Represent a quadratic polynomial in the form c_[2] * x^2 + c_[1] * x + c_[0]. More...

#include <curve.h>

Detailed Description

Represent a quadratic polynomial in the form c_[2] * x^2 + c_[1] * x + c_[0].

Public Types
typedef Range::TArray< 2 >	RootsArray

typedef Range::RangeArray< 2 >	RangeArray

Public Member Functions
	QuadraticCurve (const float c2, const float c1, const float c0)

	QuadraticCurve (const float *c)

	QuadraticCurve (const QuadraticCurve &q, const float y_scale)

	QuadraticCurve (const QuadraticInitWithStartDerivative &init)

	QuadraticCurve (const QuadraticInitWithOrigin &init)

	QuadraticCurve (const QuadraticInitWithPoint &init)

void	Init (const QuadraticInitWithStartDerivative &init)

void	Init (const QuadraticInitWithOrigin &init)

void	Init (const QuadraticInitWithPoint &init)

QuadraticCurve &	operator+= (const QuadraticCurve &rhs)
	Add the coefficients of another quadratic curve to this one.

QuadraticCurve &	operator-= (const QuadraticCurve &rhs)
	Subtract the coefficients of another quadratic curve from this one.

void	ShiftLeft (const float x_shift)

void	ShiftRight (const float x_shift)
	Shift the curve along the x-axis: x_shift to the right.

void	ShiftUp (const float y_offset)
	Shift the curve along the y-axis by y_offset: y_offset up the y-axis.

void	ScaleUp (const float y_scale)
	Scale the curve along the y-axis by a factor of y_scale.

float	Evaluate (const float x) const

float	Derivative (const float x) const

float	SecondDerivative () const

float	SecondDerivative (const float) const

float	ThirdDerivative (const float x) const

float	EpsilonInRange (const float max_x) const

float	EpsilonOfCoefficients () const

float	Epsilon (const float x) const

float	MaxCoeff () const

float	Discriminant () const

float	ReliableDiscriminant (const float epsilon) const

float	CriticalPoint () const
	Return the x at which the derivative is zero. More...

void	Roots (RootsArray *roots) const

void	RootsWithoutNormalizing (RootsArray *roots) const

void	RootsInRange (const Range &x_limits, RootsArray *roots) const

void	XsForValue (float value, RootsArray *roots) const
	Calculate the x-coordinates at the value.

void	RangesMatchingSign (const Range &x_limits, const float sign, RangeArray *matching) const

void	RangesAboveZero (const Range &x_limits, RangeArray *matching) const
	Return the ranges on which the quadratic is above zero.

void	RangesBelowZero (const Range &x_limits, RangeArray *matching) const
	Return the ranges on which the quadratic is below zero.

float	Coeff (int i) const
	Returns the coefficient for x-to-the-ith -power.

int	NumCoeff () const
	Returns the number of coefficients in this curve.

QuadraticCurve	ScaleInX (const float x_scale) const

QuadraticCurve	ScaleInXByReciprocal (const float x_scale_reciprocal) const

QuadraticCurve	ScaleInY (const float y_scale) const
	Returns the curve y_scale * f(x).

QuadraticCurve	AbsCoeff () const

bool	operator== (const QuadraticCurve &rhs) const
	Equality. Checks for exact match. Useful for testing.

bool	operator!= (const QuadraticCurve &rhs) const

std::string	Text () const
	A string with the equation for this quadratic. Useful for debugging.

Static Public Attributes
static const int	kNumCoeff = 3

Member Function Documentation

QuadraticCurve motive::QuadraticCurve::AbsCoeff ( ) const

inline

Returns the same curve but with all coefficients in absolute value. Useful for numerical precision work.

float motive::QuadraticCurve::CriticalPoint ( ) const

inline

Return the x at which the derivative is zero.

