Math Class Reference

Core class implementing some useful functions and constants. More...

#include <mathematics.h>

Static Public Member Functions
static double	Clamp (const double &, const double &=0.0, const double &=1.0)
	Clamp a double value between two bounds. More...
static int	Clamp (int, int=0, int=255)
	Clamp an integer value between two bounds. More...
static double	Sqr (const double &)
	Squares a double value. More...
static constexpr double	Cube (const double &)
	Cubes a double value. More...
static double	Sqr4 (const double &)
	Fourth power of a double value. More...
static double	SymmetricSqr (const double &)
	Symmetric of square function over unit interval. More...
static double	Pow (const double &, const double &)
	Power. More...
static double	Abs (const double &)
	Absolute value. More...
static double	Sqrt32 (const double &)
	Compute x3/2 as the square root of x3. More...
static double	Sqrt4 (const double &)
	Compute the fourth root of x3. More...
static void	Powers (const double &, int, double *)
	Compute the powers series of a real value 1, x, ... , xn-1. More...
static constexpr double	Mod (const double &, const double &)
	Modulus for reals with negative values handled properly. More...
static double	Floor (const double &)
	Floor function. More...
static double	Fract (const double &)
	Fractional part of a real. More...
static double	FractFloor (const double &, double &)
	Fractional part of a real. More...
static double	Ceil (const double &)
	Ceil function. More...
static double	Min (const double &, const double &)
	Minimum of two reals. More...
static double	Max (const double &, const double &)
	Maximum of two reals. More...
static double	Min (const double &, const double &, const double &)
	Minimum of three reals. More...
static double	Max (const double &, const double &, const double &)
	Maximum of three reals. More...
static double	Min (const double &, const double &, const double &, const double &)
	Minimum of four reals. More...
static double	Max (const double &, const double &, const double &, const double &)
	Maximum of four reals. More...
static constexpr double	DegreeToRadian (const double &)
	Convert degrees to randians. More...
static constexpr double	RadianToDegree (const double &)
	Convert radian to degrees. More...
static constexpr double	Angle (int, int)
	Compute 2 k π / n. More...
static constexpr double	Angle (const double &)
	Angle strictly in [0,2 π[. More...
static double	ArcTan (const double &, const double &)
	Inline version of the atan2() function.
static double	Step (double, double)
	Step function. More...
static double	Cycloidal (const double &)
	Sine wave over unit interval. More...
static double	Triangle (const double &)
	Triangle wave over unit interval. More...
static double	Lerp (const double &, const double &, const double &)
	Linear interpolation. More...
static double	Bilinear (const double &, const double &, const double &, const double &, const double &, const double &)
	Bi-linear interpolation between four values. More...
static double	Trilinear (const double &, const double &, const double &, const double &, const double &, const double &, const double &, const double &, const double &, const double &, const double &)
	Trilinear interpolation between eight values. More...
static double	Cubic (const double &, const double &, const double &, const double &, const double &)
	Cubic Hermite interpolation. More...
static double	CubicPoints (const double &, const double &, const double &, const double &, const double &)
	Cubic point interpolation. More...
static double	BiCubic (const double &, const double &, const double &, const double &, const double &, const double &, const double &, const double &, const double &, const double &, const double &, const double &, const double &, const double &, const double &=0.0, const double &=0.0, const double &=0.0, const double &=0.0)
	Bi-cubic interpolation between four values, given partial derivatives. More...
static double	BiCubic (const double &, const double &, const double &, const double &, const double &, const double &)
	Bi-cubic interpolation between four values, partial derivatives are implicitly defined as null. More...
static double	Sigmoid (const double &, const double &=1.0)
	Sigmoid-like function. More...
static double	SigmoidQuadric (const double &)
	Compactly supported sigmoid-like function implemented using a C1 piecewise symmetric quadric. More...
static double	Impulse (const double &, const double &=1.0)
	Impulse function. More...
static double	Warp (const double &)
	Unit interval warping function. More...
static double	Warp (const double &, const double &)
	Unit interval gain function. More...
static bool	InRange (const double &, const double &, const double &, const double &)
	Check if a values lies within a prescribed range. More...
static void	SetMinMax (const double &, double &, double &)
	Update the minimum and maximum values given a double value. More...
static int	Integer (const double &)
	Compute the integer part of a real. More...
static void	Sort (double &, double &)
	Sort two reals. More...
static void	Sort (int &, int &)
	Sort two integers. More...
static void	Sort (double &, double &, double &)
	Sort three reals. More...
static void	Swap (double &, double &)
	Swap two reals. More...
static void	Swap (int &, int &)
	Swap two integers. More...
static bool	IsNumber (double)
	Check if a real number is not NaN. More...
static bool	IsFinite (double)
	Check if a real is not infinite. More...
static int	IntegerSign (const double &, const double &=0.0)
	Compute the integer sign of a real. More...
static bool	SameSign (const double &, const double &)
	Check if two reals have the same signs. More...
static bool	SameSign (const double &, const double &, const double &)
	Check if three reals have the same signs. More...
static double	CopySign (const double &, const double &)
	Composes a real with the magnitude of x and the sign of y. More...
static long	Binomial (int, int)
	Calculate the Binomial coefficent. More...
static constexpr double	Unit (int, int)
	Unit real value in [0,1] from two integers. More...
static double	Gaussian (const double &, const double &)
	Gaussian. More...
static double	CompactGaussian (const double &, const double &)
	Compact support Gaussian approximation. More...
static double	Fract (const double &, double &)
	Compute integer and fractionnal parts. More...
static double	Fract (const double &, const double &, double &)
	Compute integer and fractionnal parts. More...
static double	Fract (const double &, const double &, const double &, double &)
	Compute integer and fractionnal parts given an interval. More...
static double	Geometric (double, int)
	Compute the sum of the terms of a geometric series. More...
static void	Swap (double *&, double *&)
	Swap two reals. More...
static void	Swap (int *&, int *&)
	Swap two integers. More...
static void	BernsteinSeries (const double &, int, double *)
	Compute the Bernstein terms.
static double	Bernstein (const double &, int, int)
	Compute the i-th Bernstein term.

