MATRIX(2)MATRIX(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 <draw.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 4×4 matrices, thus:

typedef double Matrix[4][4];

*Ident*
stores an identity matrix in its argument.
*Matmul*
stores
*a×b*
in
*a*.
*Matmulr*
stores
*b×a*
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 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 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*
increasing 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`

## SEE ALSO

*arith3*(2)