Algebraic Class Reference

#include <Algebraic.h>

Inheritance diagram for Algebraic:

[legend]Collaboration diagram for Algebraic:

[legend]List of all members.

Detailed Description

An algebraic implicit surface class.

This class implements arbitrary polynomials of three variables.

Definition at line 15 of file Algebraic.h.

Public Member Functions

Algebraic ()

Constucts a constant Algebraic.

Algebraic (int degree)

~Algebraic ()

virtual double proc (gmVector3 x)

virtual gmVector3 grad (gmVector3 v)

grad returns the gradient of the function (proc()) for a point or region.

virtual gmMatrix3 hess (gmVector3 v)

hess returns the Hessian of the function (proc()) for a point or region.

virtual Intervald proc (Box< double >)

proc of an interval-vector (aka a Box) X.

virtual Box3d grad (Box< double >)

virtual IMatrix3d hess (Box< double >)

int degree ()

Degree of polynomial.

void degree (int)

int getNumCoef ()

Number of coefficients.

int getCoefIndex (int i, int j, int k)

Return the ordinal of the x^i y^j z^k coefficient.

double getCoef (int i, int j, int k)

Returns scalar factor of the x^i y^j z^k term.

void setCoef (int i, int j, int k, double a)

Sets coefficient of x^i y^j z^k to a.

virtual void procq (gmVector3, double *)

Computes dproc()/dq for an algebraic.

virtual void procq (Box< double >, Intervald *)

Computes dproc()/dq in Interval form for an algebraic.

virtual void gradq (Box< double >, Intervald *, Intervald *, Intervald *)

Computes dgrad()/dq = d^2proc/dxdq and stores it in three arrays.

virtual void getq (double *)

Get copy of the implicit model parameters.

virtual void _setq (double *)

Overridden by subclasses to set Qs.

virtual int qlen ()

Number of implicit model parameters for this implicit object including its children.

virtual void getqname (char **qn)

The name of each parameter.

MAKE_NAME ()

Static Public Member Functions

int coefficients (int)

Compute the number of terms for a degree d trivariate polynomial.

Protected Member Functions

void initPowerArrays ()

Initialize the m_x,m_y and m_z arrays to their corresponding exponents in coefficient order.

void initXYZPowerArrays (gmVector3 v)

Fills m_[xyz]Pow cache with powers of x, y and z.

void initDerivCoefs (void)

void calcNumCoef ()

Set the number of coefficients in m_numCoef based on the current degree held in m_d.

double dx (gmVector3 v)

double dy (gmVector3 v)

double dz (gmVector3 v)

double dx2 (gmVector3 v)

double dy2 (gmVector3 v)

double dz2 (gmVector3 v)

double dxdy (gmVector3 v)

double dxdz (gmVector3 v)

double dydz (gmVector3 v)

Protected Attributes

int m_d

Degree.

int m_numCoef

Number of terms.

int * m_x

Array storing the exponents of x in coefficient order.

int * m_y

Array storing the exponents of y in coefficient order.

int * m_z

Array storing the exponents of z in coefficient order.

double * m_xPow

Cached array of powers of input value of x in coefficient order.

double * m_yPow

Cached array of powers of input value of y in coefficient order.

double * m_zPow

Cached array of powers of input value of z in coefficient order.

Private Member Functions

void init (int)

Initialization routine called by constructors.

Intervald m_alrp (int, Intervald)

Linear interpolate - 4D critical points.

Private Attributes

double * m_a

Array of coefficients.

Constructor & Destructor Documentation

Algebraic::Algebraic ( )

Constucts a constant Algebraic.

Definition at line 37 of file Algebraic.cpp.
References init().

Algebraic::Algebraic ( int degree )

Definition at line 42 of file Algebraic.cpp.
References init().

Algebraic::~Algebraic ( )

Definition at line 104 of file Algebraic.cpp.
References m_a, m_x, m_xPow, m_y, m_yPow, m_z, and m_zPow.

