Cubic Class Reference

Cubic polynomials. More...

#include <cubic.h>

Public Member Functions
	Cubic ()
	Empty.
	Cubic (const double &, const double &, const double &, const double &)
	Creates a cubic.
	Cubic (const Quadric &)
	Creates a cubic given a quadric.
	~Cubic ()
	Empty.
constexpr double &	operator[] (int)
	Access class components.
constexpr double	operator[] (int) const
	Overloaded.
int	CheckDegree () const
	Check the degree of the cubic.
Cubic	operator- () const
	Unary.
Cubic &	operator+= (const Cubic &)
	Destructive sum of two cubics.
Cubic &	operator-= (const Cubic &)
	Destructive difference of two cubics.
Cubic &	operator*= (const double &)
	Scale a cubic by a double value.
Cubic &	operator/= (const double &)
	Scale a cubic by a double value.
constexpr double	operator() (const double &) const
	Evaluates the cubic.
constexpr double	Derivative (const double &) const
	Computes the derivative of the cubic.
Quadric	Prime () const
	Computes the first derivative of a cubic, which is a quadric.
Linear	Second () const
	Computes the second derivative of a cubic, which is a linear polynomial.
int	Solve (double *) const
	Solve the cubic equation.
int	Solve (double *, const double &, const double &) const
	Search the roots of a polynomial equation over a given interval.
int	SolveNormalized (double *) const
	Solve a normalized cubic equation.
void	Range (double &, double &, const double &=0.0, const double &=1.0) const
	Compute the range of values taken by the cubic over a given interval.
double	K (const double &, const double &) const
	Compute the Lipschitz constant of the cubic.

Static Public Member Functions
static Cubic	Compose (const Cubic &, const Linear &)
	Compose the cubic by a linear.
static Cubic	Hermite (const double &, const double &, const double &, const double &)
	Creates an Hermite cubic polynomial.
static Cubic	Bezier (const double &, const double &, const double &, const double &)
	Creates a cubic Bezier polynomial.
static Cubic	Spline (const double &, const double &, const double &, const double &)
	Creates a cubic Spline polynomial.
static double	Interpolation (const double &, const double &, const double &, const double &, const double &)
	Bicubic interpolation based on four values.
static double	Smooth (const double &, const double &)
	Compute the value of a C2 smooth interpolating function (1-x/r)3.
static double	SmoothCompact (const double &, const double &)
	Compactly supported smooth interpolating function.
static double	SmoothCompact (const double &, const double &, const double &)
	Compactly supported smooth interpolating function.
static double	Smooth (const double &)
	Compute the value of a C1 smooth interpolating function.
static double	SmoothStep (const double &, const double &, const double &)
	Compute a smooth cubic step.
static double	Gaussian (const double &, const double &, const double &)
	Compute a compactly supported Gaussian-like pulse.
static double	GaussianThick (const double &, const double &, const double &, const double &)
	Compute a compactly supported Gaussian-like pulse with a thick plateau.
static double	Sigmoid (const double &, const double &, const double &)
	Compute the value of a compactly supported sigmoid-shaped symmetric cubic.
static double	Warp (const double &)
	Unit interval warping function.
static Cubic	Bernstein (int)
	Creates a cubic Bernstein polynomial.
static double	Bernstein (int, const double &)
	Compute the quadric Bernstein polynomial for a given value.

Protected Attributes
double	c [4] = { 0.0,0.0,0.0,0.0 }
	Array of coefficients.

Friends
Cubic	operator+ (const Cubic &u, const Cubic &v)
	Overloaded.
Cubic	operator- (const Cubic &v, const Cubic &u)
	Overloaded.
Cubic	operator* (const Cubic &u, const double &e)
	Multiply a cubic by a scalar value.
Cubic	operator* (const double &a, const Cubic &p)
	Overloaded.
Cubic	operator/ (const Cubic &p, const double &a)
	Overloaded.
Cubic	operator* (const Quadric &q, const Linear &l)
	Multiply a quadric by a linear polynomial.
std::ostream &	operator<< (std::ostream &s, const Cubic &c)
	Overloaded output-stream operator.

Detailed Description

Cubic polynomials.

Closed form expression of roots exist for such polynomials.

