EdenMath

Includes:

<math.h>

Introduction

Math utilities.

Discussion

EdenMath forms one part of the Eden library.

Functions

EdenMathCalcDistanceToPlane: Caclulate signed distance from point to a plane.
EdenMathIdentityMatrix: Creates a 4x4 identity matrix.
EdenMathIdentityMatrix3by3: Creates a 3x3 identity matrix.
EdenMathIntersectionLinePlane: Calculate the point of intersection (if any) between a line and a plane.
EdenMathInvertMatrix: Compute inverse of 4x4 transformation matrix
EdenMathInvertMatrix3by3: Invert a 3x3 matrix A, placing result in Ainv.
EdenMathInvertMatrixd: Compute inverse of 4x4 transformation matrix
EdenMathMultMatrix: Multiplies 4x4 matrix B into 4x4 matrix A, placing result in C.
EdenMathMultMatrix3by3: Multiplies 3x3 matrix B into 3x3 matrix A, placing result in C.
EdenMathMultMatrixByVector: Multiples 4x4 matrix A into column vector p, placing result in q.
EdenMathMultMatrixByVectord: Multiples 4x4 matrix A into column vector p, placing result in q.
EdenMathMultMatrixd: Multiplies 4x4 matrix B into 4x4 matrix A, placing result in C.
EdenMathNormalise: Normalise a vector.
EdenMathNormalised: Normalise a vector.
EdenMathPointsToPlane: Calculate the general form of a plane, given three points lying on the plane.
EdenMathPointsToPlaneHessianNormal: Calculates the Hessian normal (unit normal vector) to a plane.
EdenMathRotateMatrix: Rotate a matrix about an arbitrary axis.
EdenMathRotatePointAboutAxis: Rotate a point about an arbitrary axis.
EdenMathRotatePointAboutAxisd: Rotate a point about an arbitrary axis.
EdenMathRotationMatrix: Creates a matrix which represents the general case of a rotation about an arbitrary axis.
EdenMathRotationMatrixFromTo: Creates a rotation matrix that rotates a vector called "from" into another vector called "to".
EdenMathTranslateMatrix: Translate a matrix by a vector.
EdenMathTranslationMatrix: Creates a matrix which represents translation by a vector.
frsqrt: Fast calculation of reciprocal square root.
frsqrt3: Fast calculation of three reciprocal square roots.
fsqrt: Fast calculation of square root.
fsqrt3: Fast calculation of three square roots.

EdenMathCalcDistanceToPlane

Caclulate signed distance from point to a plane.

float EdenMathCalcDistanceToPlane(
    const float x0[3],
    const float abc[3],
    const float d);

Parameters

x0

Position of the point.

abc

The vector of values A, B, and C from the general definition of a plane Ax + By + Cz + D = 0 (see EdenMathPointsToPlane()).

The value D from the general definition of a plane Ax + By + Cz + D = 0 (see EdenMathPointsToPlane()).

Return Value

The distance from the point to the plane, which is positive if the point lies on the same side of the plane as the normal to the plane, or negative if it lies on the opposite side, or 0 if the parameters are invalid.

Discussion

See http://mathworld.wolfram.com/Point-PlaneDistance.html

EdenMathIdentityMatrix

Creates a 4x4 identity matrix.

void EdenMathIdentityMatrix(
    float mtx16[16]);

Parameters

mtx16: A 4x4 matrix that will receive the output.

EdenMathIdentityMatrix3by3

Creates a 3x3 identity matrix.

void EdenMathIdentityMatrix3by3(
    float mtx9[9]);

Parameters

mtx9: A 3x3 matrix that will receive the output.

EdenMathIntersectionLinePlane

Calculate the point of intersection (if any) between a line and a plane.