Member Function Documentation

void Algebraic::_setq ( double * ) [virtual]

Overridden by subclasses to set Qs.
_setq() is called directly only in subclasses of Implicit. Outside the Implicit class hierarchy, always call setq(). So, if you want to save the old set of Qs, a subclass should override _setq(). In fact, the default _setq() in Implicit does nothing, so you have to override it to do the work of setting the current Q array. However, on the off chance you have a subclass where you do not want to save the old Qs, then override setq() to do the work of settings the current Qs. Typically you will want to save the old Qs so subclasses will usually override (and call) _setq().
Reimplemented from Implicit.
Definition at line 592 of file Algebraic.cpp.
References m_a, and m_numCoef.

void Algebraic::calcNumCoef ( ) [protected]

Set the number of coefficients in m_numCoef based on the current degree held in m_d.

Definition at line 303 of file Algebraic.cpp.
References coefficients(), m_d, and m_numCoef.
Referenced by degree(), and init().

int Algebraic::coefficients ( int d ) [static]

Compute the number of terms for a degree d trivariate polynomial.
Degree d algebraic has (d^3 + 6d^2 + 11d + 6)/6 coefficients.
0: 1 1: 3 + 1 = 4 2: 6 + 4 = 10 3: 10 + 10 = 20
# of terms of degree d = d+1 + # of terms of degree d-1. Why? Multiply x times terms of degree d-1, then all that is left are terms containing only y and z. There are d+1 of these. E.g. yyy yyz yzz zzz
Hence terms(d) = d+1 + terms(d-1) and terms(0) = 1. This yields terms(d) = sum[i = 1..d+1] i = (d+2)*(d+1)/2 (Recall sum[i = 1..n] i = (n+1)*n/2) = 1/2 d^2 + 3/2 d + 1.
The # of terms of degree d or less is thus sum[i = 0..d] terms(i) = sum[i = 0..d] 1/2 d^2 + 3/2 d + 1 = 1/2 * d*(d+1)*(2d+1)/6 + 3/2 * d*(d+1)/2 + d+1 (Recall sum[i = 1..n] i^2 = n(n+1)(2n+1)/6) = 1/12 * (2d^3 + 3d^2 + d) + 3/4 * (d^2 + d) + d + 1 = d^3/6 + d^2 + 1 10/12 d + 1.
Definition at line 295 of file Algebraic.cpp.
Referenced by calcNumCoef(), getCoefIndex(), and ImpFileManager::readImplicit().

void Algebraic::degree ( int )

Definition at line 47 of file Algebraic.cpp.
References calcNumCoef(), initPowerArrays(), m_a, m_d, m_numCoef, m_x, m_xPow, m_y, m_yPow, m_z, and m_zPow.

int Algebraic::degree ( )

Degree of polynomial.

Definition at line 264 of file Algebraic.cpp.
References m_d.

double Algebraic::dx ( gmVector3 v ) [protected]

Definition at line 404 of file Algebraic.cpp.
References initXYZPowerArrays(), m_a, m_numCoef, m_x, m_xPow, m_y, m_yPow, m_z, and m_zPow.
Referenced by grad().

double Algebraic::dx2 ( gmVector3 v ) [protected]

Definition at line 446 of file Algebraic.cpp.
References initXYZPowerArrays(), m_a, m_numCoef, m_x, m_xPow, m_y, m_yPow, m_z, and m_zPow.
Referenced by hess().

double Algebraic::dxdy ( gmVector3 v ) [protected]

Definition at line 488 of file Algebraic.cpp.
References initXYZPowerArrays(), m_a, m_numCoef, m_x, m_xPow, m_y, m_yPow, m_z, and m_zPow.
Referenced by hess().

double Algebraic::dxdz ( gmVector3 v ) [protected]

Definition at line 502 of file Algebraic.cpp.
References initXYZPowerArrays(), m_a, m_numCoef, m_x, m_xPow, m_y, m_yPow, m_z, and m_zPow.
Referenced by hess().

double Algebraic::dy ( gmVector3 v ) [protected]

Definition at line 418 of file Algebraic.cpp.
References initXYZPowerArrays(), m_a, m_numCoef, m_x, m_xPow, m_y, m_yPow, m_z, and m_zPow.
Referenced by grad().

double Algebraic::dy2 ( gmVector3 v ) [protected]

Definition at line 460 of file Algebraic.cpp.
References initXYZPowerArrays(), m_a, m_numCoef, m_x, m_xPow, m_y, m_yPow, m_z, and m_zPow.
Referenced by hess().

double Algebraic::dydz ( gmVector3 v ) [protected]

Definition at line 516 of file Algebraic.cpp.
References initXYZPowerArrays(), m_a, m_numCoef, m_x, m_xPow, m_y, m_yPow, m_z, and m_zPow.
Referenced by hess().

