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libgeometry: simplify rframes
i got rid of redundant code, and added routines to get the xform matrix out of an rframe, so it can be stored and used separately or as part of a composition of xforms. also replaced the big example in the man page for more concise, useful ones.
This commit is contained in:
rodri
parent
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3 changed files with 99 additions and 207 deletions
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@ -124,6 +124,8 @@ Quaternion slerp(Quaternion, Quaternion, double);
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Point3 qrotate(Point3, Point3, double);
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/* RFrame */
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void rframematrix(Matrix, RFrame);
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void rframematrix3(Matrix3, RFrame3);
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Point2 rframexform(Point2, RFrame);
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Point3 rframexform3(Point3, RFrame3);
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Point2 invrframexform(Point2, RFrame);
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@ -1,6 +1,6 @@
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.TH GEOMETRY 2
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.SH NAME
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Flerp, fberp, fclamp, Pt2, Vec2, addpt2, subpt2, mulpt2, divpt2, lerp2, berp2, dotvec2, vec2len, normvec2, edgeptcmp, ptinpoly, Pt3, Vec3, addpt3, subpt3, mulpt3, divpt3, lerp3, berp3, dotvec3, crossvec3, vec3len, normvec3, identity, addm, subm, mulm, smulm, transposem, detm, tracem, minorm, cofactorm, adjm, invm, xform, identity3, addm3, subm3, mulm3, smulm3, transposem3, detm3, tracem3, minorm3, cofactorm3, adjm3, invm3, xform3, Quat, Quatvec, addq, subq, mulq, smulq, sdivq, dotq, invq, qlen, normq, slerp, qrotate, rframexform, rframexform3, invrframexform, invrframexform3, centroid, barycoords, centroid3, vfmt, Vfmt, GEOMfmtinstall \- computational geometry library
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Flerp, fberp, fclamp, Pt2, Vec2, addpt2, subpt2, mulpt2, divpt2, lerp2, berp2, dotvec2, vec2len, normvec2, edgeptcmp, ptinpoly, Pt3, Vec3, addpt3, subpt3, mulpt3, divpt3, lerp3, berp3, dotvec3, crossvec3, vec3len, normvec3, identity, addm, subm, mulm, smulm, transposem, detm, tracem, minorm, cofactorm, adjm, invm, xform, identity3, addm3, subm3, mulm3, smulm3, transposem3, detm3, tracem3, minorm3, cofactorm3, adjm3, invm3, xform3, Quat, Quatvec, addq, subq, mulq, smulq, sdivq, dotq, invq, qlen, normq, slerp, qrotate, rframematrix, rframematrix3, rframexform, rframexform3, invrframexform, invrframexform3, centroid, barycoords, centroid3, vfmt, Vfmt, GEOMfmtinstall \- computational geometry library
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.SH SYNOPSIS
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.de PB
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.PP
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@ -135,6 +135,8 @@ Quaternion slerp(Quaternion q, Quaternion r, double t);
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Point3 qrotate(Point3 p, Point3 axis, double θ);
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.PB
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/* RFrame */
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void rframematrix(Matrix, RFrame);
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void rframematrix3(Matrix3, RFrame3);
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Point2 rframexform(Point2 p, RFrame rf);
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Point3 rframexform3(Point3 p, RFrame3 rf);
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Point2 invrframexform(Point2 p, RFrame rf);
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@ -615,7 +617,7 @@ by
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.I θ
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radians.
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.SS Frames of reference
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A frame of reference in a
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A frame of reference (or rframe) in a
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.IR n -dimensional
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space is made out of n+1 points, one being the origin
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.IR p ,
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@ -623,33 +625,33 @@ and the remaining being the
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basis vectors
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.I b1,⋯,bn
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that define the metric within that frame.
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.PP
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The origin point and the bases are all defined in terms of an origin
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frame of reference O. Applying a forward transformation
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.RI ( rframexform
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and
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.IR rframexform3 )
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to a point relative to O will result in a point relative to the new frame.
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Applying an inverse transformation
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.RI ( invrframexform
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and
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.IR invrframexform3 )
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to that same point—now defined in terms of the new frame—will bring it back to O.
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.TP
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Name
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Description
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.TP
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.B rframematrix(\fIm\fP,\fIrf\fP)
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Initializes
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.I m
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as an rframe transformation matrix based on
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.IR rf .
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.TP
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.B rframematrix3(\fIm\fP,\fIrf\fP)
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Initializes
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.I m
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as an rframe transformation matrix based on
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.IR rf .
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.TP
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.B rframexform(\fIp\fP,\fIrf\fP)
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Transforms the point
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.IR p ,
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relative to some origin frame of reference O, into the frame of reference
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relative to some origin frame of reference O, into the rframe
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.IR rf .
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It then returns the new 2D point.
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.TP
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.B rframexform3(\fIp\fP,\fIrf\fP)
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Transforms the point
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.IR p ,
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relative to some origin frame of reference O, into the frame of reference
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relative to some origin frame of reference O, into the rframe
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.IR rf .
