Quadric Class Reference

Quadric polynomials. More...

#include <quadric.h>

Public Member Functions
	Quadric ()
	Empty.
	Quadric (const double &, const double &, const double &)
	Creates a quadric.
	Quadric (const Linear &)
	Creates a quadric from a Linear polynomial.
	~Quadric ()
	Empty.
constexpr double &	operator[] (int)
	Access class components.
constexpr double	operator[] (int) const
	Overloaded.
Quadric	operator- () const
	Overloaded.
Quadric &	operator+= (const Quadric &)
	Destructive sum of two polynomials.
Quadric &	operator-= (const Quadric &)
	Destructive difference of two polynomials.
Quadric &	operator*= (const double &)
	Scale a polynomial by a double value.
Quadric &	operator/= (const double &)
	Scale a polynomial by a double value.
constexpr double	operator() (const double &) const
	Evaluates the quadric.
constexpr double	Derivative (const double &) const
	Computes the derivative value at a given point.
Linear	Prime () const
	Compute the derivative of a quadric, which is a linear expression.
int	Solve (double &, double &) const
	Solve quadratic equations.
int	Solve (double *) const
	Solve quadratic equations. This function store the sorted roots in an array and returns the number of roots.
int	Solve (double *, const double &, const double &) const
	Solve the quadratic equation over a given interval.
void	Range (double &, double &, const double &=0.0, const double &=1.0) const
	This function computes the range of values taken by a quadric over a given interval.
double	Maximum (const double &=0.0, const double &=1.0) const
	This function computes the maximum value taken by a quadric over a given interval.
double	Minimum (const double &=0.0, const double &=1.0) const
	This function computes the minimum value taken by a quadric over a given interval.
double	K (const double &, const double &) const
	Compute the Lipschitz constant of the quadric.

Static Public Member Functions
static Quadric	Bezier (const double &, const double &, const double &)
	Creates a quadratic Bezier polynomial on interval [0,1].
static double	Smooth (const double &, const double &)
	Compute the value of the \(C^1\) smooth interpolating function \((1-x/r)^2\).
static double	Smooth (const double &)
	Compute the value of the \(C^1\) smooth interpolating function \((1-x)^2\).
static double	SmoothCompact (const double &, const double &)
	Compact support version of the smooth interpolating function Quadric::Smooth().
static double	SmoothCompact (const double &, const double &, const double &)
	Compact support version of the smooth interpolating function Quadric::Smooth() with interior and exterior radii.
static double	Warp (const double &)
	Unit interval warping function.
static Quadric	Bernstein (int)
	Creates a quadric Bernstein polynomial.
static double	Bernstein (int, const double &)
	Compute the quadric Bernstein polynomial for a given value.
static Quadric	FromRoots (const QVector< double > &)
	Create a quadric from a set of roots.

Static Public Attributes
static double	epsilon = 1.0e-16
	Epsilon value used to check discriminant term in the root finding process.

Protected Attributes
double	c [3] = { 0.0,0.0,0.0 }
	Array of coefficients.

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

Detailed Description

Quadric polynomials.

Constructors should provide the coefficients in descending order. Here is an example of how to code the quadric \(2 x^2 - x + 1\):

Quadric p(2.0,-1.0,1.0);

Member Function Documentation

Bernstein() [1/2]

Quadric Quadric::Bernstein ( int p )

static

Creates a quadric Bernstein polynomial.

Parameters

p	p-th polynomial.

Bernstein() [2/2]

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

static

Compute the quadric Bernstein polynomial for a given value.

Parameters

p	p-th polynomial.
x	Real.

Bezier()

Quadric Quadric::Bezier ( const double & a, const double & b, const double & c )

static

Creates a quadratic Bezier polynomial on interval [0,1].

Parameters

a,b,c

The parameters of the Bézier polynomial.

Parameters are as follows:

\(c(t)=a (1-t)^2+2b t(1-t)+c t^2=(a-2b+c) t^2+2(b-a) t+a\).

Derivative()

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

inlineconstexpr

Computes the derivative value at a given point.

If only one evaluation is to be made, this function is more efficient than Prime() and then evaluating the derivative for a given input value.

Parameters

x

Real.

See also

Quadric::Prime

FromRoots()

Quadric Quadric::FromRoots ( const QVector< double > & r )

static

Create a quadric from a set of roots.

Parameters

r	Set of roots.

Example of how to code the quadric (x-2)(x+1):

Quintic p=Quintic::FromRoots({2.0,1.0});

K()

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

Compute the Lipschitz constant of the quadric.

Parameters

a,b

Interval.

Maximum()

double Quadric::Maximum	(	const double &	a = 0.0,
		const double &	b = 1.0 ) const

This function computes the maximum value taken by a quadric over a given interval.

See also

Range

Parameters

a,b

Interval.

Minimum()

double Quadric::Minimum	(	const double &	a = 0.0,
		const double &	b = 1.0 ) const

This function computes the minimum value taken by a quadric over a given interval.

See also

Range

Parameters

a,b

Interval.

Prime()

Linear Quadric::Prime ( ) const

inline

Compute the derivative of a quadric, which is a linear expression.

See also

Quadric::Derivative

Range()

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

This function computes the range of values taken by a quadric over a given interval.

Bascially 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.

Smooth() [1/2]

double Quadric::Smooth ( const double & x )

inlinestatic

Compute the value of the \(C^1\) smooth interpolating function \((1-x)^2\).

See also

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

Parameters

x	Argument value.

Smooth() [2/2]

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

inlinestatic

Compute the value of the \(C^1\) smooth interpolating function \((1-x/r)^2\).

See also

Cubic::Smooth(), Quadric::SmoothCompact

Parameters

x	Argument value.
r	Radius.

SmoothCompact() [1/2]

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

inlinestatic

Compact support version of the smooth interpolating function Quadric::Smooth().

Return Quadric::Smooth(x/r) if x<r and 0 otherwise.

See also

Cubic::Smooth(), Quadric::Smooth()

Parameters

x	Argument value.
r	Radius.

SmoothCompact() [2/2]

double Quadric::SmoothCompact ( const double & x, const double & e, const double & i )

inlinestatic

Compact support version of the smooth interpolating function Quadric::Smooth() with interior and exterior radii.

See also

Cubic::Smooth(), Quadric::Smooth()

Parameters

x	Argument value.
i,e	Interior and exterior radii.

Solve() [1/3]

int Quadric::Solve	(	double &	u,
		double &	v ) const

Solve quadratic equations.

This function uses two double arguments to store the roots and returns the number of roots.

Parameters

u,v	The two of roots, if they exist.

Solve() [2/3]

int Quadric::Solve

(

double *

y

)

const

Solve quadratic equations. This function store the sorted roots in an array and returns the number of roots.

Parameters

y	The array of roots.

Solve() [3/3]

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

Solve the quadratic equation over a given interval.

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

Parameters

roots	The array of roots.
a,b	The interval.

Warp()

double Quadric::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<<

std::ostream & operator<< ( std::ostream & s, const Quadric & p )

friend

Overloaded output-stream operator.

Parameters

s	Stream.
p	The quadric.