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)

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 parent.  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 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  represents	planes	by  Point3
       structures, with (a, b, c, d) stored in (x, y, z, w).

       The  remaining functions transform the coordinate system represented by a Space.  Their Space * argument must be non-null -- you can't mod-
       ify the root Space.  Rot rotates by angle theta (in radians) about the given axis, which must be one of XAXIS, YAXIS or ZAXIS.  Qrot trans-
       forms by a rotation about an arbitrary axis, specified by Quaternion q.

       Scale  scales the coordinate system by the given scale factors in the directions of the three axes.  Move translates by the given displace-
       ment 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 viewport 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  increas-
       ing to the right and y increasing down.	The window's y coordinates are mapped, unchanged, into the viewport's z coordinates.

SOURCE

       /sys/src/libgeometry/matrix.c

																       MATRIX(9.2)