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It then returns the new 3D point.
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.TP
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@ -658,7 +660,7 @@ Transforms the point
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.IR p ,
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relative to
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.IR rf ,
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into a point relative to the origin frame of reference O.
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into a point relative to the origin rframe O.
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It then returns the new 2D point.
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.TP
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.B invrframexform3(\fIp\fP,\fIrf\fP)
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@ -666,7 +668,7 @@ Transforms the point
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.IR p ,
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relative to
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.IR rf ,
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into a point relative to the origin frame of reference O.
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into a point relative to the origin rframe O.
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It then returns the new 3D point.
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.SS Triangles
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.TP
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@ -689,7 +691,53 @@ relative to the triangle
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Returns the geometric center of
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.B Triangle3
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.IR t .
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.SH EXAMPLE
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.SH EXAMPLES
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.PP
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Rotate a point p by θ, scale it 2x, then translate it by vector v:
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.IP
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.EX
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Matrix R = {
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cos(θ), -sin(θ), 0,
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sin(θ), cos(θ), 0,
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0, 0, 1,
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}, S = {
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2, 0, 0,
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0, 2, 0,
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0, 0, 1,
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}, T = {
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1, 0, v.x,
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0, 1, v.y,
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0, 0, 1,
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};
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mulm(S, R);
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mulm(T, S);
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p = xform(p, T); /* p' = T·S·R·p */
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.EE
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.PP
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Given a space with two observers, A and B, and a point p relative to
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A, find its location relative to B:
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.IP
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.EX
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pb = rframexform(invrframexform(p, A), B);
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.EE
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.PP
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Now let's say observer C is located relative to A; find the point's
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location in C:
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.IP
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.EX
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pc = rframexform(p, C);
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.EE
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.PP
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Finally, to obtain its location relative to the space itself (its
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global position):
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.IP
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.EX
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pg = invrframexform(p, A);
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.EE
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.PP
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The following is a common example of an
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.B RFrame
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being used to define the coordinate system of a
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@ -701,13 +749,8 @@ pointing upwards, to contrast with the window system where
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.IR y -values
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grow downwards (see
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.IR graphics (2)).
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.PP
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.IP
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.EX
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#include <u.h>
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#include <libc.h>
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#include <draw.h>
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#include <geometry.h>
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RFrame worldrf;
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/* from screen... */
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@ -733,112 +776,6 @@ main(void)
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worldrf.by = Vec2(0,-1);
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⋯
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.EE
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.PP
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The following snippet shows how to use the
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.B RFrame
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declared earlier to locate and draw a ship based on its orientation,
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for which we use matrix translation
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.B T
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and rotation
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.BR R
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transformations.
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.PP
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.EX
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⋯
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typedef struct Ship Ship;
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typedef struct Shipmdl Shipmdl;
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struct Ship
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{
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RFrame;
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double θ; /* orientation (yaw) */
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Shipmdl mdl;
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};
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struct Shipmdl
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{
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Point2 pts[3]; /* a free-form triangle */
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};
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Ship *ship;
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void
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redraw(void)
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{
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int i;
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Point pts[3+1];
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Point2 *p;
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Matrix T = {
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1, 0, ship->p.x,
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0, 1, ship->p.y,
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0, 0, 1,
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}, R = {
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cos(ship->θ), -sin(ship->θ), 0,
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sin(ship->θ), cos(ship->θ), 0,
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0, 0, 1,
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};
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mulm(T, R); /* rotate, then translate */
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p = ship->mdl.pts;
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for(i = 0; i < nelem(pts)-1; i++)
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pts[i] = fromworld(xform(p[i], T));
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pts[i] = pts[0];
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draw(screen, screen->r, display->white, nil, ZP);
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poly(screen, pts, nelem(pts), 0, 0, 0, display->black, ZP);
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}
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⋯
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main(void)
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⋯
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ship = malloc(sizeof(Ship));
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ship->p = Pt2(0,0,1); /* place it at the origin */
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ship->θ = 45*DEG; /* counter-clockwise */
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ship->mdl.pts[0] = Pt2( 10, 0,1);
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ship->mdl.pts[1] = Pt2(-10, 5,1);
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ship->mdl.pts[2] = Pt2(-10,-5,1);
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⋯
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redraw();
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⋯
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.EE
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.PP
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Notice how we could've used the
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.B RFrame
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embedded in the
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.B ship
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to transform the
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.B Shipmdl
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into the window. Instead of applying the matrices to every point, the
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ship's local frame of reference can be rotated, effectively changing
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the model coordinates after an
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.IR invrframexform .
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We are also getting rid of the
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.B θ
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variable, since it's no longer needed.
