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| -rw-r--r-- | doc/libgeometry.ms | 429 | ||||
| -rw-r--r-- | doc/libgeometry.pdf | bin | 0 -> 35120 bytes | |||
| -rw-r--r-- | doc/libgeometry.ps | 2799 | ||||
| -rw-r--r-- | doc/mkfile | 15 | ||||
| -rw-r--r-- | geometry.h | 110 | ||||
| -rw-r--r-- | matrix.c | 348 | ||||
| -rw-r--r-- | mkfile | 15 | ||||
| -rw-r--r-- | point.c | 161 | ||||
| -rw-r--r-- | quaternion.c | 90 | ||||
| -rw-r--r-- | rframe.c | 51 | ||||
| -rw-r--r-- | triangle.c | 9 |
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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 Binary files differnew file mode 100644 index 0000000..3a49480 --- /dev/null +++ b/doc/libgeometry.pdf diff --git a/doc/libgeometry.ps b/doc/libgeometry.ps new file mode 100644 index 0000000..635b9cd --- /dev/null +++ b/doc/libgeometry.ps @@ -0,0 +1,2799 @@ +%!PS-Adobe-2.0 +%%Version: 0.1 +%%Creator: troff, Plan 9 edition +%%DocumentFonts: (atend) 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L\363pez) 2469 1480 w +(rgl@antares-labs.eu) 2377 1760 w +10 /LucidaSansUnicode00 f +(Antares Telecom Laboratories) 2156 1960 w +(Albatera, Alicante) 2451 2100 w +10 /LucidaSans-Demi f +(1.) 720 2700 w +(Data Structures) 873 2700 w +(1.1.) 720 2940 w +(Point2) 962 2940 w +9 /LucidaTypewriter f +(struct) 920 3110 w +(Point2) 1375 3110 w +({) 1830 3110 w +(double) 1440 3220 w +(x,) 1895 3220 w +(y,) 2090 3220 w +(w;) 2285 3220 w +(};) 920 3330 w +10 /LucidaSans-Italic f +(Point2) 970 3546 w +10 /LucidaSansUnicode00 f +(represents) 1324 3546 w +(a) 1886 3546 w +(point) 1989 3546 w +(in) 2289 3546 w +(two-dimensional) 2428 3546 w +(projective) 3299 3546 w +(space,) 3823 3546 w +(which) 4179 3546 w +(itself) 4508 3546 w +(is) 4795 3546 w +(an) 4923 3546 w +(extension) 720 3666 w +(of) 1240 3666 w +(the) 1383 3666 w +(two-dimensional) 1583 3666 w +(euclidean) 2451 3666 w +(space) 2959 3666 w +(that) 3280 3666 w +(allows) 3516 3666 w +(us) 3862 3666 w +(to) 4019 3666 w +(work) 4161 3666 w +(with) 4442 3666 w +(vectors) 4691 3666 w +(and compose affine transformations in a friendly manner.) 720 3786 w +(A point) 3589 3786 w +(\() 2696 3966 w +(x) 2737 3966 w +(,) 2806 3966 w +(y) 2846 3966 w +(,) 2906 3966 w +(w) 2946 3966 w +(\)) 3031 3966 w +(made) 720 4146 w +(out) 1027 4146 w +(of) 1227 4146 w +(homogenous) 1365 4146 w +(coordinates) 2036 4146 w +10 /LucidaSans-Italic f +(x) 2643 4146 w +10 /LucidaSansUnicode00 f +(,) 2697 4146 w +10 /LucidaSans-Italic f +(y) 2769 4146 w +10 /LucidaSansUnicode00 f +(,) 2824 4146 w +(and) 2896 4146 w +10 /LucidaSans-Italic f +(w) 3116 4146 w +10 /LucidaSansUnicode00 f +(,) 3194 4146 w +(yields) 3267 4146 w +(a) 3588 4146 w +(point) 3684 4146 w +(with) 3977 4146 w +(cartesian) 4223 4146 w +(coordi\255) 4701 4146 w +(nates) 720 4266 w +(\() 2588 4446 w +(x) 2629 4446 w +(/) 2698 4446 w +(w) 2759 4446 w +(,) 2844 4446 w +(y) 2884 4446 w +(/) 2944 4446 w +(w) 3005 4446 w +(\)) 3090 4446 w +(.) 3139 4446 w +10 /LucidaSans-Demi f +(1.2.) 720 4746 w +(Point3) 962 4746 w +9 /LucidaTypewriter f +(struct) 920 4916 w +(Point3) 1375 4916 w +({) 1830 4916 w +(double) 1440 5026 w +(x,) 1895 5026 w +(y,) 2090 5026 w +(z,) 2285 5026 w +(w;) 2480 5026 w +(};) 920 5136 w +10 /LucidaSans-Italic f +(Point3) 970 5352 w +10 /LucidaSansUnicode00 f +(is a point in three-dimensional projective space.) 1309 5352 w +10 /LucidaSans-Demi f +(1.3.) 720 5592 w +(Matrix) 962 5592 w +9 /LucidaTypewriter f +(typedef) 920 5762 w +(double) 1440 5762 w +(Matrix[3][3];) 1895 5762 w +10 /LucidaSans-Italic f +(Matrix) 970 5978 w +10 /LucidaSansUnicode00 f +(represents) 1341 5978 w +(a) 1914 5978 w +(3x3) 2028 5978 w +(matrix,) 2274 5978 w +(thought) 2681 5978 w +(to) 3123 5978 w +(compose) 3280 5978 w +(affine) 3775 5978 w +(transformations) 4110 5978 w +(to) 4942 5978 w +(apply to homogeneous 2D points.) 720 6098 w +10 /LucidaSans-Demi f +(1.4.) 720 6338 w +(Matrix3) 962 6338 w +9 /LucidaTypewriter f +(typedef) 920 6508 w +(double) 1440 6508 w +(Matrix3[4][4];) 1895 6508 w +10 /LucidaSans-Italic f +(Matrix3) 970 6724 w +10 /LucidaSansUnicode00 f +(represents) 1398 6724 w +(a) 1964 6724 w +(4x4) 2071 6724 w +(matrix,) 2310 6724 w +(thought) 2711 6724 w +(to) 3147 6724 w +(compose) 3298 6724 w +(affine) 3787 6724 w +(transformations) 4116 6724 w +(to) 4942 6724 w +(apply to homogeneous 3D points.) 720 6844 w +8 /S1 f +(__________________) 720 6980 w +8 /LucidaSansUnicode00 f +(ACHTUNG!) 