ref: c6a92afc7ac0382d90bf0fbd1d2e3e258a57aad4
dir: /sys/man/2/geometry/
.TH GEOMETRY 2
.SH NAME
Flerp, fclamp, Pt2, Vec2, addpt2, subpt2, mulpt2, divpt2, lerp2, dotvec2, vec2len, normvec2, edgeptcmp, ptinpoly, Pt3, Vec3, addpt3, subpt3, mulpt3, divpt3, lerp3, dotvec3, crossvec3, vec3len, normvec3, identity, addm, subm, mulm, smulm, transposem, detm, tracem, adjm, invm, xform, identity3, addm3, subm3, mulm3, smulm3, transposem3, detm3, tracem3, adjm3, invm3, xform3, Quat, Quatvec, addq, subq, mulq, smulq, sdivq, dotq, invq, qlen, normq, slerp, qrotate, rframexform, rframexform3, invrframexform, invrframexform3, centroid, barycoords, centroid3, vfmt, Vfmt, GEOMfmtinstall \- computational geometry library
.SH SYNOPSIS
.de PB
.PP
.ft L
.nf
..
.PB
#include <u.h>
#include <libc.h>
#include <geometry.h>
.PB
#define DEG 0.01745329251994330 /* π/180 */
.PB
typedef struct Point2 Point2;
typedef struct Point3 Point3;
typedef double Matrix[3][3];
typedef double Matrix3[4][4];
typedef struct Quaternion Quaternion;
typedef struct RFrame RFrame;
typedef struct RFrame3 RFrame3;
typedef struct Triangle2 Triangle2;
typedef struct Triangle3 Triangle3;
.PB
struct Point2 {
double x, y, w;
};
.PB
struct Point3 {
double x, y, z, w;
};
.PB
struct Quaternion {
double r, i, j, k;
};
.PB
struct RFrame {
Point2 p;
Point2 bx, by;
};
.PB
struct RFrame3 {
Point3 p;
Point3 bx, by, bz;
};
.PB
struct Triangle2
{
Point2 p0, p1, p2;
};
.PB
struct Triangle3 {
Point3 p0, p1, p2;
};
.PB
/* utils */
double flerp(double a, double b, double t);
double fclamp(double n, double min, double max);
.PB
/* Point2 */
Point2 Pt2(double x, double y, double w);
Point2 Vec2(double x, double y);
Point2 addpt2(Point2 a, Point2 b);
Point2 subpt2(Point2 a, Point2 b);
Point2 mulpt2(Point2 p, double s);
Point2 divpt2(Point2 p, double s);
Point2 lerp2(Point2 a, Point2 b, double t);
double dotvec2(Point2 a, Point2 b);
double vec2len(Point2 v);
Point2 normvec2(Point2 v);
int edgeptcmp(Point2 e0, Point2 e1, Point2 p);
int ptinpoly(Point2 p, Point2 *pts, ulong npts)
.PB
/* Point3 */
Point3 Pt3(double x, double y, double z, double w);
Point3 Vec3(double x, double y, double z);
Point3 addpt3(Point3 a, Point3 b);
Point3 subpt3(Point3 a, Point3 b);
Point3 mulpt3(Point3 p, double s);
Point3 divpt3(Point3 p, double s);
Point3 lerp3(Point3 a, Point3 b, double t);
double dotvec3(Point3 a, Point3 b);
Point3 crossvec3(Point3 a, Point3 b);
double vec3len(Point3 v);
Point3 normvec3(Point3 v);
.PB
/* Matrix */
void identity(Matrix m);
void addm(Matrix a, Matrix b);
void subm(Matrix a, Matrix b);
void mulm(Matrix a, Matrix b);
void smulm(Matrix m, double s);
void transposem(Matrix m);
double detm(Matrix m);
double tracem(Matrix m);
void adjm(Matrix m);
void invm(Matrix m);
Point2 xform(Point2 p, Matrix m);
.PB
/* Matrix3 */
void identity3(Matrix3 m);
void addm3(Matrix3 a, Matrix3 b);
void subm3(Matrix3 a, Matrix3 b);
void mulm3(Matrix3 a, Matrix3 b);
void smulm3(Matrix3 m, double s);
void transposem3(Matrix3 m);
double detm3(Matrix3 m);
double tracem3(Matrix3 m);
void adjm3(Matrix3 m);
void invm3(Matrix3 m);
Point3 xform3(Point3 p, Matrix3 m);
.PB
/* Quaternion */
Quaternion Quat(double r, double i, double j, double k);
Quaternion Quatvec(double r, Point3 v);
Quaternion addq(Quaternion a, Quaternion b);
Quaternion subq(Quaternion a, Quaternion b);
Quaternion mulq(Quaternion q, Quaternion r);
Quaternion smulq(Quaternion q, double s);
Quaternion sdivq(Quaternion q, double s);
double dotq(Quaternion q, Quaternion r);
Quaternion invq(Quaternion q);
double qlen(Quaternion q);
Quaternion normq(Quaternion q);
Quaternion slerp(Quaternion q, Quaternion r, double t);
Point3 qrotate(Point3 p, Point3 axis, double θ);
.PB
/* RFrame */
Point2 rframexform(Point2 p, RFrame rf);
Point3 rframexform3(Point3 p, RFrame3 rf);
Point2 invrframexform(Point2 p, RFrame rf);
Point3 invrframexform3(Point3 p, RFrame3 rf);
.PB
/* Triangle2 */
Point2 centroid(Triangle2 t);
Point3 barycoords(Triangle2 t, Point2 p);
.PB
/* Triangle3 */
Point3 centroid3(Triangle3 t);
.PB
/* Fmt */
#pragma varargck type "v" Point2
#pragma varargck type "V" Point3
int vfmt(Fmt*);
int Vfmt(Fmt*);
void GEOMfmtinstall(void);
.SH DESCRIPTION
This library provides routines to operate with homogeneous coordinates
in 2D and 3D projective spaces by means of points, matrices,
quaternions and frames of reference.
