standalone version release.

11 files changed, 4027 insertions, 0 deletions

diff --git a/doc/libgeometry.ms b/doc/libgeometry.ms
new file mode 100644
index 0000000..47c8d94
--- /dev/null
+++ b/doc/libgeometry.ms
@@ -0,0 +1,429 @@
+.TL
+libgeometry
+.AU
+Rodrigo G. López
+.sp
+rgl@antares-labs.eu
+.AI
+Antares Telecom Laboratories
+Albatera, Alicante
+.FS
+ACHTUNG! this is a
+.B "WORK IN PROGRESS"
+.FE
+.SH
+Introduction
+.PP
+.I Libgeometry is a computational geometry library that provides all
+the utilities anybody working with graphics or scientific simulations
+could need.
+.NH 1
+Data Structures
+.NH 2
+Point2
+.P1
+struct Point2 {
+	double x, y, w;
+};
+.P2
+.PP
+.I Point2
+represents a point in two-dimensional projective space, which itself
+is an extension of the two-dimensional euclidean space that allows us
+to work with vectors and compose affine transformations in a friendly
+manner.  A point
+.EQ
+gfont roman
+(x, y, w)
+.EN
+made out of homogenous coordinates
+.I x ,
+.I y ,
+and
+.I w ,
+yields a point with cartesian coordinates
+.EQ
+(x/w, y/w) .
+.EN
+.NH 2
+Point3
+.P1
+struct Point3 {
+	double x, y, z, w;
+};
+.P2
+.PP
+.I Point3
+is a point in three-dimensional projective space.
+.NH 2
+Matrix
+.P1
+typedef double Matrix[3][3];
+.P2
+.PP
+.I Matrix
+represents a 3x3 matrix, thought to compose affine transformations to
+apply to homogeneous 2D points.
+.NH 2
+Matrix3
+.P1
+typedef double Matrix3[4][4];
+.P2
+.PP
+.I Matrix3
+represents a 4x4 matrix, thought to compose affine transformations to
+apply to homogeneous 3D points.
+.NH 2
+Quaternion
+.P1
+struct Quaternion {
+	double r, i, j, k;
+};
+.P2
+.PP
+.I Quaternions
+are a numbering system that extends the complex numbers up to
+four-dimensional space, and are used to apply rotations and model
+mechanical interactions in 3D space.  Their main advantages with
+respect to their matrix relatives are increased computational and
+storage performance and gimbal lock avoidance.
+.NH 2
+RFrame
+.P1
+struct RFrame {
+	Point2 p;
+	Point2 bx, by;
+};
+.P2
+.PP
+A reference frame (or frame of reference) is
+.NH 2
+RFrame3
+.P1
+struct RFrame3 {
+	Point3 p;
+	Point3 bx, by, bz;
+};
+.P2
+.PP
+A reference frame (or frame of reference) is
+.NH 1
+Algorithms
+.NH 2
+Point2
+.SH
+Addition
+.P1
+Point2 addpt2(Point2 a, Point2 b)
+.P2
+.EQ
+a + b ~=~ left ( x sub a + x sub b ,~ y sub a + y sub b ,~ w sub a + w sub b right )
+.EN
+.SH
+Substraction
+.P1
+Point2 subpt2(Point2 a, Point2 b)
+.P2
+.EQ
+a - b ~=~ left ( x sub a - x sub b ,~ y sub a - y sub b ,~ w sub a - w sub b right )
+.EN
+.SH
+Multiplication
+.P1
+Point2 mulpt2(Point2 p, double s)
+.P2
+.EQ
+p * s ~=~ left ( xs,~ ys,~ ws right )
+.EN
+.SH
+Division
+.P1
+Point2 divpt2(Point2 p, double s)
+.P2
+.EQ
+p / s ~=~ left ( x over s ,~ y over s ,~ w over s right )
+.EN
+.SH
+Vector Dot Product
+.P1
+double dotvec2(Point2 a, Point2 b)
+.P2
+.EQ
+a vec ~•~ b vec ~=~ x sub a x sub b + y sub a y sub b
+.EN
+.SH
+Vector Magnitude/Length
+.P1
+double vec2len(Point2 v)
+.P2
+.EQ
+| v vec | ~=~ sqrt { x sup 2 + y sup 2 }
+.EN
+.SH
+Vector Normalization
+.P1
+Point2 normvec2(Point2 v)
+.P2
+.EQ
+n vec ~=~ left ( x over {| v vec |},~ y over {| v vec |} right )
+.EN
+.NH 2
+Point3
+.SH
+Addition
+.P1
+Point3 addpt3(Point3 a, Point3 b)
+.P2
+.EQ
+a + b ~=~ left ( x sub a + x sub b ,~ y sub a + y sub b ,~ z sub a + z sub b ,~ w sub a + w sub b right )