720 7080 w +(this) 1163 7080 w +(is) 1333 7080 w +(a) 1423 7080 w +8 /LucidaSans-Demi f +(WORK) 1493 7080 w +(IN) 1769 7080 w +(PROGRESS) 1883 7080 w +cleartomark +showpage +saveobj restore +%%EndPage: 1 1 +%%Page: 2 2 +/saveobj save def +mark +2 pagesetup +10 /LucidaSansUnicode00 f +(\255 2 \255) 2783 480 w +10 /LucidaSans-Demi f +(1.5.) 720 840 w +(Quaternion) 962 840 w +9 /LucidaTypewriter f +(struct) 920 1010 w +(Quaternion) 1375 1010 w +({) 2090 1010 w +(double) 1440 1120 w +(r,) 1895 1120 w +(i,) 2090 1120 w +(j,) 2285 1120 w +(k;) 2480 1120 w +(};) 920 1230 w +10 /LucidaSans-Italic f +(Quaternions) 970 1446 w +10 /LucidaSansUnicode00 f +(are) 1623 1446 w +(a) 1829 1446 w +(numbering) 1938 1446 w +(system) 2522 1446 w +(that) 2916 1446 w +(extends) 3161 1446 w +(the) 3601 1446 w +(complex) 3810 1446 w +(numbers) 4279 1446 w +(up) 4762 1446 w +(to) 4942 1446 w +(four-dimensional) 720 1566 w +(space,) 1607 1566 w +(and) 1953 1566 w +(are) 2171 1566 w +(used) 2361 1566 w +(to) 2631 1566 w +(apply) 2767 1566 w +(rotations) 3067 1566 w +(and) 3539 1566 w +(model) 3757 1566 w +(mechanical) 4096 1566 w +(interac\255) 4676 1566 w +(tions) 720 1686 w +(in) 1015 1686 w +(3D) 1161 1686 w +(space.) 1354 1686 w +(Their) 1749 1686 w +(main) 2055 1686 w +(advantages) 2349 1686 w +(with) 2952 1686 w +(respect) 3212 1686 w +(to) 3622 1686 w +(their) 3775 1686 w +(matrix) 4055 1686 w +(relatives) 4426 1686 w +(are) 4888 1686 w +(increased computational and storage performance and gimbal lock avoidance.) 720 1806 w +10 /LucidaSans-Demi f +(2.) 720 2046 w +(Algorithms) 873 2046 w +(2.1.) 720 2286 w +(Point2) 962 2286 w +(Addition) 720 2526 w +9 /LucidaTypewriter f +(Point2) 920 2696 w +(addpt2\(Point2) 1375 2696 w +(a,) 2285 2696 w +(Point2) 2480 2696 w +(b\)) 2935 2696 w +10 /LucidaSansUnicode00 f +(a) 2051 2971 w +10 /S f +(+) 2122 2971 w +10 /LucidaSansUnicode00 f +(b) 2193 2971 w +10 /S f +(=) 2312 2971 w +10 /LucidaSansUnicode00 f +10 /S f +(\354) 2431 2934 w +(\356) 2431 3034 w +10 /LucidaSansUnicode00 f +(x) 2480 2971 w +7 /LucidaSansUnicode00 f +(a) 2552 2991 w +10 /S f +(+) 2615 2971 w +10 /LucidaSansUnicode00 f +(x) 2686 2971 w +7 /LucidaSansUnicode00 f +(b) 2758 2991 w +10 /LucidaSansUnicode00 f +(,) 2818 2971 w +(y) 2898 2971 w +7 /LucidaSansUnicode00 f +(a) 2961 2991 w +10 /S f +(+) 3024 2971 w +10 /LucidaSansUnicode00 f +(y) 3095 2971 w +7 /LucidaSansUnicode00 f +(b) 3158 2991 w +10 /LucidaSansUnicode00 f +(,) 3218 2971 w +(w) 3298 2971 w +7 /LucidaSansUnicode00 f +(a) 3386 2991 w +10 /S f +(+) 3449 2971 w +10 /LucidaSansUnicode00 f +(w) 3520 2971 w +7 /LucidaSansUnicode00 f +(b) 3608 2991 w +10 /S f +(\374) 3660 2934 w +(\376) 3660 3034 w +10 /LucidaSans-Demi f +(Substraction) 720 3326 w +9 /LucidaTypewriter f +(Point2) 920 3496 w +(subpt2\(Point2) 1375 3496 w +(a,) 2285 3496 w +(Point2) 2480 3496 w +(b\)) 2935 3496 w +10 /LucidaSansUnicode00 f +(a) 2051 3771 w +10 /S f +(-) 2122 3771 w +10 /LucidaSansUnicode00 f +(b) 2193 3771 w +10 /S f +(=) 2312 3771 w +10 /LucidaSansUnicode00 f +10 /S f +(\354) 2431 3734 w +(\356) 2431 3834 w +10 /LucidaSansUnicode00 f +(x) 2480 3771 w +7 /LucidaSansUnicode00 f +(a) 2552 3791 w +10 /S f +(-) 2615 3771 w +10 /LucidaSansUnicode00 f +(x) 2686 3771 w +7 /LucidaSansUnicode00 f +(b) 2758 3791 w +10 /LucidaSansUnicode00 f +(,) 2818 3771 w +(y) 2898 3771 w +7 /LucidaSansUnicode00 f +(a) 2961 3791 w +10 /S f +(-) 3024 3771 w +10 /LucidaSansUnicode00 f +(y) 3095 3771 w +7 /LucidaSansUnicode00 f +(b) 3158 3791 w +10 /LucidaSansUnicode00 f +(,) 3218 3771 w +(w) 3298 3771 w +7 /LucidaSansUnicode00 f +(a) 3386 3791 w +10 /S f +(-) 3449 3771 w +10 /LucidaSansUnicode00 f +(w) 3520 3771 w +7 /LucidaSansUnicode00 f +(b) 3608 3791 w +10 /S f +(\374) 3660 3734 w +(\376) 3660 3834 w +10 /LucidaSans-Demi f +(Multiplication) 720 4126 w +9 /LucidaTypewriter f +(Point2) 920 4296 w +(mulpt2\(Point2) 1375 4296 w +(p,) 2285 4296 w +(double) 2480 4296 w +(s\)) 2935 4296 w +10 /LucidaSansUnicode00 f +(p) 2395 4536 w +(*) 2466 4536 w +(s) 2522 4536 w +10 /S f +(=) 2629 4536 w +10 /LucidaSansUnicode00 f +10 /S f +(\354) 2748 4499 w +(\356) 2748 4599 w +10 /LucidaSansUnicode00 f +(xs) 2797 4546 w +(,) 2917 4546 w +(ys) 2997 4546 w +(,) 3108 4546 w +(ws) 3188 4546 w +10 /S f +(\374) 3316 4499 w +(\376) 3316 4599 w +10 /LucidaSans-Demi f +(Division) 720 4841 w +9 /LucidaTypewriter f +(Point2) 920 5011 w +(divpt2\(Point2) 1375 5011 w +(p,) 2285 5011 w +(double) 2480 5011 w +(s\)) 2935 5011 w +10 /LucidaSansUnicode00 