Function

Tools for constructing and sampling continuous-valued objects. More...

Classes
struct	Dual< T, Dp >
	Class implementing the dual numbers, whose arithmetic operations perform a simultaneous calculation of the first derivative. More...
class	PerlinNoise< T, N >
	Real-valued smooth noise over N dimensions. More...
class	Raster< I, O, M, N >
	An M-dimensional grid of interpolated data which can be continuously sampled. More...
class	SphericalHarmonics< T, Bands >
	Class and methods for representing band-limited functions on the surface of a 3D sphere. More...
class	ZonalHarmonics< T, Bands >
	Represents a function on the N-sphere with axial symmetry. More...

Enumerations
enum class	DiscontinuityPolicy {   NaN , Average , Left , Right ,   Inf }
	Policy for handling discontinuous derivatives. More...
enum	EdgeBehavior { EDGE_CLAMP , EDGE_PERIODIC , EDGE_MIRROR , EDGE_CONSTANT }
	Raster edge-sampling behavior. More...
enum	Interpolation { INTERP_NEAREST , INTERP_LINEAR , INTERP_CUBIC }
	Behavior for sampling between data points. More...

Functions
template<typename T>
T	chebyshev (index_t kind, index_t n, T x)
template<typename T>
T	legendre (index_t l, index_t m, T x)
template<typename T>
T	legendre (index_t n, T x)
template<typename T, index_t N>
void	legendre (SphericalHarmonics< T, N > *sh, index_t m, T x)
template<typename T>
T	legendre_integral (index_t n, T x)
template<typename T>
T	spherical_harmonic_coeff (index_t l, index_t m, T cos_alt, T azi)
template<typename T>
T	positive_mod (T a, T b)
template<typename T>
T	diff_of_products (T a, T b, T c, T d)
template<typename T>
T	sum_of_products (T a, T b, T c, T d)
template<typename T>
T	multiply_add (T a, T b, T c)
	Fused multiply-add.
template<typename T>
T	clamp (const T &v, const T &lo, const T &hi)
template<typename S, typename T>
std::common_type< S, T >::type	mix (S t, T a, T b)
template<typename S, typename T>
T	interp_linear (const S s[], const T p[], int dim)
template<typename S, typename T>
T	interp_cubic (S s, const T p[4])
template<typename S, typename T>
T	interp_cubic (const S s[], const T p[], int dim)
template<typename T>
bool	quadratic_solve (T results[2], T a, T b, T c)
template<typename T>
T	angle_to (T radians_0, T radians_1)
template<typename T>
constexpr auto	ceil_div (T a, T b)
	Compute ceil(a/b).
template<typename T>
constexpr auto	snap (T a, T b)
	Snap a to the nearest multiple of b larger than or equal to a.
template<typename T>
constexpr auto	floor_div (T a, T b)
	Compute floor(a/b).
template<typename T> requires std::integral<T>
constexpr bool	is_power_of_two (T x)
template<typename T> requires (not std::integral<T>)
constexpr T	radians (T degrees)
	Convert degrees to radians.
template<typename T> requires (not std::integral<T>)
constexpr T	degrees (T radians)
	Convert radians to degrees.

Detailed Description

Tools for constructing and sampling continuous-valued objects.

Enumeration Type Documentation

DiscontinuityPolicy

enum class DiscontinuityPolicy

strong

Policy for handling discontinuous derivatives.

Enumerator
NaN	Return NaN at discontinuities.
Average	At discontinuities, return the average of the two boundary values.
Left	At discontinuities, return the value when approaching from the left.
Right	At discontinuities, return the value when approaching from the right.
Inf	At discontinuities, return +/- infinity, according to whether the discontinuity is an increasing or decreasing jump.

EdgeBehavior

enum EdgeBehavior

Raster edge-sampling behavior.

Enumerator
EDGE_CLAMP	Clip sample coordinates to the sampleable area, thus repeating edge values beyond the boundary.
EDGE_PERIODIC	Wrap around sample coordinates to the opposite edge, thus tiling the sampled data beyond the boundary.
EDGE_MIRROR	Mirror the sample coordinates across edges.
EDGE_CONSTANT	Regions outside the sampleable area have a uniform, defined value (zero by default).

Interpolation

enum Interpolation

Behavior for sampling between data points.

Enumerator
INTERP_NEAREST	Return the data nearest to the sample point.
INTERP_LINEAR	Linearly interpolate the nearest 2n data points.
INTERP_CUBIC	Cubically interpolate the nearest 4n data points.

Function Documentation

angle_to()

template<typename T>

T angle_to ( T radians_0, T radians_1 )

inline

Compute the shortest angle from radians_0 to radians_1.

Returns a value in the range [-π, π].

ceil_div()

template<typename T>

auto ceil_div ( T a, T b )

constexpr

Compute ceil(a/b).

Parameters

a	Numerator.
b	Denominator.

chebyshev()

template<typename T>

T chebyshev ( index_t kind, index_t n, T x )

inline

Evaluate a Chebyshev polynomial of order n at x.