+.EN
+.SH
+Substraction
+.P1
+Point3 subpt3(Point3 a, Point3 b)
+.P2
+.EQ
+a - b ~=~ left ( x sub a - x sub b ,~ y sub a - y sub b ,~ z sub a - z sub b ,~ w sub a - w sub b right )
+.EN
+.SH
+Multiplication
+.P1
+Point3 mulpt3(Point3 p, double s)
+.P2
+.EQ
+p * s ~=~ left ( xs,~ ys,~ zs,~ ws right )
+.EN
+.SH
+Division
+.P1
+Point3 divpt3(Point3 p, double s)
+.P2
+.EQ
+p / s ~=~ left ( x over s ,~ y over s ,~ z over s ,~ w over s right )
+.EN
+.SH
+Vector Dot Product
+.P1
+double dotvec3(Point3 a, Point3 b)
+.P2
+.EQ
+a vec ~•~ b vec ~=~ x sub a x sub b + y sub a y sub b + z sub a z sub b
+.EN
+.SH
+Vector Cross Product
+.P1
+double crossvec3(Point3 a, Point3 b)
+.P2
+.EQ
+a vec ~×~ b vec ~=~ left ( y sub a z sub b - z sub a y sub b ,~
+			 z sub a x sub b - x sub a z sub b ,~
+			 x sub a y sub b - y sub a x sub b right )
+.EN
+.SH
+Vector Magnitude/Length
+.P1
+double vec3len(Point3 v)
+.P2
+.EQ
+| v vec | ~=~ sqrt { x sup 2 + y sup 2 + z sup 2 }
+.EN
+.SH
+Vector Normalization
+.P1
+Point3 normvec3(Point3 v)
+.P2
+.EQ
+n vec ~=~ left ( x over {| v vec |},~ y over {| v vec |},~ z over {| v vec |} right )
+.EN
+.NH 2
+Matrix
+.SH
+Addition
+.P1
+void addm(Matrix A, Matrix B)
+.P2
+.EQ
+( bold A + bold B ) sub {i,j} ~=~ bold A sub {i,j} + bold B sub {i,j}
+.EN
+.SH
+Substraction
+.P1
+void subm(Matrix A, Matrix B)
+.P2
+.EQ
+( bold A - bold B ) sub {i,j} ~=~ bold A sub {i,j} - bold B sub {i,j}
+.EN
+.SH
+Multiplication
+.P1
+void mulm(Matrix A, Matrix B)
+.P2
+.EQ
+left [ bold A bold B right ] sub {i,j} ~=~ sum from {k = 0} to 3-1 bold A sub {i,k} bold B sub {k,j}
+.EN
+.SH
+Transpose
+.P1
+void transposem(Matrix M)
+.P2
+.EQ
+( bold M sup T ) sub {i,j} ~=~ bold A sub {j,i}
+.EN
+.SH
+Identity
+.P1
+void identity(Matrix M)
+.P2
+.EQ
+bold M ~=~ left [ rpile {
+	1 ~ 0 ~ 0
+above	0 ~ 1 ~ 0
+above	0 ~ 0 ~ 1
+} right ]
+.EN
+.SH
+Determinant
+.P1
+double detm(Matrix M)
+.P2
+.EQ
+det( bold M ) ~=~ lpile {
+			m sub 00 ( m sub 11 m sub 22 - m sub 12 m sub 21 ) +
+above			m sub 01 ( m sub 12 m sub 20 - m sub 10 m sub 22 ) +
+above			m sub 02 ( m sub 10 m sub 21 - m sub 11 m sub 20 )
+}
+.EN
+.NH 2
+Matrix3
+.SH
+Addition
+.P1
+void addm3(Matrix3 A, Matrix3 B)
+.P2
+.EQ
+( bold A + bold B ) sub {i,j} ~=~ bold A sub {i,j} + bold B sub {i,j}
+.EN
+.SH
+Substraction
+.P1
+void subm3(Matrix3 A, Matrix3 B)
+.P2
+.EQ
+( bold A - bold B ) sub {i,j} ~=~ bold A sub {i,j} - bold B sub {i,j}
+.EN
+.SH
+Multiplication
+.P1
+void mulm3(Matrix3 A, Matrix3 B)
+.P2
+.EQ
+left [ bold A bold B right ] sub {i,j} ~=~ sum from {k = 0} to 4-1 bold A sub {i,k} bold B sub {k,j}
+.EN
+.SH
+Transpose
+.P1
+void transposem3(Matrix3 M)
+.P2
+.EQ
+( bold M sup T ) sub {i,j} ~=~ bold A sub {j,i}
+.EN
+.SH
+Identity
+.P1
+void identity3(Matrix3 M)
+.P2
+.EQ
+bold M ~=~ left [ rpile {
+	1 ~ 0 ~ 0 ~ 0
+above	0 ~ 1 ~ 0 ~ 0
+above	0 ~ 0 ~ 1 ~ 0
+above	0 ~ 0 ~ 0 ~ 1
+} right ]
+.EN
+.SH
+Determinant
+.P1
+double detm3(Matrix3 M)
+.P2
+.EQ
+det( bold M ) ~=~ rpile {
+		  m sub 00 ( m sub 11 ( m sub 22 m sub 33 - m sub 23 m sub 32 ) +
+			     m sub 12 ( m sub 23 m sub 31 - m sub 21 m sub 33 ) +
+			     m sub 13 ( m sub 21 m sub 32 - m sub 22 m sub 31 ) )
+above		 -m sub 01 ( m sub 10 ( m sub 22 m sub 33 - m sub 23 m sub 32 ) +
+			     m sub 12 ( m sub 23 m sub 30 - m sub 20 m sub 33 ) +
+			     m sub 13 ( m sub 20 m sub 32 - m sub 22 m sub 30 ) )
