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MATRIX(9.2)									      MATRIX(9.2)

       ident,  matmul,	matmulr, determinant, adjoint, invertmat, xformpoint, xformpointd, xform-
       plane, pushmat, popmat, rot, qrot, scale, move, xform, ixform,  persp,  look,  viewport	-
       Geometric transformations

       #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)

       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.   Nomi-
       nally,  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 sys-
       tem.  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 struc-
       tures, 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 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 factors 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  cor-
       ner.   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  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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