double Algebraic::dz ( gmVector3 v ) [protected]

Definition at line 432 of file Algebraic.cpp.
References initXYZPowerArrays(), m_a, m_numCoef, m_x, m_xPow, m_y, m_yPow, m_z, and m_zPow.
Referenced by grad().

double Algebraic::dz2 ( gmVector3 v ) [protected]

Definition at line 474 of file Algebraic.cpp.
References initXYZPowerArrays(), m_a, m_numCoef, m_x, m_xPow, m_y, m_yPow, m_z, and m_zPow.
Referenced by hess().

double Algebraic::getCoef ( int i,

int j,

int k

)

Returns scalar factor of the x^i y^j z^k term.

Parameters:

i exponent for x.

j exponent for y.

k exponent for z.

Returns:
scalar factor of the x^i y^j z^k term.

Definition at line 243 of file Algebraic.cpp.
References getCoefIndex(), and m_a.
Referenced by Quadric::Q().

int Algebraic::getCoefIndex ( int i,

int j,

int k

)

Return the ordinal of the x^i y^j z^k coefficient.
Resulting index used for m_a, m_x, m_y, m_z, m_xPow, m_yPow and m_zPow.
Coefficient order is constant first, then x, then y, then z. For a quadric, coefficient order is: 1, x, y, x^2, xy, y^2, z, xz, yz, z^2.

Todo:
: Coefficient order should be by degree, so for a quadric it should be: 1 x y z x^2 xy y^2 xz yz z^2. This way changing the degree means reducing or extending the parameter list without necessarily reordering the lower degree elements.
1 \
x y \ z
xx xy yy \ xz yz \ zz
xxx xxy xyy yyy \ xxz xyz yyz \ xzz yzz \ zzz
Coordinates: level: d = i+j+k (degree) horizontal: j vertical: k
Answer is: coefficients(d) - sum[kk=1..d+1-k](kk) + j = coefficients(d) - sum[kk=1..i+j+1](kk) + j (since d = i+j+k) = coefficients(d) - (i+j+1)(i+j+2)/2 + j = coefficients(d) - (i*i + 2*i*j + j*j + 3*i + 3*j + 2)/2 + j

Parameters:

i exponent of x

j exponent of y

k exponent of z

Returns:
coefficient-order index for x^i y^j z^k.

Definition at line 349 of file Algebraic.cpp.
References coefficients().
Referenced by getCoef(), initPowerArrays(), and setCoef().

int Algebraic::getNumCoef ( )

Number of coefficients.

Definition at line 259 of file Algebraic.cpp.
References m_numCoef.

void Algebraic::getq ( double * q ) [virtual]

Get copy of the implicit model parameters.

Parameters:

A pointer to the array to be returned.

Reimplemented from Implicit.
Definition at line 584 of file Algebraic.cpp.
References m_a, and m_numCoef.

void Algebraic::getqname ( char ** qn ) [virtual]

The name of each parameter.
The names of the parameters of the function (if any) are put into an array which gets returned.
Parameters:

qn An array of size qlen of strings listing names of parameters

Reimplemented from Implicit.
Definition at line 603 of file Algebraic.cpp.
References m_numCoef, m_x, m_y, and m_z.

Box3d Algebraic::grad ( Box< double > ) [virtual]

Reimplemented from Implicit.
Definition at line 180 of file Algebraic.cpp.
References dx(), Intervald, m_a, m_alrp(), m_numCoef, m_x, m_y, and m_z.

gmVector3 Algebraic::grad ( gmVector3 v ) [virtual]

grad returns the gradient of the function (proc()) for a point or region.
Basically, the gradient returns a vector of partial derivatives, or more formally grad(proc()) = (dproc/dx,dproc/dy,dproc/dz).
Note:
Unless grad is overwritten, it defaults to the numerical approximation of the gradient = (proc(x+eps) - proc(x))/eps ...

Parameters:

x Where to evaluate the gradient of the function.

Returns:
The gradient result.

See also:
setEpsilon(double)

Reimplemented from Implicit.
Definition at line 143 of file Algebraic.cpp.
References dx(), dy(), and dz().

void Algebraic::gradq ( Box< double > x,

Intervald * dfdxdq,

Intervald * dfdydq,

Intervald * dfdzdq

) [virtual]

Computes dgrad()/dq = d^2proc/dxdq and stores it in three arrays.