EDEN_BOOL EdenMathIntersectionLinePlane(
    float intersection[3],
    float *u_out,
    const float p1[3],
    const float p2[3],
    const float abc[3],
    const float d);

Parameters

intersection

Pointer to an array of 3 floats which will be filled out with the coordinates of the point of intersection.

u_out

If non-null, the float pointed to will be filled out with the value 'u', representing the point of intersection, relative to the line segement between p1 and p2. I.e., when u is 0, the intersection is at p1, and when u is 1 the intersection is at p2. Values of u < 0 or > 1 indicate the intersection point is outside the line segment (but there IS still an intersection point). NULL may be passed if you are not interested in this value.

p1

A point lying on the line.

p2

Another point lying on the line.

abc

The vector of values A, B, and C from the general definition of a plane Ax + By + Cz + D = 0 (see EdenMathPointsToPlane()).

The value D from the general definition of a plane Ax + By + Cz + D = 0 (see EdenMathPointsToPlane()).

Return Value

FALSE if the normal to the plane is perpendicular to the line, or if a required value was not specified, TRUE otherwise.

Discussion

See http://mathworld.wolfram.com/Plane.html

EdenMathInvertMatrix

Compute inverse of 4x4 transformation matrix

EDEN_BOOL EdenMathInvertMatrix(
    float out[16],
    const float m[16]);

Parameters

m: Input, 4x4 column-major matrix.
out: Output, 4x4 column-major matrix.

Return Value

Return TRUE for success, FALSE for failure (singular matrix)

Discussion

Code contributed by Jacques Leroy jle@star.be

EdenMathInvertMatrix3by3

Invert a 3x3 matrix A, placing result in Ainv.

EDEN_BOOL EdenMathInvertMatrix3by3(
    float A[9],
    float Ainv[9]);

Discussion

A and Ainv are in column-major form, which is standard for OpenGL (wheras the usual mathematical matrix notation is row-major.)

EdenMathInvertMatrixd

Compute inverse of 4x4 transformation matrix

EDEN_BOOL EdenMathInvertMatrixd(
    double out[16],
    const double m[16]);

Parameters

m: Input, 4x4 column-major matrix.
out: Output, 4x4 column-major matrix.

Return Value

Return TRUE for success, FALSE for failure (singular matrix)

Discussion

Code contributed by Jacques Leroy jle@star.be

EdenMathMultMatrix

Multiplies 4x4 matrix B into 4x4 matrix A, placing result in C.

void EdenMathMultMatrix(
    float C[16],
    const float B[16],
    const float A[16]);

Discussion

A, B and C are in column-major form, which is standard for OpenGL (wheras the usual mathematical matrix notation is row-major.)

EdenMathMultMatrix3by3

Multiplies 3x3 matrix B into 3x3 matrix A, placing result in C.

void EdenMathMultMatrix3by3(
    float C[9],
    const float B[9],
    const float A[9]);

Discussion

A, B and C are in column-major form, which is standard for OpenGL (wheras the usual mathematical matrix notation is row-major.)

EdenMathMultMatrixByVector

Multiples 4x4 matrix A into column vector p, placing result in q.

void EdenMathMultMatrixByVector(
    float q[4],
    const float A[16],
    const float p[4]);

Discussion

A is in column-major form, which is standard for OpenGL (wheras the usual mathematical matrix notation is row-major.)

EdenMathMultMatrixByVectord

Multiples 4x4 matrix A into column vector p, placing result in q.

void EdenMathMultMatrixByVectord(
    double q[4],
    const double A[16],
    const double p[4]);

Discussion

A is in column-major form, which is standard for OpenGL (wheras the usual mathematical matrix notation is row-major.)

EdenMathMultMatrixd

Multiplies 4x4 matrix B into 4x4 matrix A, placing result in C.

void EdenMathMultMatrixd(
    double C[16],
    const double B[16],
    const double A[16]);

Discussion

A, B and C are in column-major form, which is standard for OpenGL (wheras the usual mathematical matrix notation is row-major.)

EdenMathNormalise

Normalise a vector.

float EdenMathNormalise(
    float v[3]);

Parameters

v: Vector to normalise.

EdenMathNormalised

Normalise a vector.

double EdenMathNormalised(
    double v[3]);

Parameters

v: Vector to normalise.