Static Public Attributes
static const double	Pi = 3.14159265358979323846
	π.
static const double	TwoPi = 6.28318530717958647693
	2π.
static const double	HalfPi = Math::Pi / 2.0
	Half of π.
static const double	e = 2.7182818284590452354
	Exponential.
static const double	TwoPiOverThree = 2.0943951023931954923084
	2/3 π.
static const double	FourPiOverThree = 4.1887902047863909846168
	4/3 π.
static const double	Sqrt5 = sqrt(5.0)
	Constant √5.
static const double	Sqrt3 = sqrt(3.0)
	Constant √3.
static const double	Sqrt2 = sqrt(2.0)
	Constant √2.
static const double	Golden = (sqrt(5.0) + 1.0) / 2.0
	Constant (√5+1)/2.
static const double	Large = 1.0e20
	Large constant 1020.
static const double	Infinity = DBL_MAX
	Infinity.
static int	binomials [16][16]
	Array of binomial coefficients.

Detailed Description

Core class implementing some useful functions and constants.

How can I use the constant Pi?
Simply use the static constant Math::Pi as follows:

double v = 4.0*Math::Pi*r*r*r/3.0; // Volume of a sphere.

Note that in this case, you could also have used:

double v = Sphere::Volume(r);

How many min/max functions have been implemented?
Up to four arguments are supported; for a larger number of arguments, a specific routine operating on an array should be written.

Is there a function to compute the square of a real sumber?
Use the following:

double s = Math::Sqr((1.0-sqrt(5.0))/2.0); // Square of golden ratio

For a Vector, use SquaredNorm(const Vector&);

Are there predefined square roots constants?
The sqrt function used not te be constexpr, so square roots of reals are not computed at compilation time. Some constants are provided, such as the following one:

double s = Math::Sqrt3; // Square root of 3

What is the relative performance of mathematical functions such as square root or cosine?
The relative cost of commonly used mathematical functions is summarized in the following table.