Parameters:

dfdxdq returns Intervald x components of dgrad()/dq

dfdydq returns Intervald y components of dgrad()/dq

dfdzdq returns Intervald z components of dgrad()/dq

Arrays should be preallocated to length qlen().
Assume f(x,y,z) = a x^i y^j z^k. Then df/dx = a i x^(i-1) y^j z^k. Then d^2f/dxda = i x^(i-1) y^j z^k.
Reimplemented from Implicit.
Definition at line 570 of file Algebraic.cpp.
References Intervald, m_numCoef, m_x, m_y, and m_z.

IMatrix3d Algebraic::hess ( Box< double > ) [virtual]

Reimplemented from Implicit.
Definition at line 201 of file Algebraic.cpp.
References Intervald, m_a, m_numCoef, m_x, m_y, and m_z.

gmMatrix3 Algebraic::hess ( gmVector3 v ) [virtual]

hess returns the Hessian of the function (proc()) for a point or region.
Basically, the Hessian is a matrix of second partial derivatives, or more formally hess(proc()) =
d^2proc/dx^2, d^2proc/dxdy d^2proc/dxdz \ d^2proc/dxdy, d^2proc/dy^2 d^2proc/dydz \ d^2proc/dxdz, d^2proc/dydz d^2proc/dz^2

Note:
Unless hess is overwritten, it defaults to the numerical approximation of the Hessian.

Parameters:

x Where to evaluate the Hessian of the function.

Returns:
The Hessian result.

See also:
setEpsilon(double)

Note:
The Hessian is symmetric for our uses.

Reimplemented from Implicit.
Definition at line 148 of file Algebraic.cpp.
References dx2(), dxdy(), dxdz(), dy2(), dydz(), and dz2().

void Algebraic::init ( int deg ) [private]

Initialization routine called by constructors.
This function allows all of the main object init code to be in one place.
Definition at line 17 of file Algebraic.cpp.
References calcNumCoef(), initPowerArrays(), m_a, m_d, m_numCoef, m_x, m_xPow, m_y, m_yPow, m_z, and m_zPow.
Referenced by Algebraic().

void Algebraic::initDerivCoefs ( void ) [protected]

void Algebraic::initPowerArrays ( ) [protected]

Initialize the m_x,m_y and m_z arrays to their corresponding exponents in coefficient order.

Definition at line 362 of file Algebraic.cpp.
References getCoefIndex(), Implicit::k(), m_d, m_x, m_y, and m_z.
Referenced by degree(), and init().

void Algebraic::initXYZPowerArrays ( gmVector3 v ) [protected]

Fills m_[xyz]Pow cache with powers of x, y and z.
Efficiently computes powers by using results of lower powers.
Parameters:

v The input x,y,z.

Note:
Remembers last v to avoid recomputation.

Todo:
Could memoize (hash) v to possibly avoid more recomputation. Need to build a general datastructure to handle this. Perhaps this should be handled at a higher level (e.g. as in Bloomenthal's marching cubes implementation).

Definition at line 387 of file Algebraic.cpp.
References m_d, m_xPow, m_yPow, and m_zPow.
Referenced by dx(), dx2(), dxdy(), dxdz(), dy(), dy2(), dydz(), dz(), dz2(), proc(), and procq().

Intervald Algebraic::m_alrp ( int i,

Intervald t

) [private]

Linear interpolate - 4D critical points.
We use the "current" and the "last" coefficients to calculate an interval of the coefficients over a given time interval. This is used for 4D critical point finding.
Definition at line 122 of file Algebraic.cpp.
References Implicit::getqold(), Intervald, m_a, and m_numCoef.
Referenced by grad(), and proc().

Algebraic::MAKE_NAME ( )

Reimplemented in Cone, Cylinder, Ellipsoid, Quadric, and Torus.

