  y - the value of the second argument
  
  Returns:
  
  the value of the first argument
- valueAt
  
  public double valueAt(double x, double y)
  
  Description copied from class: RealValuedFunctionTwo
  
  Call the function.
  
  Specified by:
  
  valueAt in interface RealValuedFunctTwoOps
  
  Overrides:
  
  valueAt in class RealValuedFunctionTwo
  
  Parameters:
  
  x - the function's first argument
  
  y - the function's second argument
  
  Returns:
  
  the value of the function for the given arguments
- deriv1At
  
  public double deriv1At(double x, double y)
  
  Description copied from class: RealValuedFunctionTwo
  
  Evaluate the partial derivative $\frac{\partial f}{\partial x_1}$ for a function f(x₁, x₂).
  
  Overrides:
  
  deriv1At in class RealValuedFunctionTwo
  
  Parameters:
  
  x - the function's first argument
  
  y - the function's second argument
  
  Returns:
  
  the value of the partial derivative for the given argument
- deriv2At
  
  public double deriv2At(double x, double y)
  
  Description copied from class: RealValuedFunctionTwo
  
  Evaluate the partial derivative $\frac{\partial f}{\partial x_2}$ for a function f(x₁,x₂).
  
  Overrides:
  
  deriv2At in class RealValuedFunctionTwo
  
  Parameters:
  
  x - the function's first argument
  
  y - the function's second argument
  
  Returns:
  
  the value of the partial derivative for the given argument
- deriv11At
  
  public double deriv11At(double x, double y)
  
  Description copied from class: RealValuedFunctionTwo
  
  Evaluate the partial derivative $ \frac{\partial^2 f}{\partial x_1^2}$ for a function f(x₁,x₂).
  
  Overrides:
  
  deriv11At in class RealValuedFunctionTwo
  
  Parameters:
  
  x - the function's first argument
  
  y - the function's second argument
  
  Returns:
  
  the value of the partial derivative for the given argument
- deriv12At
  
  public double deriv12At(double x, double y)
  
  Description copied from class: RealValuedFunctionTwo
  
  Evaluate the partial derivative $\frac{\partial^2 f}{\partial x_1 \partial x_2}$ for a function f(x₁,x₂).
  
  Overrides:
  
  deriv12At in class RealValuedFunctionTwo
  
  Parameters:
  
  x - the function's first argument
  
  y - the function's second argument
  
  Returns:
  
  the value of the partial derivative for the given argument
- deriv21At
  
  public double deriv21At(double x, double y)
  
  Description copied from class: RealValuedFunctionTwo
  
  Evaluate the partial derivative $\frac{\partial^2 f}{\partial x_2 \partial x_1}$ for a function f(x₁,x₂).
  
  Overrides:
  
  deriv21At in class RealValuedFunctionTwo
  
  Parameters:
  
  x - the function's first argument
  
  y - the function's second argument
  
  Returns:
  
  the value of the partial derivative for the given argument
- deriv22At
  
  public double deriv22At(double x, double y)
  
  Description copied from class: RealValuedFunctionTwo
  
  Evaluate the partial derivative $\frac{\partial^2 f}{\partial x_2^2}$ for a function f(x₁,x₂).
  
  Overrides:
  
  deriv22At in class RealValuedFunctionTwo
  
  Parameters:
  
  x - the function's first argument
  
  y - the function's second argument
  
  Returns:
  
  the value of the partial derivative for the given argument

Class BicubicTriangleInterp

Nested Class Summary

Nested classes/interfaces inherited from class org.bzdev.math.RealValuedFunctionVA

Constructor Summary

Method Summary

Methods inherited from class org.bzdev.math.RealValuedFunctionTwo

Methods inherited from class org.bzdev.math.RealValuedFunctionVA

Methods inherited from class java.lang.Object

Methods inherited from interface org.bzdev.math.RealValuedFunctTwoOps

Constructor Details

BicubicTriangleInterp

BicubicTriangleInterp

BicubicTriangleInterp

BicubicTriangleInterp

Method Details

getDomainMin1

getDomainMax1

getDomainMin2

getDomainMax2

isInDomain

uFromXY

vFromXY

wFromXY

xFromUV

xFromWU

xFromVW

yFromUV

yFromWU

yFromVW

yForZeroW

xForZeroW

valueAt

deriv1At

deriv2At

deriv11At

deriv12At

deriv21At

deriv22At