Cost

Function	Cost
+ - * fabs	1
/ sqrt modf	~4.8
/ sin cos	~8.9
/ exp log	~10.5

Math provides some functions such as Math::Sqrt32 or Math::Sqrt4 to avoid computationally intensive standard functions like pow:

double y = Math::Sqrt32(x); // Faster than double y = pow(x,1.5);

How are implemented the step and smooth-step functions that are often used in procedural modeling? Different smoothing kernels, such as Cubic::Smooth(), are implented in odd-degree polynomials Cubic, Quintic and Septic. The corresponding step functions, such as Cubic::SmoothStep(), are also implemented.

See also

Linear::Step, Cubic::Smooth, Quintic::Smooth, Cubic::SmoothStep, Quintic::SmoothStep, Septic::SmoothStep.

Member Function Documentation

Abs()

double Math::Abs ( const double &  x )

inlinestatic

Absolute value.

Parameters

x

Real.

Angle() [1/2]

constexpr double Math::Angle ( const double &  a )

inlinestaticconstexpr

Angle strictly in [0,2 π[.

Parameters

a

Angle.

Angle() [2/2]

constexpr double Math::Angle ( int  k, int  n  )

inlinestaticconstexpr

Compute 2 k π / n.

Parameters

k,n

Integers.

BiCubic() [1/2]

double Math::BiCubic ( const double &  u, const double &  v, const double &  a00, const double &  a10, const double &  a11, const double &  a01  )

static

Bi-cubic interpolation between four values, partial derivatives are implicitly defined as null.

This is an optimized implementation.

Parameters

u,v	Interpolation coefficients.
a00,a10,a11,a01	Interpolated values.

BiCubic() [2/2]

double Math::BiCubic ( const double &  u, const double &  v, const double &  a00, const double &  a10, const double &  a11, const double &  a01, const double &  u00, const double &  u10, const double &  u11, const double &  u01, const double &  v00, const double &  v10, const double &  v11, const double &  v01, const double &  x00 = 0.0, const double &  x10 = 0.0, const double &  x11 = 0.0, const double &  x01 = 0.0  )

static

Bi-cubic interpolation between four values, given partial derivatives.

The values are given in trigonometric order.

Parameters

u,v	Interpolation coefficients.
a00,a10,a11,a01	Interpolated values.
u00,u10,u11,u01,v00,v10,v11,v01	Partial derivatives with respect to u and v.
x00,x10,x11,x01	Cross derivatives.

Bilinear()

double Math::Bilinear ( const double &  a00, const double &  a10, const double &  a11, const double &  a01, const double &  u, const double &  v  )

inlinestatic

Bi-linear interpolation between four values.

The values are given in trigonometric order.

Parameters

a00,a10,a11,a01	Interpolated values.
u,v	Interpolation coefficients.

Binomial()

long Math::Binomial ( int  n, int  r  )

static

Calculate the Binomial coefficent.

Parameters

n,r	Binomial coefficient parameters.

Ceil()

double Math::Ceil ( const double &  x )

inlinestatic

Ceil function.

See also

Floor

Parameters

x

Real.

Clamp() [1/2]

double Math::Clamp ( const double &  x, const double &  a = 0.0, const double &  b = 1.0  )

inlinestatic

Clamp a double value between two bounds.

Parameters

x	Input value.
a,b	Lower and upper bounds.

Clamp() [2/2]

int Math::Clamp ( int  x, int  a = 0, int  b = 255  )

inlinestatic

Clamp an integer value between two bounds.

Parameters

x	Input value.
a,b	Lower and upper bounds.

CompactGaussian()

double Math::CompactGaussian ( const double &  x, const double &  s  )

inlinestatic

Compact support Gaussian approximation.

The compact support has a radius r = 3 sigma.

Parameters

x	Real.
s	Sigma.