f +(p) 2394 5336 w +(/) 2465 5336 w +(s) 2526 5336 w +10 /S f +(=) 2633 5336 w +10 /LucidaSansUnicode00 f +10 /S f +(\354) 2752 5249 w +(\357) 2752 5349 w +(\356) 2752 5449 w +10 /LucidaSansUnicode00 f +(s) 2831 5406 w +(x) 2826 5276 w +10 /S1 f +(_) 2811 5306 w +(_) 2852 5306 w +10 /LucidaSansUnicode00 f +(,) 2920 5336 w +(s) 3025 5406 w +(y) 3025 5276 w +10 /S1 f +(_) 3010 5306 w +(_) 3042 5306 w +10 /LucidaSansUnicode00 f +(,) 3110 5336 w +(s) 3228 5406 w +(w) 3215 5276 w +10 /S1 f +(_) 3200 5306 w +(__) 3207 5306 w +10 /S f +(\374) 3317 5249 w +(\357) 3317 5349 w +(\376) 3317 5449 w +10 /LucidaSans-Demi f +(Vector Dot Product) 720 5741 w +9 /LucidaTypewriter f +(double) 920 5911 w +(dotvec2\(Point2) 1375 5911 w +(a,) 2350 5911 w +(Point2) 2545 5911 w +(b\)) 3000 5911 w +10 /LucidaSansUnicode00 f +(a) 2375 6156 w +7 /LucidaSansUnicode21 f +(\222) 2370 6106 w +10 /LucidaSansUnicode00 f +10 /LucidaSansUnicode20 f +(") 2479 6156 w +10 /LucidaSansUnicode00 f +(b) 2583 6156 w +7 /LucidaSansUnicode21 f +(\222) 2582 6081 w +10 /LucidaSansUnicode00 f +10 /S f +(=) 2703 6156 w +10 /LucidaSansUnicode00 f +(x) 2814 6156 w +7 /LucidaSansUnicode00 f +(a) 2886 6176 w +10 /LucidaSansUnicode00 f +(x) 2941 6156 w +7 /LucidaSansUnicode00 f +(b) 3013 6176 w +10 /S f +(+) 3081 6156 w +10 /LucidaSansUnicode00 f +(y) 3152 6156 w +7 /LucidaSansUnicode00 f +(a) 3215 6176 w +10 /LucidaSansUnicode00 f +(y) 3270 6156 w +7 /LucidaSansUnicode00 f +(b) 3333 6176 w +10 /LucidaSans-Demi f +(Vector Magnitude/Length) 720 6456 w +9 /LucidaTypewriter f +(double) 920 6626 w +(vec2len\(Point2) 1375 6626 w +(v\)) 2350 6626 w +10 /LucidaSansUnicode00 f +(|) 2509 6901 w +(v) 2557 6884 w +7 /LucidaSansUnicode21 f +(\222) 2550 6834 w +10 /LucidaSansUnicode00 f +(|) 2609 6901 w +10 /S f +(=) 2705 6884 w +10 /LucidaSansUnicode00 f +12 /LucidaSansUnicode22 f +(\032) 2816 6884 w +12 /S f +(`) 2913 6884 w +(`````) 2943 6884 w +10 /LucidaSansUnicode00 f +(x) 2921 6884 w +7 /LucidaSansUnicode00 f +(2) 2987 6844 w +10 /S f +(+) 3055 6884 w +10 /LucidaSansUnicode00 f +(y) 3126 6884 w +7 /LucidaSansUnicode00 f +(2) 3183 6844 w +cleartomark +showpage +saveobj restore +%%EndPage: 2 2 +%%Page: 3 3 +/saveobj save def +mark +3 pagesetup +10 /LucidaSansUnicode00 f +(\255 3 \255) 2783 480 w +10 /LucidaSans-Demi f +(Vector Normalization) 720 840 w +9 /LucidaTypewriter f +(Point2) 920 1010 w +(normvec2\(Point2) 1375 1010 w +(v\)) 2415 1010 w +10 /LucidaSansUnicode00 f +(n) 2502 1335 w +7 /LucidaSansUnicode21 f +(\222) 2500 1285 w +10 /LucidaSansUnicode00 f +10 /S f +(=) 2620 1335 w +10 /LucidaSansUnicode00 f +10 /S f +(\354) 2739 1248 w +(\357) 2739 1348 w +(\356) 2739 1448 w +10 /LucidaSansUnicode00 f +(|) 2813 1447 w +(v) 2861 1430 w +7 /LucidaSansUnicode21 f +(\222) 2854 1380 w +10 /LucidaSansUnicode00 f +(|) 2913 1447 w +(x) 2841 1275 w +10 /S1 f +(_) 2799 1305 w +(__) 2845 1305 w +10 /LucidaSansUnicode00 f +(,) 2963 1335 w +(|) 3068 1447 w +(v) 3116 1430 w +7 /LucidaSansUnicode21 f +(\222) 3109 1380 w +10 /LucidaSansUnicode00 f +(|) 3168 1447 w +(y) 3100 1275 w +10 /S1 f +(_) 3053 1305 w +(__) 3099 1305 w +10 /S f +(\374) 3209 1248 w +(\357) 3209 1348 w +(\376) 3209 1448 w +10 /LucidaSans-Demi f +(2.2.) 720 1740 w +(Point3) 962 1740 w +(Addition) 720 1980 w +9 /LucidaTypewriter f +(Point3) 920 2150 w +(addpt3\(Point3) 1375 2150 w +(a,) 2285 2150 w +(Point3) 2480 2150 w +(b\)) 2935 2150 w +10 /LucidaSansUnicode00 f +(a) 1846 2425 w +10 /S f +(+) 1917 2425 w +10 /LucidaSansUnicode00 f +(b) 1988 2425 w +10 /S f +(=) 2107 2425 w +10 /LucidaSansUnicode00 f +10 /S f +(\354) 2226 2388 w +(\356) 2226 2488 w +10 /LucidaSansUnicode00 f +(x) 2275 2425 w +7 /LucidaSansUnicode00 f +(a) 2347 2445 w +10 /S f +(+) 2410 2425 w +10 /LucidaSansUnicode00 f +(x) 2481 2425 w +7 /LucidaSansUnicode00 f +(b) 2553 2445 w +10 /LucidaSansUnicode00 f +(,) 2613 2425 w +(y) 2693 2425 w +7 /LucidaSansUnicode00 f +(a) 2756 2445 w +10 /S f +(+) 2819 2425 w +10 /LucidaSansUnicode00 f +(y) 2890 2425 w +7 /LucidaSansUnicode00 f +(b) 2953 2445 w +10 /LucidaSansUnicode00 f +(,) 3013 2425 w +(z) 3093 2425 w +7 /LucidaSansUnicode00 f +(a) 3161 2445 w +10 /S f +(+) 3224 2425 w +10 /LucidaSansUnicode00 f +(z) 3295 2425 w +7 /LucidaSansUnicode00 f +(b) 3363 2445 w +10 /LucidaSansUnicode00 f +(,) 3423 2425 w +(w) 3503 2425 w +7 /LucidaSansUnicode00 f +(a) 3591 2445 w +10 /S f +(+) 3654 2425 w +10 /LucidaSansUnicode00 f +(w) 3725 2425 w +7 /LucidaSansUnicode00 f +(b) 3813 2445 w +10 /S f +(\374) 3865 2388 w +(\376) 3865 2488 w +10 /LucidaSans-Demi f +(Substraction) 720 2780 w +9 /LucidaTypewriter f +(Point3) 920 2950 w +(subpt3\(Point3) 1375 2950 w +(a,) 2285 2950 w +(Point3) 2480 2950 w +(b\)) 2935 2950 w +10 /LucidaSansUnicode00 f +(a) 1846 3225 w +10 /S f +(-) 1917 3225 w +10 /LucidaSansUnicode00 f +(b) 1988 3225 w +10 /S f +(=) 2107 3225 w +10 /LucidaSansUnicode00 f +10 /S f +(\354) 2226 3188 w +(\356) 2226 3288 w +10 /LucidaSansUnicode00 f +(x) 2275 3225 w +7 /LucidaSansUnicode00 f +(a) 2347 3245 w +10 /S f +(-) 2410 3225 w +10 /LucidaSansUnicode00 f +(x) 2481 3225 w +7 /LucidaSansUnicode00 f +(b) 2553 3245 w +10 /LucidaSansUnicode00 f +(,) 2613 3225 w +(y) 2693 3225 w +7 /LucidaSansUnicode00 f +(a) 2756 3245 w +10 /S f +(-) 2819 3225 w +10 /LucidaSansUnicode00 f +(y) 2890 3225 w +7 /LucidaSansUnicode00 f +(b) 2953 3245 w +10 /LucidaSansUnicode00 f +(,) 3013 3225 w +(z) 3093 3225 w +7 /LucidaSansUnicode00 f +(a) 3161 3245 w +10 /S f +(-) 3224 3225 w +10 /LucidaSansUnicode00 f +(z) 3295 3225 w +7 /LucidaSansUnicode00 f +(b) 3363 3245 w +10 /LucidaSansUnicode00 f +(,) 3423 3225 w +(w) 3503 3225 w +7 /LucidaSansUnicode00 f +(a) 3591 3245 w +10 /S f +(-) 3654 3225 w +10 /LucidaSansUnicode00 f +(w) 3725 3225 w +7 /LucidaSansUnicode00 f +(b) 3813 3245 w +10 /S f +(\374) 3865 3188 w +(\376) 3865 3288 w +10 /LucidaSans-Demi f +(Multiplication) 720 3580 w +9 /LucidaTypewriter f +(Point3) 920 3750 w +(mulpt3\(Point3) 1375 3750 w +(p,) 2285 3750 w +(double) 2480 3750 w +(s\)) 2935 3750 w +10 /LucidaSansUnicode00 f +(p) 2297 3990 w +(*) 2368 3990 w +(s) 2424 3990 w +10 /S f +(=) 2531 3990 w +10 /LucidaSansUnicode00 f +10 /S f +(\354) 2650 3953 w +(\356) 2650 4053 w +10 /LucidaSansUnicode00 f +(xs) 2699 4000 w +(,) 2819 4000 w +(ys) 2899 4000 w +(,) 3010 4000 w +(zs) 3090 4000 w +(,) 3206 4000 w +(ws) 3286 4000 w +10 /S f +(\374) 3414 3953 w +(\376) 3414 4053 w +10 /LucidaSans-Demi f +(Division) 720 4295 w +9 /LucidaTypewriter f +(Point3) 920 4465 w +(divpt3\(Point3) 1375 4465 w +(p,) 2285 4465 w +(double) 2480 4465 w +(s\)) 2935 4465 w +10 /LucidaSansUnicode00 f +(p) 2296 4790 w +(/) 2367 4790 w +(s) 2428 4790 w +10 /S f +(=) 2535 4790 w +10 /LucidaSansUnicode00 f +10 /S f +(\354) 2654 4703 w +(\357) 2654 4803 w +(\356) 2654 4903 w +10 /LucidaSansUnicode00 f +(s) 2733 4860 w +(x) 2728 4730 w +10 /S1 f +(_) 2713 4760 w +(_) 2754 4760 w +10 /LucidaSansUnicode00 f +(,) 2822 4790 w +(s) 2927 4860 w +(y) 2927 4730 w +10 /S1 f +(_) 2912 4760 w +(_) 2944 4760 w +10 /LucidaSansUnicode00 f +(,) 3012 4790 w +(s) 3120 4860 w +(z) 3117 4730 w +10 /S1 f +(_) 3102 4760 w +(_) 3139 4760 w +10 /LucidaSansUnicode00 f +(,) 3207 4790 w +(s) 3325 4860 w +(w) 3312 4730 w +10 /S1 f +(_) 3297 4760 w +(__) 3304 4760 w +10 /S f +(\374) 3414 4703 w +(\357) 3414 4803 w +(\376) 3414 4903 w +10 /LucidaSans-Demi f +(Vector Dot Product) 720 5195 w +9 /LucidaTypewriter f +(double) 920 5365 w +(dotvec3\(Point3) 1375 5365 w +(a,) 2350 5365 w +(Point3) 2545 5365 w +(b\)) 3000 5365 w +10 /LucidaSansUnicode00 f +(a) 2210 5610 w +7 /LucidaSansUnicode21 f +(\222) 2205 5560 w +10 /LucidaSansUnicode00 f +10 /LucidaSansUnicode20 f +(") 2314 5610 w +10 /LucidaSansUnicode00 f +(b) 2418 5610 w +7 /LucidaSansUnicode21 f +(\222) 2417 5535 w +10 /LucidaSansUnicode00 f +10 /S f +(=) 2538 5610 w +10 /LucidaSansUnicode00 f +(x) 2649 5610 w +7 /LucidaSansUnicode00 f +(a) 2721 5630 w +10 /LucidaSansUnicode00 f +(x) 2776 5610 w +7 /LucidaSansUnicode00 f +(b) 2848 5630 w +10 /S f +(+) 2916 5610 w +10 /LucidaSansUnicode00 f +(y) 2987 5610 w +7 /LucidaSansUnicode00 f +(a) 3050 5630 w +10 /LucidaSansUnicode00 f +(y) 3105 5610 w +7 /LucidaSansUnicode00 f +(b) 3168 5630 w +10 /S f +(+) 3236 5610 w +10 /LucidaSansUnicode00 f +(z) 3307 5610 w +7 /LucidaSansUnicode00 f +(a) 3375 5630 w +10 /LucidaSansUnicode00 f +(z) 3430 5610 w +7 /LucidaSansUnicode00 f +(b) 3498 5630 w +10 /LucidaSans-Demi f +(Vector Cross Product) 720 5910 w +9 /LucidaTypewriter f +(double) 920 6080 w +(crossvec3\(Point3) 1375 6080 w +(a,) 2480 6080 w +(Point3) 2675 6080 w +(b\)) 3130 6080 w +10 /LucidaSansUnicode00 f +(a) 1650 6355 w +7 /LucidaSansUnicode21 f +(\222) 1645 6305 w +10 /LucidaSansUnicode00 f +(\327) 1754 6355 w +(b) 1882 6355 w +7 /LucidaSansUnicode21 f +(\222) 1881 6280 w +10 /LucidaSansUnicode00 f +10 /S f +(=) 2002 6355 w +10 /LucidaSansUnicode00 f +10 /S f +(\354) 2121 6318 w +(\356) 2121 6418 w +10 /LucidaSansUnicode00 f +(y) 2170 6355 w +7 /LucidaSansUnicode00 f +(a) 2233 6375 w +10 /LucidaSansUnicode00 f +(z) 2288 6355 w +7 /LucidaSansUnicode00 f +(b) 2356 6375 w +10 /S f +(-) 2424 6355 w +10 /LucidaSansUnicode00 f +(z) 2495 6355 w +7 /LucidaSansUnicode00 f +(a) 2563 6375 w +10 /LucidaSansUnicode00 f +(y) 2618 6355 w +7 /LucidaSansUnicode00 f +(b) 2681 6375 w +10 /LucidaSansUnicode00 f +(,) 2741 6355 w +(z) 2821 6355 w +7 /LucidaSansUnicode00 f +(a) 2889 6375 w +10 /LucidaSansUnicode00 f +(x) 2944 6355 w +7 /LucidaSansUnicode00 f +(b) 3016 6375 w +10 /S f +(-) 3084 6355 w +10 /LucidaSansUnicode00 f +(x) 3155 6355 w +7 /LucidaSansUnicode00 f +(a) 3227 6375 w +10 /LucidaSansUnicode00 f +(z) 3282 6355 w +7 /LucidaSansUnicode00 f +(b) 3350 6375 w +10 /LucidaSansUnicode00 f +(,) 3410 6355 w +(x) 3490 6355 w +7 /LucidaSansUnicode00 f +(a) 3562 6375 w +10 /LucidaSansUnicode00 f +(y) 3617 6355 w +7 /LucidaSansUnicode00 f +(b) 3680 6375 w +10 /S f +(-) 3748 6355 w +10 /LucidaSansUnicode00 f +(y) 3819 6355 w +7 /LucidaSansUnicode00 f +(a) 3882 6375 w +10 /LucidaSansUnicode00 f +(x) 3937 6355 w +7 /LucidaSansUnicode00 f +(b) 4009 6375 w +10 /S f +(\374) 4061 6318 w +(\376) 4061 6418 w +10 /LucidaSans-Demi f +(Vector Magnitude/Length) 720 6710 w +9 /LucidaTypewriter f +(double) 920 6880 w +(vec3len\(Point3) 1375 6880 w +(v\)) 2350 6880 w +10 /LucidaSansUnicode00 f +(|) 2408 7155 w +(v) 2456 7138 w +7 /LucidaSansUnicode21 f +(\222) 2449 7088 w +10 /LucidaSansUnicode00 f +(|) 2508 7155 w +10 /S f +(=) 2604 7138 w +10 /LucidaSansUnicode00 f +12 /LucidaSansUnicode22 f +(\032) 2715 7138 w +12 /S f +(`) 2812 7138 w +(````````) 2863 7138 w +10 /LucidaSansUnicode00 f +(x) 2820 7138 w +7 /LucidaSansUnicode00 f +(2) 2886 7098 w +10 /S f +(+) 2954 7138 w +10 /LucidaSansUnicode00 f +(y) 3025 7138 w +7 /LucidaSansUnicode00 f +(2) 3082 7098 w +10 /S f +(+) 3150 7138 w +10 /LucidaSansUnicode00 f +(z) 3221 7138 w +7 /LucidaSansUnicode00 f +(2) 3283 7098 w +cleartomark +showpage +saveobj restore +%%EndPage: 3 3 +%%Page: 4 4 +/saveobj save def +mark +4 pagesetup +10 /LucidaSansUnicode00 f +(\255 4 \255) 2783 480 w +10 /LucidaSans-Demi f +(Vector Normalization) 720 840 w +9 /LucidaTypewriter f +(Point3) 920 1010 w +(normvec3\(Point3) 1375 1010 w +(v\)) 2415 1010 w +10 /LucidaSansUnicode00 f +(n) 2374 1335 w +7 /LucidaSansUnicode21 f +(\222) 2372 1285 w +10 /LucidaSansUnicode00 f +10 /S f +(=) 2492 1335 w +10 /LucidaSansUnicode00 f +10 /S f +(\354) 2611 1248 w +(\357) 2611 1348 w +(\356) 2611 1448 w +10 /LucidaSansUnicode00 f +(|) 2685 1447 w +(v) 2733 1430 w +7 /LucidaSansUnicode21 f +(\222) 2726 1380 w +10 /LucidaSansUnicode00 f +(|) 2785 1447 w +(x) 2713 1275 w +10 /S1 f +(_) 2671 1305 w +(__) 2717 1305 w +10 /LucidaSansUnicode00 f +(,) 2835 1335 w +(|) 2940 1447 w +(v) 2988 1430 w +7 /LucidaSansUnicode21 f +(\222) 2981 1380 w +10 /LucidaSansUnicode00 f +(|) 3040 1447 w +(y) 2972 1275 w +10 /S1 f +(_) 2925 1305 w +(__) 2971 1305 w +10 /LucidaSansUnicode00 f +(,) 3089 1335 w +(|) 3194 1447 w +(v) 3242 1430 w +7 /LucidaSansUnicode21 f +(\222) 3235 1380 w +10 /LucidaSansUnicode00 f +(|) 3294 1447 w +(z) 3224 1275 w +10 /S1 f +(_) 3180 1305 w +(__) 3226 1305 w +10 /S f +(\374) 3336 1248 w +(\357) 3336 1348 w +(\376) 3336 1448 w +10 /LucidaSans-Demi f +(2.3.) 720 1740 w +(Matrix) 962 1740 w +(Addition) 720 1980 w +9 /LucidaTypewriter f +(void) 920 2150 w +(addm\(Matrix) 1245 2150 w +(A,) 2025 2150 w +(Matrix) 2220 2150 w +(B\)) 2675 2150 w +10 /LucidaSansUnicode00 f +(\() 2393 2390 w +10 /LucidaSans-Demi f +(A) 2434 2390 w +10 /S f +(+) 2524 2390 w +10 /LucidaSans-Demi f +(B) 2595 2390 w +10 /LucidaSansUnicode00 f +(\)) 2666 2390 w +7 /LucidaSansUnicode00 f +(i) 2710 2410 w +(,) 2735 2410 w +(j) 2762 2410 w +10 /LucidaSansUnicode00 f +10 /S f +(=) 2847 2390 w +10 /LucidaSansUnicode00 f +10 /LucidaSans-Demi f +(A) 2958 2390 w +7 /LucidaSansUnicode00 f +(i) 3043 2410 w +(,) 3068 2410 w +(j) 3095 2410 w +10 /S f +(+) 3140 2390 w +10 /LucidaSans-Demi f +(B) 3211 2390 w +7 /LucidaSansUnicode00 f +(i) 3285 2410 w +(,) 3310 2410 w +(j) 3337 2410 w +10 /LucidaSans-Demi f +(Substraction) 720 2690 w +9 /LucidaTypewriter f +(void) 920 2860 w +(subm\(Matrix) 1245 2860 w +(A,) 2025 2860 w +(Matrix) 2220 2860 w +(B\)) 2675 2860 w +10 /LucidaSansUnicode00 f +(\() 2393 3100 w +10 /LucidaSans-Demi f +(A) 2434 3100 w +10 /S f +(-) 2524 3100 w +10 /LucidaSans-Demi f +(B) 2595 3100 w +10 /LucidaSansUnicode00 f +(\)) 2666 3100 w +7 /LucidaSansUnicode00 f +(i) 2710 3120 w +(,) 2735 