+above		 +m sub 02 ( m sub 10 ( m sub 21 m sub 33 - m sub 23 m sub 31 ) +
+			     m sub 11 ( m sub 23 m sub 30 - m sub 20 m sub 33 ) +
+			     m sub 13 ( m sub 20 m sub 31 - m sub 21 m sub 30 ) )
+above		 -m sub 03 ( m sub 10 ( m sub 21 m sub 32 - m sub 22 m sub 31 ) +
+			     m sub 11 ( m sub 22 m sub 30 - m sub 20 m sub 32 ) +
+			     m sub 12 ( m sub 20 m sub 31 - m sub 21 m sub 30 ) )
+}
+.EN
+.NH 2
+Quaternion
+.SH
+Addition
+.P1
+Quaternion addq(Quaternion q, Quaternion r)
+.P2
+.EQ
+q + r ~=~ ( r sub q + r sub r ,~ i sub q + i sub r ,~ j sub q + j sub r ,~ k sub q + k sub r )
+.EN
+.SH
+Substraction
+.P1
+Quaternion subq(Quaternion q, Quaternion r)
+.P2
+.EQ
+q - r ~=~ ( r sub q - r sub r ,~ i sub q - i sub r ,~ j sub q - j sub r ,~ k sub q - k sub r )
+.EN
+.SH
+Multiplication
+.P1
+Quaternion mulq(Quaternion q, Quaternion r)
+.P2
+.EQ
+q ~=~ [ r sub q ,~ v vec sub q ]
+r ~=~ [ r sub r ,~ v vec sub r ]
+qr ~=~ [ r sub q r sub r - v vec sub q • v vec sub r ,~ v vec sub r r sub q + v vec sub q r sub r + v vec sub q X v vec sub r ]
+.EN
+.SH
+Scalar Multiplication
+.P1
+Quaternion smulq(Quaternion q, double s)
+.P2
+.EQ
+qs ~=~ [ r sub q s ,~ i sub q s ,~ j sub q s ,~ k sub q s ]
+.EN
+.SH
+Inverse
+.P1
+Quaternion invq(Quaternion q)
+.P2
+.EQ
+q sup -1 ~=~ left ( r over {| q | sup 2} ,~ -i over {| q | sup 2} ,~ -j over {| q | sup 2} ,~ -k over {| q | sup 2} right )
+.EN
+.SH
+Magnitude/Length
+.P1
+double qlen(Quaternion q)
+.P2
+.EQ
+| q | ~=~ sqrt { r sup 2 + i sup 2 + j sup 2 + k sup 2 }
+.EN
+.NH 2
+RFrame
+.SH
+Change of reference
+.P1
+Point2 rframexform(Point2 p, RFrame rf)
+.P2
+.NH 2
+RFrame3
+.SH
+Change of reference
+.P1
+Point3 rframexform3(Point3 p, RFrame3 rf)
+.P2
diff --git a/doc/libgeometry.pdf b/doc/libgeometry.pdf
new file mode 100644
index 0000000..3a49480
--- /dev/null
+++ b/doc/libgeometry.pdf
diff --git a/doc/libgeometry.ps b/doc/libgeometry.ps
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index 0000000..635b9cd
--- /dev/null
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+cleartomark
+showpage
+saveobj restore
+%%EndPage: 6 6
+%%Trailer
+done
+%%DocumentFonts: LucidaSansUnicode20 LucidaSansUnicode21 LucidaSansUnicode22 S1 S LucidaSansUnicode00 LucidaSans-Demi LucidaSans-Italic LucidaTypewriter
+%%Pages: 6
diff --git a/doc/mkfile b/doc/mkfile
new file mode 100644
index 0000000..8fb6d1c
--- /dev/null
+++ b/doc/mkfile
@@ -0,0 +1,15 @@
+FONTS='.FP lucidasans'
+DOCNAME=libgeometry
+
+all:VQ: $DOCNAME.ps $DOCNAME.pdf
+
+clean:VQ:
+	rm -f $DOCNAME.ps $DOCNAME.pdf
+
+$DOCNAME.ps:V: $DOCNAME.ms
+	{echo $FONTS; cat $prereq}> _$prereq
+	eval `{doctype _$prereq} | lp -dstdout > $target && rm -f _$prereq
+
+$DOCNAME.pdf:V: $DOCNAME.ps
+	cat /sys/doc/docfonts $prereq > _$prereq
+	ps2pdf _$prereq $target && rm -f _$prereq
diff --git a/geometry.h b/geometry.h
new file mode 100644
index 0000000..0e4f748
--- /dev/null
+++ b/geometry.h
@@ -0,0 +1,110 @@
+#define DEG 0.01745329251994330	/* π/180 */
+
+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 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;
+};
+
+struct Triangle3 {
+	Point3 p0, p1, p2;
+};
+
+/* Point2 */
+Point2 Pt2(double, double, double);
+Point2 Vec2(double, double);
+Point2 addpt2(Point2, Point2);
+Point2 subpt2(Point2, Point2);
+Point2 mulpt2(Point2, double);
+Point2 divpt2(Point2, double);
+Point2 lerp2(Point2, Point2, double);