CopySign()

double Math::CopySign ( const double &  x, const double &  y  )

inlinestatic

Composes a real with the magnitude of x and the sign of y.

Parameters

x	Real.
y	Real whose sign will be copied.

Cube()

constexpr double Math::Cube ( const double &  x )

inlinestaticconstexpr

Cubes a double value.

Parameters

x	Real value.

Cubic()

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

static

Cubic Hermite interpolation.

Parameters

u	Interpolation coefficient.
a,b	Values for t=0 and t=1.
ta,tb	Derivatives for t=0 and t=1.

See also

Cubic::Hermite()

CubicPoints()

double Math::CubicPoints ( const double &  u, const double &  a, const double &  b, const double &  c, const double &  d  )

static

Cubic point interpolation.

Parameters

u	Interpolation coefficient.
a,b,c,d	Values for t=-1, t=0, t=1 and t=2. The derivatives at t=0 and t=1 will be derived from those

See also

Math::Cubic()

Cycloidal()

double Math::Cycloidal ( const double &  x )

static

Sine wave over unit interval.

Parameters

x	Real value.

DegreeToRadian()

constexpr double Math::DegreeToRadian ( const double &  a )

inlinestaticconstexpr

Convert degrees to randians.

Parameters

a	Angle in degrees.

Floor()

double Math::Floor ( const double &  x )

inlinestatic

Floor function.

While it is easier to use the C++ function, this has been implemented to be consistent with Fract().

Parameters

x

Real.

Fract() [1/4]

double Math::Fract ( const double &  x )

inlinestatic

Fractional part of a real.

Implemented as:

double y=x-Math::Floor(x);

Parameters

x

Real.

Fract() [2/4]

double Math::Fract ( const double &  x, const double &  a, const double &  b, double &  i  )

inlinestatic

Compute integer and fractionnal parts given an interval.

Parameters

x	Real.
a,b	Interval.
i	Returned integer part.

Returns

Fractional part.

Fract() [3/4]

double Math::Fract ( const double &  x, const double &  a, double &  i  )

inlinestatic

Compute integer and fractionnal parts.

Parameters

x	Real.
a	Modulus.
i	Returned integer part.

Returns

Fractional part.

Fract() [4/4]

double Math::Fract ( const double &  x, double &  i  )

inlinestatic

Compute integer and fractionnal parts.

Parameters

x	Real.
i	Returned integer part.

Returns

Fractional part.

FractFloor()

double Math::FractFloor ( const double &  x, double &  y  )

inlinestatic

Fractional part of a real.

Implemented as:

double y = x - Math::Floor(x);

Parameters

x	Real.
y	Return floor.

Gaussian()

double Math::Gaussian ( const double &  x, const double &  s  )

inlinestatic

Gaussian.

Parameters

x	Real.
s	Sigma.

Geometric()

double Math::Geometric ( double  x, int  n  )

static

Compute the sum of the terms of a geometric series.

Compute 1+x+x²+...+x^n-1=(1-xⁿ)/(1-x). If x=1.0, return n.

Impulse()

double Math::Impulse ( const double &  x, const double &  k = 1.0  )

inlinestatic

Impulse function.

An impulse function that doesn't use exponentials, k controls the falloff of the function.

Derivative is 2*sqrt(k)*(1-k*x*x)/(k*x*x+1), maximum is reached x = sqrt(1/k), value at maximum is 1.

Parameters

k	Control parameter.
x	Real.

InRange()

bool Math::InRange ( const double &  x, const double &  a, const double &  b, const double &  epsilon  )

inlinestatic

Check if a values lies within a prescribed range.

Parameters

x	Value.
a,b	Interval.
epsilon	Epsilon tolerance.

Integer()

int Math::Integer ( const double &  x )

inlinestatic

Compute the integer part of a real.

This function handles negative values differently by subtracting 1 from the result.

Parameters

x

Real.

IntegerSign()

int Math::IntegerSign ( const double &  x, const double &  t = 0.0  )

inlinestatic

Compute the integer sign of a real.