.PP
Besides their many mathematical properties and applications, the data
structures and algorithms used here to represent these abstractions
are specifically tailored to the world of computer graphics and
simulators, and so it uses the conventions associated with these
fields, such as the right-hand rule for coordinate systems and column
vectors for matrix operations.
.SS UTILS
These utility functions provide extra floating-point operations that
are not available in the standard libc.
.TP
Name
Description
.TP
.B flerp
Performs a linear interpolation by a factor of
.I t
between
.I a
and
.IR b ,
and returns the result.
.TP
.B fclamp
Constrains
.I n
to a value between
.I min
and
.IR max ,
and returns the result.
.SS Points
A point
.B (x,y,w)
in projective space results in the point
.B (x/w,y/w)
in Euclidean space. Vectors are represented by setting
.B w
to zero, since they don't belong to any projective plane themselves.
.TP
Name
Description
.TP
.B Pt2
Constructor function for a Point2 point.
.TP
.B Vec2
Constructor function for a Point2 vector.
.TP
.B addpt2
Creates a new 2D point out of the sum of
.IR a 's
and
.IR b 's
components.
.TP
subpt2
Creates a new 2D point out of the substraction of
.IR a 's
by
.IR b 's
components.
.TP
mulpt2
Creates a new 2D point from multiplying
.IR p 's
components by the scalar
.IR s .
.TP
divpt2
Creates a new 2D point from dividing
.IR p 's
components by the scalar
.IR s .
.TP
lerp2
Performs a linear interpolation between the 2D points
.I a
and
.I b
by a factor of
.IR t ,
and returns the result.
.TP
dotvec2
Computes the dot product of vectors
.I a
and
.IR b .
.TP
vec2len
Computes the length—magnitude—of vector
.IR v .
.TP
normvec2
Normalizes the vector
.I v
and returns a new 2D point.
.TP
edgeptcmp
Performs a comparison between an edge, defined by a directed line from
.I e0
to
.IR e1 ,
and the point
.IR p .
If the point is to the right of the line, the result is >0; if it's to
the left, the result is <0; otherwise—when the point is on the line—,
it returns 0.
.TP
ptinpoly
Returns 1 if the 2D point
.I p
lies within the
.IR npts -vertex
polygon defined by
.IR pts ,
0 otherwise.
.TP
Pt3
Constructor function for a Point3 point.
.TP
Vec3
Constructor function for a Point3 vector.
.TP
addpt3
Creates a new 3D point out of the sum of
.IR a 's
and
.IR b 's
components.
.TP
subpt3
Creates a new 3D point out of the substraction of
.IR a 's
by
.IR b 's
components.
.TP
mulpt3
Creates a new 3D point from multiplying
.IR p 's
components by the scalar
.IR s .
.TP
divpt3
Creates a new 3D point from dividing
.IR p 's
components by the scalar
.IR s .
.TP
lerp3
Performs a linear interpolation between the 3D points
.I a
and
.I b
by a factor of
.IR t ,
and returns the result.
.TP
dotvec3
Computes the dot/inner product of vectors
.I a
and
.IR b .
.TP
crossvec3
Computes the cross/outer product of vectors
.I a
and
.IR b .
.TP
vec3len
Computes the length—magnitude—of vector
.IR v .