+double dotvec2(Point2, Point2);
+double vec2len(Point2);
+Point2 normvec2(Point2);
+
+/* Point3 */
+Point3 Pt3(double, double, double, double);
+Point3 Vec3(double, double, double);
+Point3 addpt3(Point3, Point3);
+Point3 subpt3(Point3, Point3);
+Point3 mulpt3(Point3, double);
+Point3 divpt3(Point3, double);
+Point3 lerp3(Point3, Point3, double);
+double dotvec3(Point3, Point3);
+Point3 crossvec3(Point3, Point3);
+double vec3len(Point3);
+Point3 normvec3(Point3);
+
+/* Matrix */
+void identity(Matrix);
+void addm(Matrix, Matrix);
+void subm(Matrix, Matrix);
+void mulm(Matrix, Matrix);
+void smulm(Matrix, double);
+void transposem(Matrix);
+double detm(Matrix);
+double tracem(Matrix);
+void adjm(Matrix);
+void invm(Matrix);
+Point2 xform(Point2, Matrix);
+
+/* Matrix3 */
+void identity3(Matrix3);
+void addm3(Matrix3, Matrix3);
+void subm3(Matrix3, Matrix3);
+void mulm3(Matrix3, Matrix3);
+void smulm3(Matrix3, double);
+void transposem3(Matrix3);
+double detm3(Matrix3);
+double tracem3(Matrix3);
+void adjm3(Matrix3);
+void invm3(Matrix3);
+Point3 xform3(Point3, Matrix3);
+
+/* Quaternion */
+Quaternion Quat(double, double, double, double);
+Quaternion Quatvec(double, Point3);
+Quaternion addq(Quaternion, Quaternion);
+Quaternion subq(Quaternion, Quaternion);
+Quaternion mulq(Quaternion, Quaternion);
+Quaternion smulq(Quaternion, double);
+Quaternion sdivq(Quaternion, double);
+double dotq(Quaternion, Quaternion);
+Quaternion invq(Quaternion);
+double qlen(Quaternion);
+Quaternion normq(Quaternion);
+Point3 qrotate(Point3, Point3, double);
+
+/* RFrame */
+Point2 rframexform(Point2, RFrame);
+Point3 rframexform3(Point3, RFrame3);
+Point2 invrframexform(Point2, RFrame);
+Point3 invrframexform3(Point3, RFrame3);
+
+/* Triangle3 */
+Point3 centroid(Triangle3);
diff --git a/matrix.c b/matrix.c
new file mode 100644
index 0000000..d640403
--- /dev/null
+++ b/matrix.c
@@ -0,0 +1,348 @@
+#include <u.h>
+#include <libc.h>
+#include <geometry.h>
+
+/* 2D */
+
+void
+identity(Matrix m)
+{
+	memset(m, 0, 3*3*sizeof(double));
+	m[0][0] = m[1][1] = m[2][2] = 1;
+}
+
+void
+addm(Matrix a, Matrix b)
+{
+	int i, j;
+
+	for(i = 0; i < 3; i++)
+		for(j = 0; j < 3; j++)
+			a[i][j] += b[i][j];
+}
+
+void
+subm(Matrix a, Matrix b)
+{
+	int i, j;
+
+	for(i = 0; i < 3; i++)
+		for(j = 0; j < 3; j++)
+			a[i][j] -= b[i][j];
+}
+
+void
+mulm(Matrix a, Matrix b)
+{
+	int i, j, k;
+	Matrix tmp;
+
+	for(i = 0; i < 3; i++)
+		for(j = 0; j < 3; j++){
+			tmp[i][j] = 0;
+			for(k = 0; k < 3; k++)
+				tmp[i][j] += a[i][k]*b[k][j];
+		}
+	memmove(a, tmp, 3*3*sizeof(double));
+}
+
+void
+smulm(Matrix m, double s)
+{
+	int i, j;
+
+	for(i = 0; i < 3; i++)
+		for(j = 0; j < 3; j++)
+			m[i][j] *= s;
+}
+
+void
+transposem(Matrix m)
+{
+	int i, j;
+	double tmp;
+
+	for(i = 0; i < 3; i++)
+		for(j = i; j < 3; j++){
+			tmp = m[i][j];
+			m[i][j] = m[j][i];
+			m[j][i] = tmp;
+		}
+}
+
+double
+detm(Matrix m)
+{
+	return m[0][0]*(m[1][1]*m[2][2] - m[1][2]*m[2][1])+
+	       m[0][1]*(m[1][2]*m[2][0] - m[1][0]*m[2][2])+
+	       m[0][2]*(m[1][0]*m[2][1] - m[1][1]*m[2][0]);
+}
+
+double
+tracem(Matrix m)
+{
+	return m[0][0] + m[1][1] + m[2][2];
+}
+
+void
+adjm(Matrix m)
+{
+	Matrix tmp;
+
+	tmp[0][0] =  m[1][1]*m[2][2] - m[1][2]*m[2][1];
+	tmp[0][1] = -m[0][1]*m[2][2] + m[0][2]*m[2][1];
+	tmp[0][2] =  m[0][1]*m[1][2] - m[0][2]*m[1][1];
+	tmp[1][0] = -m[1][0]*m[2][2] + m[1][2]*m[2][0];