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.PP
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.EX
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⋯
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struct Ship
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{
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RFrame;
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Shipmdl mdl;
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};
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⋯
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redraw(void)
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⋯
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pts[i] = fromworld(invrframexform(p[i], *ship));
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⋯
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main(void)
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⋯
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Matrix R = {
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cos(45*DEG), -sin(45*DEG), 0,
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sin(45*DEG), cos(45*DEG), 0,
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0, 0, 1,
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};
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⋯
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//ship->θ = 45*DEG; /* counter-clockwise */
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ship->bx = xform(ship->bx, R);
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ship->by = xform(ship->by, R);
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⋯
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.EE
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.SH SOURCE
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.B /sys/src/libgeometry
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.SH SEE ALSO
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@ -857,6 +794,8 @@ Jonathan Blow,
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The Inner Product, April 2004.
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.br
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https://www.3dgep.com/understanding-quaternions/
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.br
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https://motion.cs.illinois.edu/RoboticSystems/CoordinateTransformations.html
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.SH BUGS
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No care is taken to avoid numeric overflows.
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.SH HISTORY
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@ -2,106 +2,57 @@
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#include <libc.h>
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#include <geometry.h>
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/*
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* implicit identity origin rframes
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*
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* static RFrame IRF2 = {
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* .p = {0,0,1},
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* .bx = {1,0,0},
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* .by = {0,1,0},
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* };
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*
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* static RFrame3 IRF3 = {
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* .p = {0,0,0,1},
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* .bx = {1,0,0,0},
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* .by = {0,1,0,0},
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* .bz = {0,0,1,0},
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* };
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*
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* these rframes are used on every xform to keep the points in the
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* correct plane (i.e. with proper w values); they are written here as a
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* reference for future changes. the bases are ignored since they turn
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* into an unnecessary identity xform.
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*
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* the implicitness comes from the fact that using the _irf* filters
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* makes the rframexform equivalent to:
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* rframexform(invrframexform(p, IRF), rf);
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* and the invrframexform to:
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* rframexform(invrframexform(p, rf), IRF);
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*/
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static Point2
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_irfxform(Point2 p)
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void
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rframematrix(Matrix m, RFrame rf)
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{
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p.w--;
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return p;
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m[0][0] = rf.bx.x; m[0][1] = rf.by.x; m[0][2] = rf.p.x;
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m[1][0] = rf.bx.y; m[1][1] = rf.by.y; m[1][2] = rf.p.y;
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m[2][0] = 0; m[2][1] = 0; m[2][2] = 1;
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}
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static Point2
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_irfxform⁻¹(Point2 p)
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void
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rframematrix3(Matrix3 m, RFrame3 rf)
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{
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p.w++;
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return p;
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}
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static Point3
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_irfxform3(Point3 p)
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{
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p.w--;
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return p;
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}
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static Point3
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_irfxform3⁻¹(Point3 p)
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{
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p.w++;
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return p;
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m[0][0] = rf.bx.x; m[0][1] = rf.by.x; m[0][2] = rf.bz.x; m[0][3] = rf.p.x;
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m[1][0] = rf.bx.y; m[1][1] = rf.by.y; m[1][2] = rf.bz.y; m[1][3] = rf.p.y;
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m[2][0] = rf.bx.z; m[2][1] = rf.by.z; m[2][2] = rf.bz.z; m[2][3] = rf.p.z;
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m[3][0] = 0; m[3][1] = 0; m[3][2] = 0; m[3][3] = 1;
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}
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Point2
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rframexform(Point2 p, RFrame rf)
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{
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Matrix m = {
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rf.bx.x, rf.by.x, 0,
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rf.bx.y, rf.by.y, 0,
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0, 0, 1,
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};
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Matrix m;
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rframematrix(m, rf);
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invm(m);
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return xform(subpt2(_irfxform⁻¹(p), rf.p), m);
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return xform(p, m);
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}
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Point3
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rframexform3(Point3 p, RFrame3 rf)
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{
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Matrix3 m = {
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rf.bx.x, rf.by.x, rf.bz.x, 0,
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rf.bx.y, rf.by.y, rf.bz.y, 0,
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rf.bx.z, rf.by.z, rf.bz.z, 0,
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0, 0, 0, 1,
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};
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Matrix3 m;
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rframematrix3(m, rf);
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invm3(m);
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return xform3(subpt3(_irfxform3⁻¹(p), rf.p), m);
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return xform3(p, m);
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}
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Point2
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invrframexform(Point2 p, RFrame rf)
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{
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Matrix m = {
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rf.bx.x, rf.by.x, 0,
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rf.bx.y, rf.by.y, 0,
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0, 0, 1,
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};
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return _irfxform(addpt2(xform(p, m), rf.p));
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Matrix m;
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rframematrix(m, rf);
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return xform(p, m);
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}
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Point3
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invrframexform3(Point3 p, RFrame3 rf)
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{
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Matrix3 m = {
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rf.bx.x, rf.by.x, rf.bz.x, 0,
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rf.bx.y, rf.by.y, rf.bz.y, 0,
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rf.bx.z, rf.by.z, rf.bz.z, 0,
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0, 0, 0, 1,
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};
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return _irfxform3(addpt3(xform3(p, m), rf.p));
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Matrix3 m;
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rframematrix3(m, rf);
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return xform3(p, m);
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}
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