MATRIX(9.2) MATRIX(9.2)
NAME
ident, matmul, matmulr, determinant, adjoint, invertmat,
xformpoint, xformpointd, xformplane, pushmat, popmat, rot,
qrot, scale, move, xform, ixform, persp, look, viewport -
Geometric transformations
SYNOPSIS
#include <libg.h>
#include <geometry.h>
void ident(Matrix m)
void matmul(Matrix a, Matrix b)
void matmulr(Matrix a, Matrix b)
double determinant(Matrix m)
void adjoint(Matrix m, Matrix madj)
double invertmat(Matrix m, Matrix inv)
Point3 xformpoint(Point3 p, Space *to, Space *from)
Point3 xformpointd(Point3 p, Space *to, Space *from)
Point3 xformplane(Point3 p, Space *to, Space *from)
Space *pushmat(Space *t)
Space *popmat(Space *t)
void rot(Space *t, double theta, int axis)
void qrot(Space *t, Quaternion q)
void scale(Space *t, double x, double y, double z)
void move(Space *t, double x, double y, double z)
void xform(Space *t, Matrix m)
void ixform(Space *t, Matrix m, Matrix inv)
int persp(Space *t, double fov, double n, double f)
void look(Space *t, Point3 eye, Point3 look, Point3 up)
void viewport(Space *t, Rectangle r, double aspect)
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MATRIX(9.2) MATRIX(9.2)
DESCRIPTION
These routines manipulate 3-space affine and projective
transformations, represented as 4x4 matrices, thus:
typedef double Matrix[4][4];
Ident stores an identity matrix in its argument. Matmul
stores axb in a. Matmulr stores bxa in b. Determinant
returns the determinant of matrix m. Adjoint stores the
adjoint (matrix of cofactors) of m in madj. Invertmat stores
the inverse of matrix m in minv, returning m's determinant.
Should m be singular (determinant zero), invertmat stores
its adjoint in minv.
The rest of the routines described here manipulate Spaces
and transform Point3s. A Point3 is a point in three-space,
represented by its homogeneous coordinates:
typedef struct Point3 Point3;
struct Point3{
double x, y, z, w;
};
The homogeneous coordinates (x, y, z, w) represent the
Euclidean point (x/w, y/w, z/w) if w≠0, and a ``point at
infinity'' if w=0.
A Space is just a data structure describing a coordinate
system:
typedef struct Space Space;
struct Space{
Matrix t;
Matrix tinv;
Space *next;
};
It contains a pair of transformation matrices and a pointer
to the Space's parent. The matrices transform points to and
from the ``root coordinate system,'' which is represented by
a null Space pointer.
Pushmat creates a new Space. Its argument is a pointer to
the parent space. Its result is a newly allocated copy of
the parent, but with its next pointer pointing at the par-
ent. Popmat discards the Space that is its argument,
returning a pointer to the stack. Nominally, these two
functions define a stack of transformations, but pushmat can
be called multiple times on the same Space multiple times,
creating a transformation tree.
Xformpoint and Xformpointd both transform points from the
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MATRIX(9.2) MATRIX(9.2)
Space pointed to by from to the space pointed to by to.
Either pointer may be null, indicating the root coordinate
system. The difference between the two functions is that
xformpointd divides x, y, z, and w by w, if w≠0, making (x,
y, z) the Euclidean coordinates of the point.
Xformplane transforms planes or normal vectors. A plane is
specified by the coefficients (a, b, c, d) of its implicit
equation ax+by+cz+d=0. Since this representation is dual to
the homogeneous representation of points, libgeometry repre-
sents planes by Point3 structures, with (a, b, c, d) stored
in (x, y, z, w).
The remaining functions transform the coordinate system rep-
resented by a Space. Their Space * argument must be non-
null - you can't modify the root Space. Rot rotates by
angle theta (in radians) about the given axis, which must be
one of XAXIS, YAXIS or ZAXIS. Qrot transforms by a rotation
about an arbitrary axis, specified by Quaternion q.
Scale scales the coordinate system by the given scale fac-
tors in the directions of the three axes. Move translates
by the given displacement in the three axial directions.
Xform transforms the coordinate system by the given Matrix.
If the matrix's inverse is known a priori, calling ixform
will save the work of recomputing it.
Persp does a perspective transformation. The transformation
maps the frustum with apex at the origin, central axis down
the positive y axis, and apex angle fov and clipping planes
y=n and y=f into the double-unit cube. The plane y=n maps
to y'=-1, y=f maps to y'=1.
Look does a view-pointing transformation. The eye point is
moved to the origin. The line through the eye and look
points is aligned with the y axis, and the plane containing
the eye, look and up points is rotated into the x-y plane.
Viewport maps the unit-cube window into the given screen
viewport. The viewport rectangle r has r.min at the top
left-hand corner, and r.max just outside the lower right-
hand corner. Argument aspect is the aspect ratio (dx/dy) of
the viewport's pixels (not of the whole viewport). The
whole window is transformed to fit centered inside the view-
port with equal slop on either top and bottom or left and
right, depending on the viewport's aspect ratio. The window
is viewed down the y axis, with x to the left and z up. The
viewport has x increasing to the right and y increasing
down. The window's y coordinates are mapped, unchanged,
into the viewport's z coordinates.
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MATRIX(9.2) MATRIX(9.2)
SOURCE
/sys/src/libgeometry/matrix.c
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