EdenMathPointsToPlane

Calculate the general form of a plane, given three points lying on the plane.

void EdenMathPointsToPlane(
    float abc[3],
    float *d,
    const float p1[3],
    const float p2[3],
    const float p3[3]);

Parameters

abc: Vector describing the plane.
d: Value describing the plane.
p1: Vector describing point on the plane.
p2: Vector describing point on the plane.
p3: Vector describing point on the plane.

Discussion

The general form of a plane is the equation Ax + By + Cz + D = 0. The plane is specified by three vectors which are points on the plane. The vector ABC is normal to the plane (not unit normal though).

See http://mathworld.wolfram.com/Plane.html

EdenMathPointsToPlaneHessianNormal

Calculates the Hessian normal (unit normal vector) to a plane.

void EdenMathPointsToPlaneHessianNormal(
    float n[3],
    float *p,
    const float p1[3],
    const float p2[3],
    const float p3[3]);

Parameters

n: Vector representing Hessian normal (unit normal vector) to plane.
p: Value of distance of plane from origin. The sign of *p determines the side of the plane on which the origin is located. If > 0, it is in the half-space determined by the direction of n, and if < 0, it is in the other half-space.
p1: Vector describing point on the plane.
p2: Vector describing point on the plane.
p3: Vector describing point on the plane.

Discussion

The plane is specified by three vectors which are points on the plane.

The normal calculated is of unit length (i.e. it is the Hessian normal).

See http://mathworld.wolfram.com/HessianNormalForm.html

EdenMathRotateMatrix

Rotate a matrix about an arbitrary axis.

void EdenMathRotateMatrix(
    float B[16],
    const float A[16],
    const float q,
    const float x,
    const float y,
    const float z);

Parameters

B: A 4x4 matrix that will receive the output in column major form.
A: A 4x4 matrix that will supply the input in column major form.
q: The angle of rotation measured in a right-hand sense, in radians.
x: X component of the normalised non-zero vector representing the axis of rotation.
y: Y component of the normalised non-zero vector representing the axis of rotation.
z: Z component of the normalised non-zero vector representing the axis of rotation.

Discussion

(description)

EdenMathRotatePointAboutAxis

Rotate a point about an arbitrary axis.

void EdenMathRotatePointAboutAxis(
    float p2[3],
    const float p1[3],
    const float q,
    const float a[3]);

Parameters

p2: Rotated point.
p1: Point to rotate.
q: The angle of rotation measured in a right-hand sense, in radians.
a: Normalised non-zero vector representing the axis of rotation.

EdenMathRotatePointAboutAxisd

Rotate a point about an arbitrary axis.

void EdenMathRotatePointAboutAxisd(
    double p2[3],
    const double p1[3],
    const double q,
    const double a[3]);

Parameters

p2: Rotated point.
p1: Point to rotate.
q: The angle of rotation measured in a right-hand sense, in radians.
a: Normalised non-zero vector representing the axis of rotation.

EdenMathRotationMatrix

Creates a matrix which represents the general case of a rotation about an arbitrary axis.

void EdenMathRotationMatrix(
    float mtx16[16],
    const float q,
    const float x,
    const float y,
    const float z);

Parameters

mtx16: A 4x4 matrix that will receive the output in column major form.
q: The angle of rotation measured in a right-hand sense, in radians.
x: X component of the normalised non-zero vector representing the axis of rotation.
y: Y component of the normalised non-zero vector representing the axis of rotation.
z: Z component of the normalised non-zero vector representing the axis of rotation.

EdenMathRotationMatrixFromTo

Creates a rotation matrix that rotates a vector called "from" into another vector called "to".

void EdenMathRotationMatrixFromTo(
    const float from[3],
    const float to[3],
    float mtx9[9]);

Parameters

from: Normalised non-zero vector.
to: Normalised non-zero vector.
mtx9: A 3x3 matrix in column-major form which receives the result.