+	tmp[1][1] =  m[0][0]*m[2][2] - m[0][2]*m[2][0];
+	tmp[1][2] = -m[0][0]*m[1][2] + m[0][2]*m[1][0];
+	tmp[2][0] =  m[1][0]*m[2][1] - m[1][1]*m[2][0];
+	tmp[2][1] = -m[0][0]*m[2][1] + m[0][1]*m[2][0];
+	tmp[2][2] =  m[0][0]*m[1][1] - m[0][1]*m[1][0];
+	memmove(m, tmp, 3*3*sizeof(double));
+}
+
+/* Cayley-Hamilton */
+//void
+//invertm(Matrix m)
+//{
+//	Matrix m², r;
+//	double det, trm, trm²;
+//
+//	det = detm(m);
+//	if(det == 0)
+//		return;
+//	trm = tracem(m);
+//	memmove(m², m, 3*3*sizeof(double));
+//	mulm(m², m²);
+//	trm² = tracem(m²);
+//	identity(r);
+//	smulm(r, (trm*trm - trm²)/2);
+//	smulm(m, trm);
+//	subm(r, m);
+//	addm(r, m²);
+//	smulm(r, 1/det);
+//	memmove(m, r, 3*3*sizeof(double));
+//}
+
+/* Cramer's */
+void
+invm(Matrix m)
+{
+	double det;
+
+	det = detm(m);
+	if(det == 0)
+		return; /* singular matrices are not invertible */
+	adjm(m);
+	smulm(m, 1/det);
+}
+
+Point2
+xform(Point2 p, Matrix m)
+{
+	return (Point2){
+		p.x*m[0][0] + p.y*m[0][1] + p.w*m[0][2],
+		p.x*m[1][0] + p.y*m[1][1] + p.w*m[1][2],
+		p.x*m[2][0] + p.y*m[2][1] + p.w*m[2][2]
+	};
+}
+
+/* 3D */
+
+void
+identity3(Matrix3 m)
+{
+	memset(m, 0, 4*4*sizeof(double));
+	m[0][0] = m[1][1] = m[2][2] = m[3][3] = 1;
+}
+
+void
+addm3(Matrix3 a, Matrix3 b)
+{
+	int i, j;
+
+	for(i = 0; i < 4; i++)
+		for(j = 0; j < 4; j++)
+			a[i][j] += b[i][j];
+}
+
+void
+subm3(Matrix3 a, Matrix3 b)
+{
+	int i, j;
+
+	for(i = 0; i < 4; i++)
+		for(j = 0; j < 4; j++)
+			a[i][j] -= b[i][j];
+}
+
+void
+mulm3(Matrix3 a, Matrix3 b)
+{
+	int i, j, k;
+	Matrix3 tmp;
+
+	for(i = 0; i < 4; i++)
+		for(j = 0; j < 4; j++){
+			tmp[i][j] = 0;
+			for(k = 0; k < 4; k++)
+				tmp[i][j] += a[i][k]*b[k][j];
+		}
+	memmove(a, tmp, 4*4*sizeof(double));
+}
+
+void
+smulm3(Matrix3 m, double s)
+{
+	int i, j;
+
+	for(i = 0; i < 4; i++)
+		for(j = 0; j < 4; j++)
+			m[i][j] *= s;
+}
+
+void
+transposem3(Matrix3 m)
+{
+	int i, j;
+	double tmp;
+
+	for(i = 0; i < 4; i++)
+		for(j = i; j < 4; j++){
+			tmp = m[i][j];
+			m[i][j] = m[j][i];
+			m[j][i] = tmp;
+		}
+}
+
+double
+detm3(Matrix3 m)
+{
+	return m[0][0]*(m[1][1]*(m[2][2]*m[3][3] - m[2][3]*m[3][2])+
+			m[1][2]*(m[2][3]*m[3][1] - m[2][1]*m[3][3])+
+			m[1][3]*(m[2][1]*m[3][2] - m[2][2]*m[3][1]))
+	      -m[0][1]*(m[1][0]*(m[2][2]*m[3][3] - m[2][3]*m[3][2])+
+			m[1][2]*(m[2][3]*m[3][0] - m[2][0]*m[3][3])+
+			m[1][3]*(m[2][0]*m[3][2] - m[2][2]*m[3][0]))
+	      +m[0][2]*(m[1][0]*(m[2][1]*m[3][3] - m[2][3]*m[3][1])+
+			m[1][1]*(m[2][3]*m[3][0] - m[2][0]*m[3][3])+
+			m[1][3]*(m[2][0]*m[3][1] - m[2][1]*m[3][0]))
+	      -m[0][3]*(m[1][0]*(m[2][1]*m[3][2] - m[2][2]*m[3][1])+
+			m[1][1]*(m[2][2]*m[3][0] - m[2][0]*m[3][2])+
+			m[1][2]*(m[2][0]*m[3][1] - m[2][1]*m[3][0]));
+}
+
+double
+tracem3(Matrix3 m)
+{
+	return m[0][0] + m[1][1] + m[2][2] + m[3][3];
+}
+
+void
+adjm3(Matrix3 m)
+{
+	Matrix3 tmp;
+
+	tmp[0][0]=m[1][1]*(m[2][2]*m[3][3] - m[2][3]*m[3][2])+
+		  m[2][1]*(m[1][3]*m[3][2] - m[1][2]*m[3][3])+
+		  m[3][1]*(m[1][2]*m[2][3] - m[1][3]*m[2][2]);
+	tmp[0][1]=m[0][1]*(m[2][3]*m[3][2] - m[2][2]*m[3][3])+
+		  m[2][1]*(m[0][2]*m[3][3] - m[0][3]*m[3][2])+
+		  m[3][1]*(m[0][3]*m[2][2] - m[0][2]*m[2][3]);
+	tmp[0][2]=m[0][1]*(m[1][2]*m[3][3] - m[1][3]*m[3][2])+
+		  m[1][1]*(m[0][3]*m[3][2] - m[0][2]*m[3][3])+
+		  m[3][1]*(m[0][2]*m[1][3] - m[0][3]*m[1][2]);
+	tmp[0][3]=m[0][1]*(m[1][3]*m[2][2] - m[1][2]*m[2][3])+