.TP
normvec3
Normalizes the vector
.I v
and returns a new 3D point.
.SS Matrices
.TP
Name
Description
.TP
identity
Initializes
.I m
into an identity, 3x3 matrix.
.TP
addm
Sums the matrices
.I a
and
.I b
and stores the result back in
.IR a .
.TP
subm
Substracts the matrix
.I a
by
.I b
and stores the result back in
.IR a .
.TP
mulm
Multiplies the matrices
.I a
and
.I b
and stores the result back in
.IR a .
.TP
smulm
Multiplies every element of
.I m
by the scalar
.IR s ,
storing the result in m.
.TP
transposem
Transforms the matrix
.I m
into its transpose.
.TP
detm
Computes the determinant of
.I m
and returns the result.
.TP
tracem
Computes the trace of
.I m
and returns the result.
.TP
adjm
Transforms the matrix
.I m
into its adjoint.
.TP
invm
Transforms the matrix
.I m
into its inverse.
.TP
xform
Transforms the point
.I p
by the matrix
.I m
and returns the new 2D point.
.TP
identity3
Initializes
.I m
into an identity, 4x4 matrix.
.TP
addm3
Sums the matrices
.I a
and
.I b
and stores the result back in
.IR a .
.TP
subm3
Substracts the matrix
.I a
by
.I b
and stores the result back in
.IR a .
.TP
mulm3
Multiplies the matrices
.I a
and
.I b
and stores the result back in
.IR a .
.TP
smulm3
Multiplies every element of
.I m
by the scalar
.IR s ,
storing the result in m.
.TP
transposem3
Transforms the matrix
.I m
into its transpose.
.TP
detm3
Computes the determinant of
.I m
and returns the result.
.TP
tracem3
Computes the trace of
.I m
and returns the result.
.TP
adjm3
Transforms the matrix
.I m
into its adjoint.
.TP
invm3
Transforms the matrix
.I m
into its inverse.
.TP
xform3
Transforms the point
.I p
by the matrix
.I m
and returns the new 3D point.
.SS Quaternions
Quaternions are an extension of the complex numbers conceived as a
tool to analyze 3-dimensional points. They are most commonly used to
orient and rotate objects in 3D space.
.TP
Name
Description
.TP
Quat
Constructor function for a Quaternion.
.TP
Quatvec
Constructor function for a Quaternion that takes the imaginary part in
the form of a vector
.IR v .
.TP
addq
Creates a new quaternion out of the sum of
.IR a 's
and
.IR b 's
components.
.TP
subq
Creates a new quaternion from the substraction of
.IR a 's
by
.IR b 's
components.
.TP
mulq
Multiplies
.I a
and
.I b
and returns a new quaternion.
.TP
smulq
Multiplies each of the components of
.I q
by the scalar
.IR s ,
returning a new quaternion.
.TP
sdivq
Divides each of the components of
.I q
by the scalar
.IR s ,
returning a new quaternion.
.TP
dotq
Computes the dot-product of
.I q
and
.IR r ,
and returns the result.
.TP
invq
Computes the inverse of
.I q
and returns a new quaternion out of it.
.TP
qlen
Computes
.IR q 's
length—magnitude—and returns the result.
.TP
normq
Normalizes
.I q
and returns a new quaternion out of it.
.TP
slerp
Performs a spherical linear interpolation between the quaternions
.I q
and
.I r
by a factor of
.IR t ,
and returns the result.
.TP
qrotate
Returns the result of rotating the point
.I p
around the vector
.I axis
by
.I θ
radians.
.SS Frames of reference
A frame of reference in a
.IR n -dimensional
space is made out of n+1 points, one being the origin
.IR p ,
relative to some other frame of reference, and the remaining being the
basis vectors
.I b1,⋯,bn
that define the metric within that frame.
.PP
Every one of these routines assumes the origin reference frame
.B O
has an orthonormal basis when performing an inverse transformation;
it's up to the user to apply a forward transformation to the resulting
point with the proper reference frame if that's not the case.
.TP
Name
Description
.TP
rframexform
Transforms the point
.IR p ,
relative to origin O, into the frame of reference
.I rf
with origin in
.BR rf.p ,
which is itself also relative to O. It then returns the new 2D point.
.TP
rframexform3
Transforms the point
.IR p ,
relative to origin O, into the frame of reference
.I rf
with origin in
.BR rf.p ,
which is itself also relative to O. It then returns the new 3D point.
.TP
invrframexform
Transforms the point
.IR p ,
relative to
.BR rf.p ,
into the frame of reference O, assumed to have an orthonormal basis.
.TP
invrframexform3
Transforms the point
.IR p ,
relative to
.BR rf.p ,
into the frame of reference O, assumed to have an orthonormal basis.
