# MAN.9FRONT.ORG RTFM

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GEOMETRY(2)                                           GEOMETRY(2)

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,
mulq, smulq, sdivq, dotq, invq, qlen, normq, slerp, qrotate,
rframexform, rframexform3, invrframexform, invrframexform3,
centroid, barycoords, centroid3, vfmt, Vfmt, GEOMfmtinstall
- computational geometry library

SYNOPSIS
#include <u.h>
#include <libc.h>
#include <geometry.h>

#define DEG 0.01745329251994330    /* π/180 */

typedef struct Point2 Point2;
typedef struct Point3 Point3;
typedef double Matrix;
typedef double Matrix3;
typedef struct Quaternion Quaternion;
typedef struct RFrame RFrame;
typedef struct RFrame3 RFrame3;
typedef struct Triangle2 Triangle2;
typedef struct Triangle3 Triangle3;

struct Point2 {
double x, y, w;
};

struct Point3 {
double x, y, z, w;
};

struct Quaternion {
double r, i, j, k;
};

struct RFrame {
Point2 p;
Point2 bx, by;
};

struct RFrame3 {
Point3 p;
Point3 bx, by, bz;

GEOMETRY(2)                                           GEOMETRY(2)

};

struct Triangle2
{
Point2 p0, p1, p2;
};

struct Triangle3 {
Point3 p0, p1, p2;
};

/* utils */
double flerp(double a, double b, double t);
double fclamp(double n, double min, double max);

/* Point2 */
Point2 Pt2(double x, double y, double w);
Point2 Vec2(double x, double y);
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)

/* Point3 */
Point3 Pt3(double x, double y, double z, double w);
Point3 Vec3(double x, double y, double z);
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);

/* Matrix */
void identity(Matrix m);
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);

GEOMETRY(2)                                           GEOMETRY(2)

void invm(Matrix m);
Point2 xform(Point2 p, Matrix m);

/* Matrix3 */
void identity3(Matrix3 m);
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 invm3(Matrix3 m);
Point3 xform3(Point3 p, Matrix3 m);

/* Quaternion */
Quaternion Quat(double r, double i, double j, double k);
Quaternion Quatvec(double r, Point3 v);
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);

/* 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);

/* Triangle2 */
Point2 centroid(Triangle2 t);
Point3 barycoords(Triangle2 t, Point2 p);

/* Triangle3 */
Point3 centroid3(Triangle3 t);

/* Fmt */
#pragma varargck type "v" Point2
#pragma varargck type "V" Point3
int vfmt(Fmt*);
int Vfmt(Fmt*);
void GEOMfmtinstall(void);

DESCRIPTION

GEOMETRY(2)                                           GEOMETRY(2)

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.

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.

UTILS
These utility functions provide extra floating-point
operations that are not available in the standard libc.

Name Description

flerp
Performs a linear interpolation by a factor of t
between a and b, and returns the result.

fclamp
Constrains n to a value between min and max, and
returns the result.

Points
A point (x,y,w) in projective space results in the point
(x/w,y/w) in Euclidean space. Vectors are represented by
setting w to zero, since they don't belong to any projective
plane themselves.

Name Description

Pt2  Constructor function for a Point2 point.

Vec2 Constructor function for a Point2 vector.

Creates a new 2D point out of the sum of a's and b's
components.

subpt2
Creates a new 2D point out of the substraction of a's
by b's components.

mulpt2
Creates a new 2D point from multiplying p's components
by the scalar s.

divpt2
Creates a new 2D point from dividing p's components by

GEOMETRY(2)                                           GEOMETRY(2)

the scalar s.

lerp2
Performs a linear interpolation between the 2D points a
and b by a factor of t, and returns the result.

dotvec2
Computes the dot product of vectors a and b.

vec2len
Computes the lengthmagnitudeof vector v.

normvec2
Normalizes the vector v and returns a new 2D point.

edgeptcmp
Performs a comparison between an edge, defined by a
directed line from e0 to e1, and the point 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; otherwisewhen the
point is on the line, it returns 0.

ptinpoly
Returns 1 if the 2D point p lies within the npts-vertex
polygon defined by pts, 0 otherwise.

Pt3  Constructor function for a Point3 point.

Vec3 Constructor function for a Point3 vector.

Creates a new 3D point out of the sum of a's and b's
components.

subpt3
Creates a new 3D point out of the substraction of a's
by b's components.

mulpt3
Creates a new 3D point from multiplying p's components
by the scalar s.

divpt3
Creates a new 3D point from dividing p's components by
the scalar s.

lerp3
Performs a linear interpolation between the 3D points a
and b by a factor of t, and returns the result.

dotvec3
Computes the dot/inner product of vectors a and b.

GEOMETRY(2)                                           GEOMETRY(2)

crossvec3
Computes the cross/outer product of vectors a and b.

vec3len
Computes the lengthmagnitudeof vector v.

normvec3
Normalizes the vector v and returns a new 3D point.

Matrices
Name Description

identity
Initializes m into an identity, 3x3 matrix.

addm Sums the matrices a and b and stores the result back in
a.

subm Substracts the matrix a by b and stores the result back
in a.

mulm Multiplies the matrices a and b and stores the result
back in a.

smulm
Multiplies every element of m by the scalar s, storing
the result in m.

transposem
Transforms the matrix m into its transpose.

detm Computes the determinant of m and returns the result.

tracem
Computes the trace of m and returns the result.

invm Transforms the matrix m into its inverse.

xform
Transforms the point p by the matrix m and returns the
new 2D point.

identity3
Initializes m into an identity, 4x4 matrix.