Constructors should provide the coefficients in descending order. Here is an example of how to code the cubic 2 x³-x+1:

Cubic p(2.0,0.0,-1.0,1.0);

How do I compute the interpolating smooth cubic (1-x/r)³ for values x within [0,r]?
A first solution is to use the Cubic class as follows:

double y=Cubic(-1.0/(r*r*r),3.0/(r*r),-3.0/r,1.0)(x)

Cubic also implements some inline static member function such as :

inline double Cubic::Smooth(const double& x, const double& r)
{
return (1.0-x/r)*(1.0-x/r)*(1.0-x/r);
}

This function is more efficient if only one or two function calls are needed.

How do I create a Hermite cubic spline curve?
A simple way to do this is to use the static member function of the CubicCurve class, for example:

CubicCurve c=CubicCurve::Hermite(Vector(0,0,0),Vector(2,1,0),Vector(0,1,0),Vector(1,0,0));

Constructor & Destructor Documentation

Cubic()

Cubic::Cubic ( const double & a, const double & b, const double & c, const double & d )

inlineexplicit

Creates a cubic.

Coefficients start from the highest degree to the lowest.

Parameters

a,b,c,d

Coefficients of the cubic a x3+b x2+c x+d.

Member Function Documentation

Bernstein() [1/2]

Cubic Cubic::Bernstein ( int p )

static

Creates a cubic Bernstein polynomial.

Parameters

p	p-th polynomial.

Bernstein() [2/2]

double Cubic::Bernstein ( int p, const double & x )

static

Compute the quadric Bernstein polynomial for a given value.

Parameters

p	p-th polynomial.
x	Real.

Bezier()

Cubic Cubic::Bezier ( const double & a, const double & b, const double & c, const double & d )

static

Creates a cubic Bezier polynomial.

Interval parameterization is unit.

Parameters

a,b,c,d

Control values.

Compose()

Cubic Cubic::Compose ( const Cubic & p, const Linear & q )

static

Compose the cubic by a linear.

This function computes p ( q(x) ), where p denotes the cubic and q the linear polynomial.

Parameters

p	The cubic.
q	The linear polynomial.

Derivative()

double Cubic::Derivative ( const double & x ) const

inlineconstexpr

Computes the derivative of the cubic.

This function is more efficient than using Prime() and then evaluating the derivative for a given input value.

Parameters

x

Real.

Gaussian()

double Cubic::Gaussian ( const double & c, const double & r, const double & x )

inlinestatic

Compute a compactly supported Gaussian-like pulse.

The function has C¹ continuity.

See also

Quintic::Gaussian()

Parameters

c	Center.
r	Radius.
x	Value.

GaussianThick()

double Cubic::GaussianThick ( const double & c, const double & t, const double & r, const double & x )

inlinestatic

Compute a compactly supported Gaussian-like pulse with a thick plateau.

The function has C¹ continuity.

See also

Quintic::GaussianThick()

The support has 2 ( r + t ) extent.

Parameters

c	Center.
t,r	Thickness (plateau) and radius.
x	Value.

Hermite()

Cubic Cubic::Hermite ( const double & a, const double & b, const double & ta, const double & tb )

static

Creates an Hermite cubic polynomial.

Interval parameterization is unit.

Parameters

a,b	Values for t=0 and t=1.
ta,tb	Derivative values for t=0 and t=1.

See also

Math::Cubic()

Interpolation()

double Cubic::Interpolation ( const double & x, const double & a, const double & b, const double & c, const double & d )

static

Bicubic interpolation based on four values.

Parameters

a,b,c,d	Four input values.
x	Interpolant.

Author

Lois Paulin

K()

double Cubic::K	(	const double &	a,
		const double &	b ) const

Compute the Lipschitz constant of the cubic.

Parameters

a,b

Interval.

operator()()

double Cubic::operator() ( const double & x ) const

inlineconstexpr

Evaluates the cubic.

Parameters

x

Value.

Range()

void Cubic::Range	(	double &	x,
		double &	y,
		const double &	a = 0.0,
		const double &	b = 1.0 ) const

Compute the range of values taken by the cubic over a given interval.

Computes the roots of the first derivative, and evaluates the polynomial at the roots if they are within the interval bounds.

Parameters

a,b	Interval.
x,y	Returned range.