3120 w +(j) 2762 3120 w +10 /LucidaSansUnicode00 f +10 /S f +(=) 2847 3100 w +10 /LucidaSansUnicode00 f +10 /LucidaSans-Demi f +(A) 2958 3100 w +7 /LucidaSansUnicode00 f +(i) 3043 3120 w +(,) 3068 3120 w +(j) 3095 3120 w +10 /S f +(-) 3140 3100 w +10 /LucidaSans-Demi f +(B) 3211 3100 w +7 /LucidaSansUnicode00 f +(i) 3285 3120 w +(,) 3310 3120 w +(j) 3337 3120 w +10 /LucidaSans-Demi f +(Multiplication) 720 3400 w +9 /LucidaTypewriter f +(void) 920 3570 w +(mulm\(Matrix) 1245 3570 w +(A,) 2025 3570 w +(Matrix) 2220 3570 w +(B\)) 2675 3570 w +10 /S f +(\351) 2370 3823 w +(\353) 2370 3923 w +10 /LucidaSans-Demi f +(AB) 2420 3870 w +10 /S f +(\371) 2557 3823 w +(\373) 2557 3923 w +7 /LucidaSansUnicode00 f +(i) 2606 3905 w +(,) 2631 3905 w +(j) 2658 3905 w +10 /LucidaSansUnicode00 f +10 /S f +(=) 2743 3860 w +10 /LucidaSansUnicode00 f +7 /LucidaSansUnicode00 f +(k) 2847 3960 w +7 /S f +(=) 2899 3960 w +7 /LucidaSansUnicode00 f +(0) 2949 3960 w +15 /S f +(S) 2876 3890 w +7 /LucidaSansUnicode00 f +(3) 2846 3760 w +7 /S f +(-) 2901 3760 w +7 /LucidaSansUnicode00 f +(1) 2951 3760 w +10 /LucidaSans-Demi f +(A) 3027 3860 w +7 /LucidaSansUnicode00 f +(i) 3112 3880 w +(,) 3137 3880 w +(k) 3164 3880 w +10 /LucidaSans-Demi f +(B) 3221 3860 w +7 /LucidaSansUnicode00 f +(k) 3295 3880 w +(,) 3341 3880 w +(j) 3368 3880 w +10 /LucidaSans-Demi f +(Transpose) 720 4240 w +9 /LucidaTypewriter f +(void) 920 4410 w +(transposem\(Matrix) 1245 4410 w +(M\)) 2415 4410 w +10 /LucidaSansUnicode00 f +(\() 2552 4650 w +10 /LucidaSans-Demi f +(M) 2593 4650 w +7 /LucidaSansUnicode00 f +(T) 2689 4610 w +10 /LucidaSansUnicode00 f +(\)) 2749 4650 w +7 /LucidaSansUnicode00 f +(i) 2793 4670 w +(,) 2818 4670 w +(j) 2845 4670 w +10 /LucidaSansUnicode00 f +10 /S f +(=) 2930 4650 w +10 /LucidaSansUnicode00 f +10 /LucidaSans-Demi f +(A) 3041 4650 w +7 /LucidaSansUnicode00 f +(j) 3126 4670 w +(,) 3152 4670 w +(i) 3179 4670 w +10 /LucidaSans-Demi f +(Identity) 720 4950 w +9 /LucidaTypewriter f +(void) 920 5120 w +(identity\(Matrix) 1245 5120 w +(M\)) 2285 5120 w +10 /LucidaSans-Demi f +(M) 2560 5495 w +10 /LucidaSansUnicode00 f +10 /S f +(=) 2707 5495 w +10 /LucidaSansUnicode00 f +10 /S f +(\351) 2826 5358 w +(\357) 2826 5458 w +(\357) 2826 5558 w +(\353) 2826 5658 w +10 /LucidaSansUnicode00 f +(0) 2876 5645 w +(0) 2987 5645 w +(1) 3098 5645 w +(0) 2876 5505 w +(1) 2987 5505 w +(0) 3098 5505 w +(1) 2876 5365 w +(0) 2987 5365 w +(0) 3098 5365 w +10 /S f +(\371) 3161 5358 w +(\357) 3161 5458 w +(\357) 3161 5558 w +(\373) 3161 5658 w +10 /LucidaSans-Demi f +(Determinant) 720 5950 w +9 /LucidaTypewriter f +(double) 920 6120 w +(detm\(Matrix) 1375 6120 w +(M\)) 2155 6120 w +10 /LucidaSansUnicode00 f +(det) 1992 6500 w +(\() 2156 6500 w +10 /LucidaSans-Demi f +(M) 2197 6500 w +10 /LucidaSansUnicode00 f +(\)) 2296 6500 w +10 /S f +(=) 2393 6500 w +10 /LucidaSansUnicode00 f +(m) 2504 6660 w +7 /LucidaSansUnicode00 f +(02) 2608 6680 w +10 /LucidaSansUnicode00 f +(\() 2712 6660 w +(m) 2753 6660 w +7 /LucidaSansUnicode00 f +(10) 2857 6680 w +10 /LucidaSansUnicode00 f +(m) 2961 6660 w +7 /LucidaSansUnicode00 f +(21) 3065 6680 w +10 /S f +(-) 3177 6660 w +10 /LucidaSansUnicode00 f +(m) 3248 6660 w +7 /LucidaSansUnicode00 f +(11) 3352 6680 w +10 /LucidaSansUnicode00 f +(m) 3456 6660 w +7 /LucidaSansUnicode00 f +(20) 3560 6680 w +10 /LucidaSansUnicode00 f +(\)) 3664 6660 w +(m) 2504 6500 w +7 /LucidaSansUnicode00 f +(01) 2608 6520 w +10 /LucidaSansUnicode00 f +(\() 2712 6500 w +(m) 2753 6500 w +7 /LucidaSansUnicode00 f +(12) 2857 6520 w +10 /LucidaSansUnicode00 f +(m) 2961 6500 w +7 /LucidaSansUnicode00 f +(20) 3065 6520 w +10 /S f +(-) 3177 6500 w +10 /LucidaSansUnicode00 f +(m) 3248 6500 w +7 /LucidaSansUnicode00 f +(10) 3352 6520 w +10 /LucidaSansUnicode00 f +(m) 3456 6500 w +7 /LucidaSansUnicode00 f +(22) 3560 6520 w +10 /LucidaSansUnicode00 f +(\)) 3664 6500 w +10 /S f +(+) 3713 6500 w +10 /LucidaSansUnicode00 f +(m) 2504 6340 w +7 /LucidaSansUnicode00 f +(00) 2608 6360 w +10 /LucidaSansUnicode00 f +(\() 2712 6340 w +(m) 2753 6340 w +7 /LucidaSansUnicode00 f +(11) 2857 6360 w +10 /LucidaSansUnicode00 f +(m) 2961 6340 w +7 /LucidaSansUnicode00 f +(22) 3065 6360 w +10 /S f +(-) 3177 6340 w +10 /LucidaSansUnicode00 f +(m) 3248 6340 w +7 /LucidaSansUnicode00 f +(12) 3352 6360 w +10 /LucidaSansUnicode00 f +(m) 3456 6340 w +7 /LucidaSansUnicode00 f +(21) 3560 6360 w +10 /LucidaSansUnicode00 f +(\)) 3664 6340 w +10 /S f +(+) 3713 6340 w +cleartomark +showpage +saveobj restore +%%EndPage: 4 4 +%%Page: 5 5 +/saveobj save def +mark +5 pagesetup +10 /LucidaSansUnicode00 f +(\255 5 \255) 2783 480 w +10 /LucidaSans-Demi f +(2.4.) 