Discussion

Author: Tomas Moller, 1999

EdenMathTranslateMatrix

Translate a matrix by a vector.

void EdenMathTranslateMatrix(
    float B[16],
    const float A[16],
    const float x,
    const float y,
    const float z);

Parameters

B: A 4x4 matrix that will receive the output in column major form.
A: A 4x4 matrix that will supply the input in column major form.
x: X component of the translation vector.
y: Y component of the translation vector.
z: Z component of the translation vector.

EdenMathTranslationMatrix

Creates a matrix which represents translation by a vector.

void EdenMathTranslationMatrix(
    float mtx16[16],
    const float x,
    const float y,
    const float z);

Parameters

mtx16: A 4x4 matrix that will receive the output in column major form.
x: X component of the translation vector.
y: Y component of the translation vector.
z: Z component of the translation vector.

frsqrt

Fast calculation of reciprocal square root.

void frsqrt(
    double *arg);

Parameters

arg: Pointer to double-precision floating point number. Will be replaced with the reciprocal of the root of the number. (*arg = (*arg)^-0.5).

Discussion

Uses a ppc-only instruction to get an estimate of the reciprocal square root, then performs 4 iterations of Newton-Rhapson refinement to give better precision. Note: these functions are not IEEE-754 compliant and do not calculate the last bit correctly. For correct accuary, refer to libm on MacOS X. Written by A. Sazegari, started on February 2002. Copyright 2002 Apple Computer, Inc. All rights reserved.

Availability: Available on PowerPC CPUs only.

frsqrt3

Fast calculation of three reciprocal square roots.

void frsqrt3(
    double *arg1,
    double *arg2,
    double *arg3);

Parameters

arg1: Pointer to double-precision floating point number. Will be replaced with the reciprocal of the root of the number.
arg2: Pointer to double-precision floating point number. Will be replaced with the reciprocal of the root of the number.
arg3: Pointer to double-precision floating point number. Will be replaced with the reciprocal of the root of the number.

Discussion

Availability: Available on PowerPC CPUs only.

fsqrt

Fast calculation of square root.

void fsqrt(
    double *arg);

Parameters

arg: Pointer to double-precision floating point number. Will be replaced with the root of the number. (*arg = (*arg)^0.5).

Discussion

Uses a ppc-only instruction to get an estimate of the square root, then performs 4 iterations of Newton-Rhapson refinement to give better precision. Note: these functions are not IEEE-754 compliant and do not calculate the last bit correctly. For correct accuary, refer to libm on MacOS X. Written by A. Sazegari, started on February 2002. Copyright 2002 Apple Computer, Inc. All rights reserved.

Availability: Available on PowerPC CPUs only.

fsqrt3

Fast calculation of three square roots.

void fsqrt3(
    double *arg1,
    double *arg2,
    double *arg3);

Parameters

arg1: Pointer to double-precision floating point number. Will be replaced with the root of the number.
arg2: Pointer to double-precision floating point number. Will be replaced with the root of the number.
arg3: Pointer to double-precision floating point number. Will be replaced with the root of the number.

Discussion

Availability: Available on PowerPC CPUs only.

Macro Definitions

ADD: (description)
AVERAGE: (description)
COPY: (description)
CROSS: Vector cross-product in R3.
DOT: Vector dot-product in R3.
DTOR: Convert degrees to radians.
HALFPI: (description)
LENGTH: (description)
M_PI: Value of PI.
MAX: Determine maximum of two values.
MIN: Determine minimum of two values.
PI: Value of PI.
RTOD: Convert radians to degrees.
SUB: (description)
TWOPI: (description)

ADD

(description)

#define ADD(dest,v1,v2)

Discussion

(description)

AVERAGE

(description)

#define AVERAGE(dest,v1,v2)

Discussion

(description)

COPY

(description)

#define COPY(dest, v)

Discussion

(description)

CROSS

Vector cross-product in R3.

#define CROSS(dest,v1,v2)

Discussion

6 multiplies + 3 subtracts. Vector cross product calculates a vector with direction orthogonal to plane formed by the other two vectors, and length equal to the area of the parallelogram formed by the other two vectors. Right hand rule for vector cross products: Point thumb of right hand in direction of v1, fingers together in direction of v2, then palm faces in the direction of dest.