Intervald Algebraic::proc ( Box< double > x ) [virtual]

proc of an interval-vector (aka a Box) X.
If X is a 4D box then we assume q is lerp between qold and q.
x[3] = [told,t] m_alrp (i,x[3]) returns [(a_i)old,a_i]
Reimplemented from Implicit.
Definition at line 169 of file Algebraic.cpp.
References Intervald, m_a, m_alrp(), m_numCoef, m_x, m_y, and m_z.

double Algebraic::proc ( gmVector3 x ) [virtual]

Implements Implicit.
Definition at line 132 of file Algebraic.cpp.
References initXYZPowerArrays(), m_a, m_numCoef, m_x, m_xPow, m_y, m_yPow, m_z, and m_zPow.

void Algebraic::procq ( Box< double > x,

Intervald * dfdq

) [virtual]

Computes dproc()/dq in Interval form for an algebraic.
Each term of the algebraic is of the form a x^i y^j z^k so its derivative with respect to a is just x^i y^j z^k.
Reimplemented from Implicit.
Definition at line 548 of file Algebraic.cpp.
References Intervald, m_numCoef, m_x, m_y, and m_z.

void Algebraic::procq ( gmVector3 x,

double * dfdq

) [virtual]

Computes dproc()/dq for an algebraic.
Each term of the algebraic is of the form a x^i y^j z^k so its derivative with respect to a is just x^i y^j z^k.
Reimplemented from Implicit.
Definition at line 534 of file Algebraic.cpp.
References initXYZPowerArrays(), m_numCoef, m_x, m_xPow, m_y, m_yPow, m_z, and m_zPow.

int Algebraic::qlen ( ) [virtual]

Number of implicit model parameters for this implicit object including its children.

Reimplemented from Implicit.
Definition at line 598 of file Algebraic.cpp.
References m_numCoef.

void Algebraic::setCoef ( int i,

int j,

int k,

double a

)

Sets coefficient of x^i y^j z^k to a.

Parameters:

i exponent for x.

j exponent for y.

k exponent for z.

a Coefficient for x^i y^j z^k.

Definition at line 254 of file Algebraic.cpp.
References getCoefIndex(), and m_a.
Referenced by Quadric::Quadric(), Torus::Torus(), and Quadric::Transform().

Member Data Documentation

double* Algebraic::m_a [private]

Array of coefficients.
Should be accessed through getCoef and setCoef.
Definition at line 65 of file Algebraic.h.
Referenced by _setq(), degree(), dx(), dx2(), dxdy(), dxdz(), dy(), dy2(), dydz(), dz(), dz2(), getCoef(), getq(), grad(), hess(), init(), m_alrp(), proc(), setCoef(), and ~Algebraic().

int Algebraic::m_d [protected]

Degree.

Definition at line 72 of file Algebraic.h.
Referenced by calcNumCoef(), degree(), init(), initPowerArrays(), and initXYZPowerArrays().

int Algebraic::m_numCoef [protected]

Number of terms.
Initialized by calcNumCoef. Accessed via getNumCoef.
Definition at line 78 of file Algebraic.h.
Referenced by _setq(), calcNumCoef(), degree(), dx(), dx2(), dxdy(), dxdz(), dy(), dy2(), dydz(), dz(), dz2(), getNumCoef(), getq(), getqname(), grad(), gradq(), hess(), init(), m_alrp(), proc(), procq(), and qlen().

int* Algebraic::m_x [protected]

Array storing the exponents of x in coefficient order.

Definition at line 81 of file Algebraic.h.
Referenced by degree(), dx(), dx2(), dxdy(), dxdz(), dy(), dy2(), dydz(), dz(), dz2(), getqname(), grad(), gradq(), hess(), init(), initPowerArrays(), proc(), procq(), and ~Algebraic().

double* Algebraic::m_xPow [protected]

Cached array of powers of input value of x in coefficient order.

Definition at line 90 of file Algebraic.h.
Referenced by degree(), dx(), dx2(), dxdy(), dxdz(), dy(), dy2(), dydz(), dz(), dz2(), init(), initXYZPowerArrays(), proc(), procq(), and ~Algebraic().

int* Algebraic::m_y [protected]

Array storing the exponents of y in coefficient order.

Definition at line 84 of file Algebraic.h.
Referenced by degree(), dx(), dx2(), dxdy(), dxdz(), dy(), dy2(), dydz(), dz(), dz2(), getqname(), grad(), gradq(), hess(), init(), initPowerArrays(), proc(), procq(), and ~Algebraic().

double* Algebraic::m_yPow [protected]

Cached array of powers of input value of y in coefficient order.