0 = f'(x) = 2*c2*x + c1 ==> x = -c1 / 2c2

float motive::QuadraticCurve::Derivative ( const float x ) const

inline

Return the quadratic function's slope at x. f'(x) = 2*c2*x + c1

float motive::QuadraticCurve::Discriminant ( ) const

inline

Used for finding roots, and more. See http://en.wikipedia.org/wiki/Discriminant

float motive::QuadraticCurve::Epsilon ( const float x ) const

inline

Given values in the range of x, returns a value below which should be considered zero.

float motive::QuadraticCurve::EpsilonInRange ( const float max_x ) const

inline

Returns a value below which floating point precision is unreliable. If we're testing for zero, for instance, we should test against this value. Pass in the x-range as well, in case that coefficient dominates the others.

float motive::QuadraticCurve::EpsilonOfCoefficients ( ) const

inline

Returns a value below which floating point precision is unreliable. If we're testing for zero, for instance, we should test against this value. We don't consider the x-range here, so this value is useful only when dealing with the equation itself.

float motive::QuadraticCurve::Evaluate ( const float x ) const

inline

Return the quadratic function's value at x. f(x) = c2*x^2 + c1*x + c0

float motive::QuadraticCurve::MaxCoeff ( ) const

inline

Returns the largest absolute value of the coefficients. Gives a sense of the scale of the quaternion. If all the coefficients are tiny or huge, we'll need to renormalize around 1, since we take reciprocals of the coefficients, and floating point precision is poor for floats.

void motive::QuadraticCurve::RangesMatchingSign	(	const Range &	x_limits,
		const float	sign,
		RangeArray *	matching
	)		const

inline

Get ranges above or below zero. Since a quadratic can cross zero at most twice, there can be at most two ranges. Ranges are clamped to x_limits. sign is used to determine above or below zero only–actual value is ignored.

float motive::QuadraticCurve::ReliableDiscriminant ( const float epsilon ) const

When Discriminant() is close to zero, set to zero. Often floating point precision problems can make the discriminant very slightly non-zero, even though mathematically it should be zero.

void motive::QuadraticCurve::Roots ( RootsArray * roots ) const

inline

Calculate the x-coordinates where this quadratic is zero, and put them into roots, in ascending order. Returns the 0, 1, or 2, the number of unique values put into roots.

void motive::QuadraticCurve::RootsInRange	(	const Range &	x_limits,
		RootsArray *	roots
	)		const

inline

Calculate the x-coordinates within the range [start_x, end_x] where the quadratic is zero. Put them into roots. Return the number of unique values put into roots.

void motive::QuadraticCurve::RootsWithoutNormalizing ( RootsArray * roots ) const

inline

Same as Roots() but do not normalize the quadratic for precision. This function is not recommended, in general. Normalizing does not take much time, and floating point precision is something that we only like to think we can ignore. It may be safe to call this function if you are certain that your x^2 coefficient is near 1.

QuadraticCurve motive::QuadraticCurve::ScaleInX ( const float x_scale ) const

inline

Returns the curve f(x / x_scale), where f is the current quadratic. This stretches the curve along the x-axis by x_scale.

QuadraticCurve motive::QuadraticCurve::ScaleInXByReciprocal ( const float x_scale_reciprocal ) const

inline

Returns the curve f(x * x_scale_reciprocal), where f is the current quadratic. This stretches the curve along the x-axis by 1/x_scale_reciprocal

float motive::QuadraticCurve::SecondDerivative ( ) const

inline

Return the quadratic function's constant second derivative. f''(x) = 2*c2

float motive::QuadraticCurve::SecondDerivative ( const float ) const

inline

Return the quadratic function's constant second derivative. Even though x is unused, we pass it in for consistency with other curve classes.

void motive::QuadraticCurve::ShiftLeft ( const float x_shift )

Shift the curve along the x-axis: x_shift to the left. That is x_shift becomes the curve's x=0.

float motive::QuadraticCurve::ThirdDerivative ( const float x ) const

inline

Return the quadratic function's constant third derivative: 0. Even though x is unused, we pass it in for consistency with other curve classes. f'''(x) = 0

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

curve.h

Detailed Description

Public Types

Public Member Functions

Static Public Attributes

Member Function Documentation