TH MATRIX 2
SH NAME
ident, matmul, matmulr, determinant, adjoint, invertmat, xformpoint, xformpointd, xformplane, pushmat, popmat, rot, qrot, scale, move, xform, ixform, persp, look, viewport \- Geometric transformations
SH SYNOPSIS
PP
B
#include <draw.h>
PP
B
#include <geometry.h>
PP
B
void ident(Matrix m)
PP
B
void matmul(Matrix a, Matrix b)
PP
B
void matmulr(Matrix a, Matrix b)
PP
B
double determinant(Matrix m)
PP
B
void adjoint(Matrix m, Matrix madj)
PP
B
double invertmat(Matrix m, Matrix inv)
PP
B
Point3 xformpoint(Point3 p, Space *to, Space *from)
PP
B
Point3 xformpointd(Point3 p, Space *to, Space *from)
PP
B
Point3 xformplane(Point3 p, Space *to, Space *from)
PP
B
Space *pushmat(Space *t)
PP
B
Space *popmat(Space *t)
PP
B
void rot(Space *t, double theta, int axis)
PP
B
void qrot(Space *t, Quaternion q)
PP
B
void scale(Space *t, double x, double y, double z)
PP
B
void move(Space *t, double x, double y, double z)
PP
B
void xform(Space *t, Matrix m)
PP
B
void ixform(Space *t, Matrix m, Matrix inv)
PP
B
int persp(Space *t, double fov, double n, double f)
PP
B
void look(Space *t, Point3 eye, Point3 look, Point3 up)
PP
B
void viewport(Space *t, Rectangle r, double aspect)
SH DESCRIPTION
These routines manipulate 3-space affine and projective transformations,
represented as 4\(mu4 matrices, thus:
IP
EX
ta 6n
typedef double Matrix[4][4];
EE
PP
I Ident
stores an identity matrix in its argument.
I Matmul
stores
I a\(mub
in
IR a .
I Matmulr
stores
I b\(mua
in
IR b .
I Determinant
returns the determinant of matrix
IR m .
I Adjoint
stores the adjoint (matrix of cofactors) of
I m
in
IR madj .
I Invertmat
stores the inverse of matrix
I m
in
IR minv ,
returning
IR m 's
determinant.
Should
I m
be singular (determinant zero),
I invertmat
stores its
adjoint in
IR minv .
PP
The rest of the routines described here
manipulate
I Spaces
and transform
IR Point3s .
A
I Point3
is a point in three-space, represented by its
homogeneous coordinates:
IP
EX
typedef struct Point3 Point3;
struct Point3{
       double x, y, z, w;
};
EE
PP
The homogeneous coordinates
RI ( x ,
IR y ,
IR z ,
IR w )
represent the Euclidean point
RI ( x / w ,
IR y / w ,
IR z / w )
if
IR w ≠0,
and a ``point at infinity'' if
IR w =0.
PP
A
I Space
is just a data structure describing a coordinate system:
IP
EX
typedef struct Space Space;
struct Space{
       Matrix t;
       Matrix tinv;
       Space *next;
};
EE
PP
It contains a pair of transformation matrices and a pointer
to the
IR Space 's
parent.  The matrices transform points to and from the ``root
coordinate system,'' which is represented by a null
I Space
pointer.
PP
I Pushmat
creates a new
IR Space .
Its argument is a pointer to the parent space.  Its result
is a newly allocated copy of the parent, but with its
B next
pointer pointing at the parent.
I Popmat
discards the
B Space
that is its argument, returning a pointer to the stack.
Nominally, these two functions define a stack of transformations,
but
B pushmat
can be called multiple times
on the same
B Space
multiple times, creating a transformation tree.
PP
I Xformpoint
and
I Xformpointd
both transform points from the
B Space
pointed to by
I from
to the space pointed to by
IR to .
Either pointer may be null, indicating the root coordinate system.
The difference between the two functions is that
B xformpointd
divides
IR x ,
IR y ,
IR z ,
and
I w
by
IR w ,
if
IR w ≠0,
making
RI ( x ,
IR y ,
IR z )
the Euclidean coordinates of the point.
PP
I Xformplane
transforms planes or normal vectors.  A plane is specified by the
coefficients
RI ( a ,
IR b ,
IR c ,
IR d )
of its implicit equation
IR ax+by+cz+d =0.
Since this representation is dual to the homogeneous representation of points,
B libgeometry
represents planes by
B Point3
structures, with
RI ( a ,
IR b ,
IR c ,
IR d )
stored in
RI ( x ,
IR y ,
IR z ,
IR w ).
PP
The remaining functions transform the coordinate system represented
by a
BR Space .
Their
B Space *
argument must be non-null \(em you can't modify the root
BR Space .
I Rot
rotates by angle
I theta
(in radians) about the given
IR axis ,
which must be one of
BR XAXIS ,
B YAXIS
or
BR ZAXIS .
I Qrot
transforms by a rotation about an arbitrary axis, specified by
B Quaternion
IR q .
PP
I Scale
scales the coordinate system by the given scale factors in the directions of the three axes.
IB Move
translates by the given displacement in the three axial directions.
PP
I Xform
transforms the coordinate system by the given
BR Matrix .
If the matrix's inverse is known
I a
IR priori ,
calling
I ixform
will save the work of recomputing it.
PP
I Persp
does a perspective transformation.
The transformation maps the frustum with apex at the origin,
central axis down the positive
I y
axis, and apex angle
I fov
and clipping planes
IR y = n
and
IR y = f
into the double-unit cube.
The plane
IR y = n
maps to
IR y '=-1,
IR y = f
maps to
IR y '=1.
PP
I Look
does a view-pointing transformation.  The
B eye
point is moved to the origin.
The line through the
I eye
and
I look
points is aligned with the y axis,
and the plane containing the
BR eye ,
B look
and
B up
points is rotated into the
IR x - y
plane.
PP
I Viewport
maps the unit-cube window into the given screen viewport.
The viewport rectangle
I r
has
IB r .min
at the top left-hand corner, and
IB r .max
just outside the lower right-hand corner.
Argument
I aspect
is the aspect ratio
RI ( 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
I y
axis, with
I x
to the left and
I z
up.  The viewport
has
I x
increasing to the right and
I y
increasing down.  The window's
I y
coordinates are mapped, unchanged, into the viewport's
I z
coordinates.
SH SOURCE
B /sys/src/libgeometry/matrix.c
SH "SEE ALSO
IR arith3 (2)

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