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 C¹ smooth interpolating function (1-x/r)².

static double	Smooth (const double &)
	Compute the value of the C¹ 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 b²-4ac 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. is an example of how to code the quadric 2 x²-x+1:

Quadric p(2.0,-1.0,1.0);

Quadric::Quadric

Quadric()

Empty.

Definition quadric.h:13

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 B�zier polynomial.

Parameters are as follows:

c(t)=a (1-t)²+2b t(1-t)+c t²=(a-2b+c) t²+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});

Quintic

Quintic polynomials.

Definition quintic.h:7

Quintic::FromRoots

static Quintic FromRoots(const QVector< double > &)

Create a quintic from a set of roots.

Definition quintic.cpp:235

◆ 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 C¹ 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 C¹ 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.

Public Member Functions

Static Public Member Functions

Static Public Attributes

Protected Attributes

Friends

Detailed Description

Member Function Documentation

◆ Bernstein() [1/2]

◆ Bernstein() [2/2]

◆ Bezier()

◆ Derivative()

◆ FromRoots()

◆ K()

◆ Maximum()

◆ Minimum()

◆ Prime()

◆ Range()

◆ Smooth() [1/2]

◆ Smooth() [2/2]

◆ SmoothCompact() [1/2]

◆ SmoothCompact() [2/2]

◆ Solve() [1/3]

◆ Solve() [2/3]

◆ Solve() [3/3]

◆ Warp()

Friends And Related Symbol Documentation

◆ operator<<