Definition at line 93 of file Algebraic.h.
Referenced by degree(), dx(), dx2(), dxdy(), dxdz(), dy(), dy2(), dydz(), dz(), dz2(), init(), initXYZPowerArrays(), proc(), procq(), and ~Algebraic().

int* Algebraic::m_z [protected]

Array storing the exponents of z in coefficient order.

Definition at line 87 of file Algebraic.h.
Referenced by degree(), dx(), dx2(), dxdy(), dxdz(), dy(), dy2(), dydz(), dz(), dz2(), getqname(), grad(), gradq(), hess(), init(), initPowerArrays(), proc(), procq(), and ~Algebraic().

double* Algebraic::m_zPow [protected]

Cached array of powers of input value of z in coefficient order.

Definition at line 96 of file Algebraic.h.
Referenced by degree(), dx(), dx2(), dxdy(), dxdz(), dy(), dy2(), dydz(), dz(), dz2(), init(), initXYZPowerArrays(), proc(), procq(), and ~Algebraic().

The documentation for this class was generated from the following files:

Generated on Mon Jun 28 15:02:00 2004 for Advanced Surface Library by

1.3.4


Public Member Functions
	Algebraic ()
	Constucts a constant Algebraic.
	Algebraic (int degree)
	~Algebraic ()
virtual double	proc (gmVector3 x)
virtual gmVector3	grad (gmVector3 v)
	grad returns the gradient of the function (proc()) for a point or region.
virtual gmMatrix3	hess (gmVector3 v)
	hess returns the Hessian of the function (proc()) for a point or region.
virtual Intervald	proc (Box< double >)
	proc of an interval-vector (aka a Box) X.
virtual Box3d	grad (Box< double >)
virtual IMatrix3d	hess (Box< double >)
int	degree ()
	Degree of polynomial.
void	degree (int)
int	getNumCoef ()
	Number of coefficients.
int	getCoefIndex (int i, int j, int k)
	Return the ordinal of the x^i y^j z^k coefficient.
double	getCoef (int i, int j, int k)
	Returns scalar factor of the x^i y^j z^k term.
void	setCoef (int i, int j, int k, double a)
	Sets coefficient of x^i y^j z^k to a.
virtual void	procq (gmVector3, double *)
	Computes dproc()/dq for an algebraic.
virtual void	procq (Box< double >, Intervald *)
	Computes dproc()/dq in Interval form for an algebraic.
virtual void	gradq (Box< double >, Intervald , Intervald , Intervald *)
	Computes dgrad()/dq = d^2proc/dxdq and stores it in three arrays.
virtual void	getq (double *)
	Get copy of the implicit model parameters.
virtual void	_setq (double *)
	Overridden by subclasses to set Qs.
virtual int	qlen ()
	Number of implicit model parameters for this implicit object including its children.
virtual void	getqname (char **qn)
	The name of each parameter.
	MAKE_NAME ()
Static Public Member Functions
int	coefficients (int)
	Compute the number of terms for a degree d trivariate polynomial.
Protected Member Functions
void	initPowerArrays ()
	Initialize the m_x,m_y and m_z arrays to their corresponding exponents in coefficient order.
void	initXYZPowerArrays (gmVector3 v)
	Fills m_[xyz]Pow cache with powers of x, y and z.
void	initDerivCoefs (void)
void	calcNumCoef ()
	Set the number of coefficients in m_numCoef based on the current degree held in m_d.
double	dx (gmVector3 v)
double	dy (gmVector3 v)
double	dz (gmVector3 v)
double	dx2 (gmVector3 v)
double	dy2 (gmVector3 v)
double	dz2 (gmVector3 v)
double	dxdy (gmVector3 v)
double	dxdz (gmVector3 v)
double	dydz (gmVector3 v)
Protected Attributes
int	m_d
	Degree.
int	m_numCoef
	Number of terms.
int *	m_x
	Array storing the exponents of x in coefficient order.
int *	m_y
	Array storing the exponents of y in coefficient order.
int *	m_z
	Array storing the exponents of z in coefficient order.
double *	m_xPow
	Cached array of powers of input value of x in coefficient order.
double *	m_yPow
	Cached array of powers of input value of y in coefficient order.
double *	m_zPow
	Cached array of powers of input value of z in coefficient order.
Private Member Functions
void	init (int)
	Initialization routine called by constructors.
Intervald	m_alrp (int, Intervald)
	Linear interpolate - 4D critical points.
Private Attributes
double *	m_a
	Array of coefficients.