Parameters

x	Real.
t	Threshold.

IsFinite()

bool Math::IsFinite ( double  x )

static

Check if a real is not infinite.

Parameters

x	Real value.

IsNumber()

bool Math::IsNumber ( double  x )

static

Check if a real number is not NaN.

The code is simply:

return (x == x);

This looks like it should always be true, but it's false if x is a NaN.

Parameters

x	Real value.

Lerp()

double Math::Lerp ( const double &  a, const double &  b, const double &  t  )

inlinestatic

Linear interpolation.

Returns (1-t)a+tb.

Parameters

a,b	Interpolated values.
t	Interpolant.

Max() [1/3]

double Math::Max ( const double &  a, const double &  b  )

inlinestatic

Maximum of two reals.

Parameters

a,b	Real values.

Max() [2/3]

double Math::Max ( const double &  a, const double &  b, const double &  c  )

inlinestatic

Maximum of three reals.

Parameters

a,b,c

Real values.

Max() [3/3]

double Math::Max ( const double &  a, const double &  b, const double &  c, const double &  d  )

inlinestatic

Maximum of four reals.

Parameters

a,b,c,d

Real values.

Min() [1/3]

double Math::Min ( const double &  a, const double &  b  )

inlinestatic

Minimum of two reals.

Parameters

a,b	Real values.

Min() [2/3]

double Math::Min ( const double &  a, const double &  b, const double &  c  )

inlinestatic

Minimum of three reals.

Parameters

a,b,c

Real values.

Min() [3/3]

double Math::Min ( const double &  a, const double &  b, const double &  c, const double &  d  )

inlinestatic

Minimum of four reals.

Parameters

a,b,c,d

Real values.

Mod()

constexpr double Math::Mod ( const double &  x, const double &  a  )

inlinestaticconstexpr

Modulus for reals with negative values handled properly.

Parameters

x	Real.
a	Modulo.

Pow()

double Math::Pow ( const double &  x, const double &  e  )

inlinestatic

Power.

Returns base x raised to the power exponent e if x>0, and -x raised to the power exponent e otherwise.

Parameters

x	Base.
e	Exponent.

Powers()

void Math::Powers ( const double &  x, int  n, double *  a  )

inlinestatic

Compute the powers series of a real value 1, x, ... , x^n-1.

Parameters

x	Real.
n	Maximum power.
a	Array of real, should be at least of size n.

RadianToDegree()

constexpr double Math::RadianToDegree ( const double &  a )

inlinestaticconstexpr

Convert radian to degrees.

Parameters

a	Angle in radian.

SameSign() [1/2]

bool Math::SameSign ( const double &  a, const double &  b  )

inlinestatic

Check if two reals have the same signs.

Parameters

a,b

Reals.

SameSign() [2/2]

bool Math::SameSign ( const double &  a, const double &  b, const double &  c  )

inlinestatic

Check if three reals have the same signs.

Parameters

a,b,c

Reals.

SetMinMax()

void Math::SetMinMax ( const double &  x, double &  a, double &  b  )

inlinestatic

Update the minimum and maximum values given a double value.

Parameters

x	Input value.
a,b	Lower and upper bounds that will be updated according to x.

Sigmoid()

double Math::Sigmoid ( const double &  x, const double &  s = 1.0  )

inlinestatic

Sigmoid-like function.

This function is more efficient than the real sigmoid function which requires the computation of the exponential.

Parameters

s	Sigma.
x	Real.

SigmoidQuadric()

double Math::SigmoidQuadric ( const double &  x )

inlinestatic

Compactly supported sigmoid-like function implemented using a C¹ piecewise symmetric quadric.

The compact support is [-2,2] and the range [-1,1].

The quadric was obtaied by solving the Hermite Cubic contraints:

Cubic c=2.0*Cubic::Compose(Cubic::Hermite(0, 0.5, 1.0, 0.0), Linear(0.5, 0.0)) <<std::endl;

See also

Math::Sigmoid

Parameters

x

Real.