+		  m[1][1]*(m[0][2]*m[2][3] - m[0][3]*m[2][2])+
+		  m[2][1]*(m[0][3]*m[1][2] - m[0][2]*m[1][3]);
+	tmp[1][0]=m[1][0]*(m[2][3]*m[3][2] - m[2][2]*m[3][3])+
+		  m[2][0]*(m[1][2]*m[3][3] - m[1][3]*m[3][2])+
+		  m[3][0]*(m[1][3]*m[2][2] - m[1][2]*m[2][3]);
+	tmp[1][1]=m[0][0]*(m[2][2]*m[3][3] - m[2][3]*m[3][2])+
+		  m[2][0]*(m[0][3]*m[3][2] - m[0][2]*m[3][3])+
+		  m[3][0]*(m[0][2]*m[2][3] - m[0][3]*m[2][2]);
+	tmp[1][2]=m[0][0]*(m[1][3]*m[3][2] - m[1][2]*m[3][3])+
+		  m[1][0]*(m[0][2]*m[3][3] - m[0][3]*m[3][2])+
+		  m[3][0]*(m[0][3]*m[1][2] - m[0][2]*m[1][3]);
+	tmp[1][3]=m[0][0]*(m[1][2]*m[2][3] - m[1][3]*m[2][2])+
+		  m[1][0]*(m[0][3]*m[2][2] - m[0][2]*m[2][3])+
+		  m[2][0]*(m[0][2]*m[1][3] - m[0][3]*m[1][2]);
+	tmp[2][0]=m[1][0]*(m[2][1]*m[3][3] - m[2][3]*m[3][1])+
+		  m[2][0]*(m[1][3]*m[3][1] - m[1][1]*m[3][3])+
+		  m[3][0]*(m[1][1]*m[2][3] - m[1][3]*m[2][1]);
+	tmp[2][1]=m[0][0]*(m[2][3]*m[3][1] - m[2][1]*m[3][3])+
+		  m[2][0]*(m[0][1]*m[3][3] - m[0][3]*m[3][1])+
+		  m[3][0]*(m[0][3]*m[2][1] - m[0][1]*m[2][3]);
+	tmp[2][2]=m[0][0]*(m[1][1]*m[3][3] - m[1][3]*m[3][1])+
+		  m[1][0]*(m[0][3]*m[3][1] - m[0][1]*m[3][3])+
+		  m[3][0]*(m[0][1]*m[1][3] - m[0][3]*m[1][1]);
+	tmp[2][3]=m[0][0]*(m[1][3]*m[2][1] - m[1][1]*m[2][3])+
+		  m[1][0]*(m[0][1]*m[2][3] - m[0][3]*m[2][1])+
+		  m[2][0]*(m[0][3]*m[1][1] - m[0][1]*m[1][3]);
+	tmp[3][0]=m[1][0]*(m[2][2]*m[3][1] - m[2][1]*m[3][2])+
+		  m[2][0]*(m[1][1]*m[3][2] - m[1][2]*m[3][1])+
+		  m[3][0]*(m[1][2]*m[2][1] - m[1][1]*m[2][2]);
+	tmp[3][1]=m[0][0]*(m[2][1]*m[3][2] - m[2][2]*m[3][1])+
+		  m[2][0]*(m[0][2]*m[3][1] - m[0][1]*m[3][2])+
+		  m[3][0]*(m[0][1]*m[2][2] - m[0][2]*m[2][1]);
+	tmp[3][2]=m[0][0]*(m[1][2]*m[3][1] - m[1][1]*m[3][2])+
+		  m[1][0]*(m[0][1]*m[3][2] - m[0][2]*m[3][1])+
+		  m[3][0]*(m[0][2]*m[1][1] - m[0][1]*m[1][2]);
+	tmp[3][3]=m[0][0]*(m[1][1]*m[2][2] - m[1][2]*m[2][1])+
+		  m[1][0]*(m[0][2]*m[2][1] - m[0][1]*m[2][2])+
+		  m[2][0]*(m[0][1]*m[1][2] - m[0][2]*m[1][1]);
+	memmove(m, tmp, 4*4*sizeof(double));
+}
+
+/* Cayley-Hamilton */
+//void
+//invertm3(Matrix3 m)
+//{
+//	Matrix3 m², m³, r;
+//	double det, trm, trm², trm³;
+//
+//	det = detm3(m);
+//	if(det == 0)
+//		return;
+//	trm = tracem3(m);
+//	memmove(m³, m, 4*4*sizeof(double));
+//	mulm(m³, m³);
+//	mulm(m³, m);
+//	trm³ = tracem3(m³);
+//	memmove(m², m, 4*4*sizeof(double));
+//	mulm(m², m²);
+//	trm² = tracem3(m²);
+//	identity3(r);
+//	smulm3(r, (trm*trm*trm - 3*trm*trm² + 2*trm³)/6);
+//	smulm3(m, (trm*trm - trm²)/2);
+//	smulm3(m², trm);
+//	subm(r, m);
+//	addm(r, m²);
+//	subm(r, m³);
+//	smulm(r, 1/det);
+//	memmove(m, r, 4*4*sizeof(double));
+//}
+
+/* Cramer's */
+void
+invm3(Matrix3 m)
+{
+	double det;
+
+	det = detm3(m);
+	if(det == 0)
+		return; /* singular matrices are not invertible */
+	adjm3(m);
+	smulm3(m, 1/det);
+}
+
+Point3
+xform3(Point3 p, Matrix3 m)
+{
+	return (Point3){
+		p.x*m[0][0] + p.y*m[0][1] + p.z*m[0][2] + p.w*m[0][3],
+		p.x*m[1][0] + p.y*m[1][1] + p.z*m[1][2] + p.w*m[1][3],
+		p.x*m[2][0] + p.y*m[2][1] + p.z*m[2][2] + p.w*m[2][3],
+		p.x*m[3][0] + p.y*m[3][1] + p.z*m[3][2] + p.w*m[3][3],
+	};
+}
diff --git a/mkfile b/mkfile
new file mode 100644
index 0000000..85f7c76
--- /dev/null
+++ b/mkfile
@@ -0,0 +1,15 @@
+</$objtype/mkfile
+
+LIB=libgeometry.a$O
+OFILES=\
+	point.$O\
+	matrix.$O\
+	quaternion.$O\
+	rframe.$O\
+	triangle.$O\
+
+HFILES=geometry.h
+
+CFLAGS=$CFLAGS -I.
