Subversion Repositories planix.SVN

.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)