.SS Triangles
.TP
Name
Description
.TP
centroid
Returns the geometric center of
.B Triangle2
.IR t .
.TP
barycoords
Returns a 3D point that represents the barycentric coordinates of the
2D point
.I p
relative to the triangle
.IR t .
.TP
centroid3
Returns the geometric center of
.B Triangle3
.IR t .
.SH EXAMPLE
The following is a common example of an
.B RFrame
being used to define the coordinate system of a
.IR rio (3)
window. It places the origin at the center of the window and sets up
an orthonormal basis with the
.IR y -axis
pointing upwards, to contrast with the window system where
.IR y -values
grow downwards (see
.IR graphics (2)).
.PP
.EX
#include <u.h>
#include <libc.h>
#include <draw.h>
#include <geometry.h>
RFrame screenrf;
Point
toscreen(Point2 p)
{
p = invrframexform(p, screenrf);
return Pt(p.x,p.y);
}
Point2
fromscreen(Point p)
{
return rframexform(Pt2(p.x,p.y,1), screenrf);
}
void
main(void)
⋯
screenrf.p = Pt2(screen->r.min.x+Dx(screen->r)/2,screen->r.max.y-Dy(screen->r)/2,1);
screenrf.bx = Vec2(1, 0);
screenrf.by = Vec2(0,-1);
⋯
.EE
.PP
The following snippet shows how to use the
.B RFrame
declared earlier to locate and draw a ship based on its orientation,
for which we use matrix translation
.B T
and rotation
.BR R
transformations.
.PP
.EX
⋯
typedef struct Ship Ship;
typedef struct Shipmdl Shipmdl;
struct Ship
{
RFrame;
double θ; /* orientation (yaw) */
Shipmdl mdl;
};
struct Shipmdl
{
Point2 pts[3]; /* a free-form triangle */
};
Ship *ship;
void
redraw(void)
{
int i;
Point pts[3+1];
Point2 *p;
Matrix T = {
1, 0, ship->p.x,
0, 1, ship->p.y,
0, 0, 1,
}, R = {
cos(ship->θ), -sin(ship->θ), 0,
sin(ship->θ), cos(ship->θ), 0,
0, 0, 1,
};
mulm(T, R); /* rotate, then translate */
p = ship->mdl.pts;
for(i = 0; i < nelem(pts)-1; i++)
pts[i] = toscreen(xform(p[i], T));
pts[i] = pts[0];
draw(screen, screen->r, display->white, nil, ZP);
poly(screen, pts, nelem(pts), 0, 0, 0, display->black, ZP);
}
⋯
main(void)
⋯
ship = malloc(sizeof(Ship));
ship->p = Pt2(0,0,1); /* place it at the origin */
ship->θ = 45*DEG; /* counter-clockwise */
ship->mdl.pts[0] = Pt2( 10, 0,1);
ship->mdl.pts[1] = Pt2(-10, 5,1);
ship->mdl.pts[2] = Pt2(-10,-5,1);
⋯
redraw();
⋯
.EE
.PP
Notice how we could've used the
.B RFrame
embedded in the
.B ship
to transform the
.B Shipmdl
into the window. Instead of applying the matrices to every point, the
ship's local frame of reference can be rotated, effectively changing
the model coordinates after an
.IR invrframexform .
We are also getting rid of the
.B θ
variable, since it's no longer needed.
.PP
.EX
⋯
struct Ship
{
RFrame;
Shipmdl mdl;
};
⋯
redraw(void)
⋯
pts[i] = toscreen(invrframexform(p[i], *ship));
⋯
main(void)
⋯
Matrix R = {
cos(45*DEG), -sin(45*DEG), 0,
sin(45*DEG), cos(45*DEG), 0,
0, 0, 1,
};
⋯
//ship->θ = 45*DEG; /* counter-clockwise */
ship->bx = xform(ship->bx, R);
ship->by = xform(ship->by, R);
⋯
.EE
.SH SOURCE
.B /sys/src/libgeometry
.SH SEE ALSO
.IR sin (2),
.IR floor (2),
.IR graphics (2)
.br
Philip J. Schneider, David H. Eberly,
“Geometric Tools for Computer Graphics”,
.I
Morgan Kaufmann Publishers, 2003.
.br
Jonathan Blow,
“Understanding Slerp, Then Not Using it”,
.I
The Inner Product, April 2004.
.br
https://www.3dgep.com/understanding-quaternions/
.SH BUGS
No care is taken to avoid numeric overflows.
.SH HISTORY
Libgeometry first appeared in Plan 9 from Bell Labs. It was revamped
for 9front in January of 2023.