+
+</sys/src/cmd/mklib
diff --git a/point.c b/point.c
new file mode 100644
index 0000000..4c0731d
--- /dev/null
+++ b/point.c
@@ -0,0 +1,161 @@
+#include <u.h>
+#include <libc.h>
+#include <geometry.h>
+
+/* 2D */
+
+Point2
+Pt2(double x, double y, double w)
+{
+	return (Point2){x, y, w};
+}
+
+Point2
+Vec2(double x, double y)
+{
+	return (Point2){x, y, 0};
+}
+
+Point2
+addpt2(Point2 a, Point2 b)
+{
+	return (Point2){a.x+b.x, a.y+b.y, a.w+b.w};
+}
+
+Point2
+subpt2(Point2 a, Point2 b)
+{
+	return (Point2){a.x-b.x, a.y-b.y, a.w-b.w};
+}
+
+Point2
+mulpt2(Point2 p, double s)
+{
+	return (Point2){p.x*s, p.y*s, p.w*s};
+}
+
+Point2
+divpt2(Point2 p, double s)
+{
+	return (Point2){p.x/s, p.y/s, p.w/s};
+}
+
+Point2
+lerp2(Point2 a, Point2 b, double t)
+{
+	if(t < 0) t = 0;
+	if(t > 1) t = 1;
+	return (Point2){
+		(1 - t)*a.x + t*b.x,
+		(1 - t)*a.y + t*b.y,
+		(1 - t)*a.w + t*b.w
+	};
+}
+
+double
+dotvec2(Point2 a, Point2 b)
+{
+	return a.x*b.x + a.y*b.y;
+}
+
+double
+vec2len(Point2 v)
+{
+	return sqrt(dotvec2(v, v));
+}
+
+Point2
+normvec2(Point2 v)
+{
+	double len;
+
+	len = vec2len(v);
+	if(len == 0)
+		return (Point2){0, 0, 0};
+	return (Point2){v.x/len, v.y/len, 0};
+}
+
+/* 3D */
+
+Point3
+Pt3(double x, double y, double z, double w)
+{
+	return (Point3){x, y, z, w};
+}
+
+Point3
+Vec3(double x, double y, double z)
+{
+	return (Point3){x, y, z, 0};
+}
+
+Point3
+addpt3(Point3 a, Point3 b)
+{
+	return (Point3){a.x+b.x, a.y+b.y, a.z+b.z, a.w+b.w};
+}
+
+Point3
+subpt3(Point3 a, Point3 b)
+{
+	return (Point3){a.x-b.x, a.y-b.y, a.z-b.z, a.w-b.w};
+}
+
+Point3
+mulpt3(Point3 p, double s)
+{
+	return (Point3){p.x*s, p.y*s, p.z*s, p.w*s};
+}
+
+Point3
+divpt3(Point3 p, double s)
+{
+	return (Point3){p.x/s, p.y/s, p.z/s, p.w/s};
+}
+
+Point3
+lerp3(Point3 a, Point3 b, double t)
+{
+	if(t < 0) t = 0;
+	if(t > 1) t = 1;
+	return (Point3){
+		(1 - t)*a.x + t*b.x,
+		(1 - t)*a.y + t*b.y,
+		(1 - t)*a.z + t*b.z,
+		(1 - t)*a.w + t*b.w
+	};
+}
+
+double
+dotvec3(Point3 a, Point3 b)
+{
+	return a.x*b.x + a.y*b.y + a.z*b.z;
+}
+
+Point3
+crossvec3(Point3 a, Point3 b)
+{
+	return (Point3){
+		a.y*b.z - a.z*b.y,
+		a.z*b.x - a.x*b.z,
+		a.x*b.y - a.y*b.x,
+		0
+	};
+}
+
+double
+vec3len(Point3 v)
+{
+	return sqrt(dotvec3(v, v));
+}
+
+Point3
+normvec3(Point3 v)
+{
+	double len;
+
+	len = vec3len(v);
+	if(len == 0)
+		return (Point3){0, 0, 0, 0};
+	return (Point3){v.x/len, v.y/len, v.z/len, 0};
+}
diff --git a/quaternion.c b/quaternion.c
new file mode 100644
index 0000000..3297788
--- /dev/null
+++ b/quaternion.c
@@ -0,0 +1,90 @@
+#include <u.h>
+#include <libc.h>
+#include <geometry.h>
+
+Quaternion
+Quat(double r, double i, double j, double k)
+{
+	return (Quaternion){r, i, j, k};
+}
+
+Quaternion
+Quatvec(double s, Point3 v)
+{
+	return (Quaternion){s, v.x, v.y, v.z};
+}
+