Sort() [1/3]

void Math::Sort ( double &  a, double &  b  )

inlinestatic

Sort two reals.

See also

Math::Swap(double&,double&)

Parameters

a,b	Real arguments, which will be swapped so that a<=b.

Sort() [2/3]

void Math::Sort ( double &  a, double &  b, double &  c  )

inlinestatic

Sort three reals.

See also

Math::Sort(double&,double&)

Parameters

a,b,c

Real arguments, which will be swapped so that a<=b<=c.

Sort() [3/3]

void Math::Sort ( int &  a, int &  b  )

inlinestatic

Sort two integers.

See also

Math::Swap(int&,int&)

Parameters

a,b	Real arguments, which will be swapped so that a<=b.

Sqr()

double Math::Sqr ( const double &  x )

inlinestatic

Squares a double value.

See also

Math::SymmetricSqr

Parameters

x	Real value.

Sqr4()

double Math::Sqr4 ( const double &  x )

inlinestatic

Fourth power of a double value.

Parameters

x	Real value.

Sqrt32()

double Math::Sqrt32 ( const double &  x )

inlinestatic

Compute x^3/2 as the square root of x³.

This function is more efficient than pow() which is extremely slow.

double y=Math::Sqrt32(7.0); // Equivalent but faster than y=pow(7.0,1.5);

Parameters

x	Real value.

Sqrt4()

double Math::Sqrt4 ( const double &  x )

inlinestatic

Compute the fourth root of x³.

This function is more efficient than pow() which is extremely slow.

double y=Math::Sqrt4(7.0); // Equivalent to y=sqrt(sqrt(7.0));

Parameters

x	Real value.

Step()

double Math::Step ( double  e, double  x  )

inlinestatic

Step function.

Parameters

e	Step position.
x	Variable.

Swap() [1/4]

void Math::Swap ( double &  a, double &  b  )

inlinestatic

Swap two reals.

See also

Sort(double&,double&)

Parameters

a,b	Arguments.

Swap() [2/4]

void Math::Swap ( double *&  a, double *&  b  )

inlinestatic

Swap two reals.

Parameters

a,b	Arguments.

Swap() [3/4]

void Math::Swap ( int &  a, int &  b  )

inlinestatic

Swap two integers.

See also

Sort(int&,int&)

Parameters

a,b	Arguments.

Swap() [4/4]

void Math::Swap ( int *&  a, int *&  b  )

inlinestatic

Swap two integers.

Parameters

a,b	Arguments.

SymmetricSqr()

double Math::SymmetricSqr ( const double &  x )

inlinestatic

Symmetric of square function over unit interval.

Simply compute 1-(1-x)².

See also

Math::Sqr

Parameters

x	Real value.

Triangle()

double Math::Triangle ( const double &  x )

static

Triangle wave over unit interval.

Parameters

x	Real value.

Trilinear()

double Math::Trilinear ( const double &  a, const double &  b, const double &  c, const double &  d, const double &  e, const double &  f, const double &  g, const double &  h, const double &  u, const double &  v, const double &  w  )

inlinestatic

Trilinear interpolation between eight values.

See also

Math::Bilinear()

Parameters

a,b,c,d,e,f,g,h	Interpolated values.
u,v,w	Interpolation coefficients.

Unit()

constexpr double Math::Unit ( int  i, int  n  )

inlinestaticconstexpr

Unit real value in [0,1] from two integers.

Usefull for loops, same as:

for (int i=0;i<n;i++)
{
double t=double(i)/double(n-1); // t=Math::Unit(i,n);
}

Parameters

i,n

Integers.

Warp() [1/2]

double Math::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].

Warp() [2/2]

double Math::Warp ( const double &  x, const double &  k  )

inlinestatic

Unit interval gain function.

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

This function uses calls to pow and is computationally intensive.

See also

Quadric::Warp(const double&)

Parameters

x	Real in [0,1].
k	Exponent.