Sums the matrices a and b and stores the result back in
a.

subm3

GEOMETRY(2)                                           GEOMETRY(2)

Substracts the matrix a by b and stores the result back
in a.

mulm3
Multiplies the matrices a and b and stores the result
back in a.

smulm3
Multiplies every element of m by the scalar s, storing
the result in m.

transposem3
Transforms the matrix m into its transpose.

detm3
Computes the determinant of m and returns the result.

tracem3
Computes the trace of m and returns the result.

Transforms the matrix m into its adjoint.

invm3
Transforms the matrix m into its inverse.

xform3
Transforms the point p by the matrix m and returns the
new 3D point.

Quaternions
Quaternions are an extension of the complex numbers con-
ceived as a tool to analyze 3-dimensional points.  They are
most commonly used to orient and rotate objects in 3D space.

Name Description

Quat Constructor function for a Quaternion.

Quatvec
Constructor function for a Quaternion that takes the
imaginary part in the form of a vector v.

addq Creates a new quaternion out of the sum of a's and b's
components.

subq Creates a new quaternion from the substraction of a's
by b's components.

mulq Multiplies a and b and returns a new quaternion.

smulq

GEOMETRY(2)                                           GEOMETRY(2)

Multiplies each of the components of q by the scalar s,
returning a new quaternion.

sdivq
Divides each of the components of q by the scalar s,
returning a new quaternion.

dotq Computes the dot-product of q and r, and returns the
result.

invq Computes the inverse of q and returns a new quaternion
out of it.

qlen Computes q's lengthmagnitudeand returns the result.

normq
Normalizes q and returns a new quaternion out of it.

slerp
Performs a spherical linear interpolation between the
quaternions q and r by a factor of t, and returns the
result.

qrotate
Returns the result of rotating the point p around the

Frames of reference
A frame of reference in a n-dimensional space is made out of
n+1 points, one being the origin p, relative to some other
frame of reference, and the remaining being the basis vec-
tors b1,⋯,bn that define the metric within that frame.

Every one of these routines assumes the origin reference
frame 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 refer-
ence frame if that's not the case.

Name Description

rframexform
Transforms the point p, relative to origin O, into the
frame of reference rf with origin in rf.p, which is
itself also relative to O. It then returns the new 2D
point.

rframexform3
Transforms the point p, relative to origin O, into the
frame of reference rf with origin in rf.p, which is
itself also relative to O. It then returns the new 3D
point.

GEOMETRY(2)                                           GEOMETRY(2)

invrframexform
Transforms the point p, relative to rf.p, into the
frame of reference O, assumed to have an orthonormal
basis.

invrframexform3
Transforms the point p, relative to rf.p, into the
frame of reference O, assumed to have an orthonormal
basis.

Triangles
Name Description

centroid
Returns the geometric center of Triangle2 t.

barycoords
Returns a 3D point that represents the barycentric
coordinates of the 2D point p relative to the triangle
t.

centroid3
Returns the geometric center of Triangle3 t.

EXAMPLE
The following is a common example of an RFrame being used to
define the coordinate system of a rio(3) window.  It places
the origin at the center of the window and sets up an
orthonormal basis with the y-axis pointing upwards, to con-
trast with the window system where y-values grow downwards
(see graphics(2)).

#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);
}

GEOMETRY(2)                                           GEOMETRY(2)

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);
⋯

The following snippet shows how to use the RFrame declared
earlier to locate and draw a ship based on its orientation,
for which we use matrix translation T and rotation R trans-
formations.

⋯
typedef struct Ship Ship;
typedef struct Shipmdl Shipmdl;

struct Ship
{
RFrame;
double θ; /* orientation (yaw) */
Shipmdl mdl;
};

struct Shipmdl
{
Point2 pts; /* 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;

GEOMETRY(2)                                           GEOMETRY(2)

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 = Pt2( 10, 0,1);
ship->mdl.pts = Pt2(-10, 5,1);
ship->mdl.pts = Pt2(-10,-5,1);
⋯
redraw();
⋯

Notice how we could've used the RFrame embedded in the ship
to transform the Shipmdl into the window.  Instead of apply-
ing the matrices to every point, the ship's local frame of
reference can be rotated, effectively changing the model
coordinates after an invrframexform. We are also getting rid
of the θ variable, since it's no longer needed.

⋯
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);
⋯

SOURCE
/sys/src/libgeometry

sin(2), floor(2), graphics(2)

GEOMETRY(2)                                           GEOMETRY(2)

Philip J. Schneider, David H. Eberly, Geometric Tools for
Computer Graphics, Morgan Kaufmann Publishers, 2003.
Jonathan Blow, Understanding Slerp, Then Not Using it, The
Inner Product, April 2004.
https://www.3dgep.com/understanding-quaternions/

BUGS
No care is taken to avoid numeric overflows.

HISTORY
Libgeometry first appeared in Plan 9 from Bell Labs.  It was
revamped for 9front in January of 2023.

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