720 840 w +(Matrix3) 962 840 w +(Addition) 720 1080 w +9 /LucidaTypewriter f +(void) 920 1250 w +(addm3\(Matrix3) 1245 1250 w +(A,) 2155 1250 w +(Matrix3) 2350 1250 w +(B\)) 2870 1250 w +10 /LucidaSansUnicode00 f +(\() 2393 1490 w +10 /LucidaSans-Demi f +(A) 2434 1490 w +10 /S f +(+) 2524 1490 w +10 /LucidaSans-Demi f +(B) 2595 1490 w +10 /LucidaSansUnicode00 f +(\)) 2666 1490 w +7 /LucidaSansUnicode00 f +(i) 2710 1510 w +(,) 2735 1510 w +(j) 2762 1510 w +10 /LucidaSansUnicode00 f +10 /S f +(=) 2847 1490 w +10 /LucidaSansUnicode00 f +10 /LucidaSans-Demi f +(A) 2958 1490 w +7 /LucidaSansUnicode00 f +(i) 3043 1510 w +(,) 3068 1510 w +(j) 3095 1510 w +10 /S f +(+) 3140 1490 w +10 /LucidaSans-Demi f +(B) 3211 1490 w +7 /LucidaSansUnicode00 f +(i) 3285 1510 w +(,) 3310 1510 w +(j) 3337 1510 w +10 /LucidaSans-Demi f +(Substraction) 720 1790 w +9 /LucidaTypewriter f +(void) 920 1960 w +(subm3\(Matrix3) 1245 1960 w +(A,) 2155 1960 w +(Matrix3) 2350 1960 w +(B\)) 2870 1960 w +10 /LucidaSansUnicode00 f +(\() 2393 2200 w +10 /LucidaSans-Demi f +(A) 2434 2200 w +10 /S f +(-) 2524 2200 w +10 /LucidaSans-Demi f +(B) 2595 2200 w +10 /LucidaSansUnicode00 f +(\)) 2666 2200 w +7 /LucidaSansUnicode00 f +(i) 2710 2220 w +(,) 2735 2220 w +(j) 2762 2220 w +10 /LucidaSansUnicode00 f +10 /S f +(=) 2847 2200 w +10 /LucidaSansUnicode00 f +10 /LucidaSans-Demi f +(A) 2958 2200 w +7 /LucidaSansUnicode00 f +(i) 3043 2220 w +(,) 3068 2220 w +(j) 3095 2220 w +10 /S f +(-) 3140 2200 w +10 /LucidaSans-Demi f +(B) 3211 2200 w +7 /LucidaSansUnicode00 f +(i) 3285 2220 w +(,) 3310 2220 w +(j) 3337 2220 w +10 /LucidaSans-Demi f +(Multiplication) 720 2500 w +9 /LucidaTypewriter f +(void) 920 2670 w +(mulm3\(Matrix3) 1245 2670 w +(A,) 2155 2670 w +(Matrix3) 2350 2670 w +(B\)) 2870 2670 w +10 /S f +(\351) 2374 2923 w +(\353) 2374 3023 w +10 /LucidaSans-Demi f +(AB) 2424 2970 w +10 /S f +(\371) 2561 2923 w +(\373) 2561 3023 w +7 /LucidaSansUnicode00 f +(i) 2610 3005 w +(,) 2635 3005 w +(j) 2662 3005 w +10 /LucidaSansUnicode00 f +10 /S f +(=) 2747 2960 w +10 /LucidaSansUnicode00 f +7 /LucidaSansUnicode00 f +(k) 2859 3060 w +7 /S f +(=) 2911 3060 w +7 /LucidaSansUnicode00 f +(0) 2961 3060 w +15 /S f +(S) 2888 2990 w +7 /LucidaSansUnicode00 f +(4) 2858 2860 w +7 /S f +(-) 2913 2860 w +7 /LucidaSansUnicode00 f +(1) 2963 2860 w +10 /LucidaSans-Demi f +(A) 3023 2960 w +7 /LucidaSansUnicode00 f +(i) 3108 2980 w +(,) 3133 2980 w +(k) 3160 2980 w +10 /LucidaSans-Demi f +(B) 3217 2960 w +7 /LucidaSansUnicode00 f +(k) 3291 2980 w +(,) 3337 2980 w +(j) 3364 2980 w +10 /LucidaSans-Demi f +(Transpose) 720 3340 w +9 /LucidaTypewriter f +(void) 920 3510 w +(transposem3\(Matrix3) 1245 3510 w +(M\)) 2545 3510 w +10 /LucidaSansUnicode00 f +(\() 2552 3750 w +10 /LucidaSans-Demi f +(M) 2593 3750 w +7 /LucidaSansUnicode00 f +(T) 2689 3710 w +10 /LucidaSansUnicode00 f +(\)) 2749 3750 w +7 /LucidaSansUnicode00 f +(i) 2793 3770 w +(,) 2818 3770 w +(j) 2845 3770 w +10 /LucidaSansUnicode00 f +10 /S f +(=) 2930 3750 w +10 /LucidaSansUnicode00 f +10 /LucidaSans-Demi f +(A) 3041 3750 w +7 /LucidaSansUnicode00 f +(j) 3126 3770 w +(,) 3152 3770 w +(i) 3179 3770 w +10 /LucidaSans-Demi f +(Identity) 720 4050 w +9 /LucidaTypewriter f +(void) 920 4220 w +(identity3\(Matrix3) 1245 4220 w +(M\)) 2415 4220 w +10 /LucidaSans-Demi f +(M) 2505 4695 w +10 /LucidaSansUnicode00 f +10 /S f +(=) 2652 4695 w +10 /LucidaSansUnicode00 f +10 /S f +(\351) 2771 4458 w +(\357) 2771 4558 w +(\357) 2771 4658 w +(\357) 2771 4758 w +(\357) 2771 4858 w +(\353) 2771 4958 w +10 /LucidaSansUnicode00 f +(0) 2821 4915 w +(0) 2932 4915 w +(0) 3043 4915 w +(1) 3154 4915 w +(0) 2821 4775 w +(0) 2932 4775 w +(1) 3043 4775 w +(0) 3154 4775 w +(0) 2821 4635 w +(1) 2932 4635 w +(0) 3043 4635 w +(0) 3154 4635 w +(1) 2821 4495 w +(0) 2932 4495 w +(0) 3043 4495 w +(0) 3154 4495 w +10 /S f +(\371) 3217 4458 w +(\357) 3217 4558 w +(\357) 3217 4658 w +(\357) 3217 4758 w +(\357) 3217 4858 w +(\373) 3217 4958 w +10 /LucidaSans-Demi f +(Determinant) 720 5250 w +9 /LucidaTypewriter f +(double) 920 5420 w +(detm3\(Matrix3) 1375 5420 w +(M\)) 2285 5420 w +10 /LucidaSansUnicode00 f +(det) 567 5910 w +(\() 731 5910 w +10 /LucidaSans-Demi f +(M) 772 5910 w +10 /LucidaSansUnicode00 f +(\)) 871 5910 w +10 /S f +(=) 968 5910 w +10 /LucidaSansUnicode00 