Sigmoid()

double Cubic::Sigmoid ( const double & x, const double & r, const double & t )

static

Compute the value of a compactly supported sigmoid-shaped symmetric cubic.

The cubic was obtaied by solving the Hermite Cubic contraints: f(0)=0, f'(0)=1, f(R)=T and f'(R)=0. Coefficients are: a₀=0, a₁=1, a₂=(3T-4R)/4R^2, a₃=(R-T)/4R^3.

See also

Math::SigmoidQuadric

Parameters

x	Real.
r	Radius of influence.
t	Value.

Smooth() [1/2]

double Cubic::Smooth ( const double & x )

inlinestatic

Compute the value of a C¹ smooth interpolating function.

The cubic is defined as x²(3 - 2 x). Its first derivatives at 0.0 and 1.0 are 0.0.

The Lipschitz constant of the smooth quintic over [0,1] is λ=3/2.

Parameters

x	Argument in [0,1].

See also

Quintic::Smooth(), Septic::Smooth()

Smooth() [2/2]

double Cubic::Smooth ( const double & x, const double & r )

inlinestatic

Compute the value of a C² smooth interpolating function (1-x/r)³.

It is possible to implement Wyvill's (Galin's in fact) smoothing kernel by passing the squared distance to the skeleton and the squared radius to this function. For example, given a box skeleton:

Vector p(-2.0,3.0,5.0);
double d=Box(1.0).R(p); // Squared distance to a box
double r=4.0; // Radius;
double d=d>r*r?0.0:Cubic::Smooth(d,r*r); // Direct implementation of Wyvill's smoothing kernel

See also

Smooth(const double&, const double&, const double&)

SmoothCompact(const double&, const double&)

Parameters

x	Squared distance.
r	Squared radius.

SmoothCompact() [1/2]

double Cubic::SmoothCompact ( const double & x, const double & r )

inlinestatic

Compactly supported smooth interpolating function.

See also

Cubic::Smooth(const double&, const double&), Quadric::SmoothCompact()

Parameters

x	Squared distance.
r	Squared radius.

SmoothCompact() [2/2]

double Cubic::SmoothCompact ( const double & x, const double & ra, const double & rb )

inlinestatic

Compactly supported smooth interpolating function.

See also

Cubic::SmoothCompact(const double&, const double&)

Parameters

x	Distance (not the squared distance).
ra,rb	Radius interval.

SmoothStep()

double Cubic::SmoothStep ( const double & x, const double & a, const double & b )

inlinestatic

Compute a smooth cubic step.

The code is slightly more efficient than:

double y=Cubic::Smooth(Linear::Step(x,a,b));

Parameters

a,b	Interval values.
x	Input value.

See also

Quintic::SmoothStep(), Septic::SmoothStep()

Solve() [1/2]

int Cubic::Solve

(

double *

y

)

const

Solve the cubic equation.

This function stores the sorted roots in an array and returns the number of roots.

Parameters

y	Array for storing the roots.

Solve() [2/2]

int Cubic::Solve	(	double *	roots,
		const double &	a,
		const double &	b ) const

Search the roots of a polynomial equation over a given interval.

Parameters

roots	Array for storing the roots.
a,b	Interval limits.

SolveNormalized()

int Cubic::SolveNormalized

(

double *

y

)

const

Solve a normalized cubic equation.

The highest degree coefficient is 1.0.

This function stores the sorted roots in an array and returns the number of roots.

Parameters

y	The array of roots.

Spline()

Cubic Cubic::Spline ( const double & a, const double & b, const double & c, const double & d )

static

Creates a cubic Spline polynomial.

Interval parameterization is unit.

Parameters

a,b,c,d

Control values.

Warp()

double Cubic::Warp ( const double & x )

inlinestatic

Unit interval warping function.

Remaps the unit interval into the unit interval by expanding the sides and compressing the center, keeping 1/2 mapped to 1/2.

Parameters

x	Real in [0,1].

Friends And Related Symbol Documentation

operator*

Cubic operator* ( const Quadric & q, const Linear & l )

friend

Multiply a quadric by a linear polynomial.

Parameters

q	Quadric.
l	Linear.

operator<<

std::ostream & operator<< ( std::ostream & s, const Cubic & c )

friend

Overloaded output-stream operator.

Parameters

s	Stream.
c	Cubic.