#include <geomc/function/Basis.h>

Parameters

kind	1 or 2 for the Chebyshev polynomial of the first or second kind, respectively.
n	Order of the polynomial.
x	Value of x at which to evaluate.

clamp()

template<typename T>

T clamp ( const T & v, const T & lo, const T & hi )

inline

Clamp v between lo and hi.

Parameters

v	Value to clamp.
lo	Minimum value of v.
hi	Maximum value of v.

diff_of_products()

template<typename T>

T diff_of_products ( T a, T b, T c, T d )

inline

A high-precision method for computing (a * b) - (c * d). In cases where the two products are large and close in value, the result can be wildly wrong due to catastrophic cancellation when using the "naive" method. This function will return a result within 1.5 ULP of the exact answer.

A high-precision method is available for float, double, long double, and Dual numbers composed of those types. Other types will fall back to the "naive" method.

floor_div()

template<typename T>

auto floor_div ( T a, T b )

constexpr

Compute floor(a/b).

Rounds towards negative infinity (rather than towards zero).

Parameters

a	Numerator.
b	Denominator.

interp_cubic() [1/2]

template<typename S, typename T>

T interp_cubic ( const S s[], const T p[], int dim )

inline

Cubic interpolation across dim input dimensions.

Parameters

s	An array of dim values; a point on (0, 1)dim representing the position in the center grid cell at which to interpolate.
p	An array of the 4dim values adjacent to s.
dim	Number of input dimensions.

Returns

Interpolated value.

interp_cubic() [2/2]

template<typename S, typename T>

T interp_cubic ( S s, const T p[4] )

inline

Cubic interpolation in 1 dimension.

Parameters

s	Interpolation parameter between 0 and 1, representing the point between p[1] and p[2] at which to interpolate.
p	An array of 4 values surrounding s.

Returns

Interpolated value.

interp_linear()

template<typename S, typename T>

T interp_linear ( const S s[], const T p[], int dim )

inline

Linearly interpolate across dim input dimensions.

Parameters

s	An array of dim values; a point on (0, 1)dim at which to interpolate.
p	An array of the 2dim values adjacent to s.
dim	Number of input dimensions.

Returns

Interpolated value.

legendre() [1/2]

template<typename T>

T legendre ( index_t l, index_t m, T x )

inline

Evaluate an associated Legendre polynomial.

#include <geomc/function/Basis.h>

Parameters

l	Band index.
m	Sub-band.
x	Value of x at which to evaluate the polynomial.

legendre() [2/2]

template<typename T>

T legendre ( index_t n, T x )

inline

Evaluate a Legendre polynomal of order n at x. Equivalent to the associated legendre polynomial with m = 0.

#include <geomc/function/Basis.h>

Parameters

n	Order of polynomial.
x	Value of x at which to evaluate the polynomial.

legendre_integral()

template<typename T>

T legendre_integral ( index_t n, T x )

inline

Evaluate the integral of the Legendre polynomal of order n between -1 and x.

#include <geomc/function/Basis.h>

Parameters

n	Order of polynomial.
x	Value of x at which to evaluate the integral.

mix()

template<typename S, typename T>

std::common_type< S, T >::type mix ( S t, T a, T b )

inline

Linearly interpolate between a and b.

Parameters

t	Interpolation parameter, between 0 and 1.
a	Value at t = 0.
b	Value at t = 1.

Returns

Interpolated value.

multiply_add()

template<typename T>

T multiply_add ( T a, T b, T c )

inline

Fused multiply-add.

Compute a * b + c using a fused multiply add operation (if available), which may be both more performant and more precise than multiple instructions.

Overloaded such that the calculation falls back to an ordinary composite multiply-add if a fused instruction is not available for T. (Compare to std::fma() which will promote to floating point, possibly losing precision).

quadratic_solve()

template<typename T>

bool quadratic_solve ( T results[2], T a, T b, T c )

inline

Solve the quadratic equation ax²+ bx + c = 0. Formulated to preserve precision and avoid any divisions by zero.

In the case where there is exactly one real solution, the two roots will be set equal to each other.

Parameters

[out]	results	Roots of the specified quadratic equation.
	a	Coefficient of x2.
	b	Coefficient of x.
	c	Constant.

Returns

true if real roots exist and the coefficients are not all zero; false otherwise (in which case the contents of results will be unaltered).

spherical_harmonic_coeff()

template<typename T>

T spherical_harmonic_coeff ( index_t l, index_t m, T cos_alt, T azi )

inline

Evaluate a spherical harmonic coefficient.

#include <geomc/function/SphericalHarmonics.h>

Parameters

l	Band index.
m	Sub-band.
cos_alt	Cosine of angle from polar axis.
azi	Azimuthal (longitude) angle.

sum_of_products()

template<typename T>

T sum_of_products ( T a, T b, T c, T d )

inline

A high-precision method for computing (a * b) + (c * d). In cases where the two products are large and similar in magnitude but opposite in sign, the result can be wildly wrong due to catastrophic cancellation when using the "naive" method. This function will return a result within 1.5 ULP of the exact answer.

A high-precision method is available for float, double, long double, and Dual numbers composed of those types. Other types will fall back to the "naive" method.