f +10 /S f +(-) 1079 6120 w +10 /LucidaSansUnicode00 f +(m) 1150 6120 w +7 /LucidaSansUnicode00 f +(03) 1254 6140 w +10 /LucidaSansUnicode00 f +(\() 1358 6120 w +(m) 1399 6120 w +7 /LucidaSansUnicode00 f +(10) 1503 6140 w +10 /LucidaSansUnicode00 f +(\() 1607 6120 w +(m) 1648 6120 w +7 /LucidaSansUnicode00 f +(21) 1752 6140 w +10 /LucidaSansUnicode00 f +(m) 1856 6120 w +7 /LucidaSansUnicode00 f +(32) 1960 6140 w +10 /S f +(-) 2072 6120 w +10 /LucidaSansUnicode00 f +(m) 2143 6120 w +7 /LucidaSansUnicode00 f +(22) 2247 6140 w +10 /LucidaSansUnicode00 f +(m) 2351 6120 w +7 /LucidaSansUnicode00 f +(31) 2455 6140 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/LucidaSansUnicode00 f +(q) 3500 1270 w +10 /S f +(-) 3568 1250 w +10 /LucidaSansUnicode00 f +(k) 3639 1250 w +7 /LucidaSansUnicode00 f +(r) 3708 1270 w +10 /LucidaSansUnicode00 f +(\)) 3753 1250 w +10 /LucidaSans-Demi f +(Multiplication) 720 1550 w +9 /LucidaTypewriter f +(Quaternion) 920 1720 w +(mulq\(Quaternion) 1635 1720 w +(q,) 2675 1720 w +(Quaternion) 2870 1720 w +(r\)) 3585 1720 w +10 /LucidaSansUnicode00 f +(q) 1321 1965 w +10 /S f +(=) 1440 1965 w +10 /LucidaSansUnicode00 f +([) 1551 1965 w +(r) 1592 1965 w +7 /LucidaSansUnicode00 f +(q) 1644 1985 w +10 /LucidaSansUnicode00 f +(,) 1704 1965 w +(v) 1784 1965 w +7 /LucidaSansUnicode21 f +(\222) 1777 1915 w +7 /LucidaSansUnicode00 f +(q) 1847 1985 w +10 /LucidaSansUnicode00 f +(]) 1907 1965 w +(r) 1956 1965 w +10 /S f +(=) 2053 1965 w +10 /LucidaSansUnicode00 f +([) 2164 1965 w +(r) 2205 1965 w +7 /LucidaSansUnicode00 f +(r) 2257 1985 w +10 /LucidaSansUnicode00 f +(,) 2302 1965 w +(v) 2382 1965 w +7 /LucidaSansUnicode21 f 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2695 w +10 /LucidaSansUnicode00 f +(s) 2916 2675 w +(,) 2975 2675 w +(j) 3055 2675 w +7 /LucidaSansUnicode00 f +(q) 3096 2695 w +10 /LucidaSansUnicode00 f +(s) 3156 2675 w +(,) 3215 2675 w +(k) 3295 2675 w +7 /LucidaSansUnicode00 f +(q) 3364 2695 w +10 /LucidaSansUnicode00 f +(s) 3424 2675 w +(]) 3483 2675 w +10 /LucidaSans-Demi f +(Inverse) 720 2975 w +9 /LucidaTypewriter f +(Quaternion) 920 3145 w +(invq\(Quaternion) 1635 3145 w +(q\)) 2675 3145 w +10 /LucidaSansUnicode00 f +(q) 2009 3470 w +7 /S f +(-) 2083 3430 w +7 /LucidaSansUnicode00 f +(1) 2133 3430 w +10 /LucidaSansUnicode00 f +10 /S f +(=) 2241 3470 w +10 /LucidaSansUnicode00 f +10 /S f +(\354) 2360 3383 w +(\357) 2360 3483 w +(\356) 2360 3583 w +10 /LucidaSansUnicode00 f +(|) 2434 3567 w +(q) 2482 3550 w +(|) 2545 3567 w +7 /LucidaSansUnicode00 f +(2) 2583 3510 w +10 /LucidaSansUnicode00 f +(r) 2514 3410 w +10 /S1 f +(_) 2419 3440 w +(____) 2450 3440 w +10 /LucidaSansUnicode00 f +(,) 2668 3470 w +(|) 2781 3567 w +(q) 2829 3550 w +(|) 2892 3567 w +7 /LucidaSansUnicode00 f +(2) 2930 3510 w +10 /S f +(-) 2832 3410 w +10 /LucidaSansUnicode00 f +(i) 2903 3410 w +10 /S1 f +(_) 2767 3440 w +(____) 2798 3440 w +10 /LucidaSansUnicode00 f +(,) 3016 3470 w +(|) 3129 3567 w +(q) 3177 3550 w +(|) 3240 3567 w +7 /LucidaSansUnicode00 f +(2) 3278 3510 w +10 /S f +(-) 3179 3410 w +10 /LucidaSansUnicode00 f +(j) 3250 3410 w +10 /S1 f +(_) 3114 3440 w +(____) 3145 3440 w +10 /LucidaSansUnicode00 f +(,) 3363 3470 w +(|) 3476 3567 w +(q) 3524 3550 w +(|) 3587 3567 w +7 /LucidaSansUnicode00 f +(2) 3625 3510 w +10 /S f +(-) 3512 3410 w +10 /LucidaSansUnicode00 f +(k) 3583 3410 w +10 /S1 f +(_) 3461 3440 w +(____) 3492 3440 w +10 /S f +(\374) 3702 3383 w +(\357) 3702 3483 w +(\376) 3702 3583 w +10 /LucidaSans-Demi f +(Magnitude/Length) 720 3875 w +9 /LucidaTypewriter f +(double) 920 4045 w +(qlen\(Quaternion) 1375 4045 w +(q\)) 2415 4045 w +10 /LucidaSansUnicode00 f +(|) 2337 4320 w +(q) 2385 4303 w +(|) 2448 4320 w +10 /S f +(=) 2544 4303 w +10 /LucidaSansUnicode00 f +12 /LucidaSansUnicode22 f +(\032) 2655 4303 w +12 /S f +(`) 2752 4303 w +(```````````) 2755 4303 w +10 /LucidaSansUnicode00 f +(r) 2760 4303 w +7 /LucidaSansUnicode00 f +(2) 2806 4263 w +10 /S f +(+) 2874 4303 w +10 /LucidaSansUnicode00 f +(i) 2945 4303 w +7 /LucidaSansUnicode00 f +(2) 2979 4263 w +10 /S f +(+) 3047 4303 w +10 /LucidaSansUnicode00 f +(j) 3118 4303 w +7 /LucidaSansUnicode00 f +(2) 3153 4263 w +10 /S f +(+) 3221 4303 w +10 /LucidaSansUnicode00 f +(k) 3292 4303 w +7 /LucidaSansUnicode00 f +(2) 3355 4263 w +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], + }; +} @@ -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 @@ -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); +} |