+Quaternion
+addq(Quaternion a, Quaternion b)
+{
+	return (Quaternion){a.r+b.r, a.i+b.i, a.j+b.j, a.k+b.k};
+}
+
+Quaternion
+subq(Quaternion a, Quaternion b)
+{
+	return (Quaternion){a.r-b.r, a.i-b.i, a.j-b.j, a.k-b.k};
+}
+
+Quaternion
+mulq(Quaternion q, Quaternion r)
+{
+	Point3 qv, rv, tmp;
+
+	qv = Vec3(q.i, q.j, q.k);
+	rv = Vec3(r.i, r.j, r.k);
+	tmp = addpt3(addpt3(mulpt3(rv, q.r), mulpt3(qv, r.r)), crossvec3(qv, rv));
+	return (Quaternion){q.r*r.r - dotvec3(qv, rv), tmp.x, tmp.y, tmp.z};
+}
+
+Quaternion
+smulq(Quaternion q, double s)
+{
+	return (Quaternion){q.r*s, q.i*s, q.j*s, q.k*s};
+}
+
+Quaternion
+sdivq(Quaternion q, double s)
+{
+	return (Quaternion){q.r/s, q.i/s, q.j/s, q.k/s};
+}
+
+double
+dotq(Quaternion q, Quaternion r)
+{
+	return q.r*r.r + q.i*r.i + q.j*r.j + q.k*r.k;
+}
+
+Quaternion
+invq(Quaternion q)
+{
+	double len²;
+
+	len² = dotq(q, q);
+	if(len² == 0)
+		return (Quaternion){0, 0, 0, 0};
+	return (Quaternion){q.r/len², -q.i/len², -q.j/len², -q.k/len²};
+}
+
+double
+qlen(Quaternion q)
+{
+	return sqrt(q.r*q.r + q.i*q.i + q.j*q.j + q.k*q.k);
+}
+
+Quaternion
+normq(Quaternion q)
+{
+	return sdivq(q, qlen(q));
+}
+
+Point3
+qrotate(Point3 p, Point3 axis, double angle)
+{
+	Quaternion qaxis, qr;
+
+	angle /= 2;
+	qaxis = Quatvec(cos(angle), mulpt3(axis, sin(angle)));
+	qr = mulq(mulq(qaxis, Quatvec(0, p)), invq(qaxis));
+	return Vec3(qr.i, qr.j, qr.k);
+}
diff --git a/rframe.c b/rframe.c
new file mode 100644
index 0000000..2f7c682
--- /dev/null
+++ b/rframe.c
@@ -0,0 +1,51 @@
+#include <u.h>
+#include <libc.h>
+#include <geometry.h>
+
+Point2
+rframexform(Point2 p, RFrame rf)
+{
+	Matrix m = {
+		rf.bx.x, rf.bx.y, -dotvec2(rf.bx, rf.p),
+		rf.by.x, rf.by.y, -dotvec2(rf.by, rf.p),
+		0, 0, 1,
+	};
+	return xform(p, m);
+}
+
+Point3
+rframexform3(Point3 p, RFrame3 rf)
+{
+	Matrix3 m = {
+		rf.bx.x, rf.bx.y, rf.bx.z, -dotvec3(rf.bx, rf.p),
+		rf.by.x, rf.by.y, rf.by.z, -dotvec3(rf.by, rf.p),
+		rf.bz.x, rf.bz.y, rf.bz.z, -dotvec3(rf.bz, rf.p),
+		0, 0, 0, 1,
+	};
+	return xform3(p, m);
+}
+
+Point2
+invrframexform(Point2 p, RFrame rf)
+{
+	Matrix m = {
+		rf.bx.x, rf.bx.y, -dotvec2(rf.bx, rf.p),
+		rf.by.x, rf.by.y, -dotvec2(rf.by, rf.p),
+		0, 0, 1,
+	};
+	invm(m);
+	return xform(p, m);
+}
+
+Point3
+invrframexform3(Point3 p, RFrame3 rf)
+{
+	Matrix3 m = {
+		rf.bx.x, rf.bx.y, rf.bx.z, -dotvec3(rf.bx, rf.p),
+		rf.by.x, rf.by.y, rf.by.z, -dotvec3(rf.by, rf.p),
+		rf.bz.x, rf.bz.y, rf.bz.z, -dotvec3(rf.bz, rf.p),
+		0, 0, 0, 1,
+	};
+	invm3(m);
+	return xform3(p, m);
+}
diff --git a/triangle.c b/triangle.c
new file mode 100644
index 0000000..3fc0cc9
--- /dev/null
+++ b/triangle.c
@@ -0,0 +1,9 @@
+#include <u.h>
+#include <libc.h>
+#include <geometry.h>
+
+Point3
+centroid(Triangle3 t)
+{
+	return divpt3(addpt3(t.p0, addpt3(t.p1, t.p2)), 3);
+}