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STYLE: remove momentOfInertia postscript and example files
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Mark Olesen
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Andrew Heather
Andrew Heather
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commit
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@ -31,6 +31,17 @@ Description
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polyhedra/cells/triSurfaces. Inertia can either be of the solid body or
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of a thin shell.
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References:
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\verbatim
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"Fast and Accurate Computation of Polyhedral Mass Properties",
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Brian Mirtich,
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Journal of Graphics Tools, Volume 1, Number 2, 1996.
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\endverbatim
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https://people.eecs.berkeley.edu/~jfc/mirtich/massProps.html
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https://people.eecs.berkeley.edu/~jfc/mirtich/papers/volInt.ps.gz
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https://people.eecs.berkeley.edu/~jfc/mirtich/code/volumeIntegration.tar
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SourceFiles
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momentOfInertia.H
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Binary file not shown.
@ -1,146 +0,0 @@
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1. OVERVIEW
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This code accompanies the paper:
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Brian Mirtich, "Fast and Accurate Computation of
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Polyhedral Mass Properties," journal of graphics
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tools, volume 1, number 2, 1996.
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It computes the ten volume integrals needed for
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determining the center of mass, moments of
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inertia, and products of inertia for a uniform
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density polyhedron. From this information, a
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body frame can be computed.
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To compile the program, use an ANSI compiler, and
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type something like
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% cc volInt.c -O2 -lm -o volInt
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Revision history
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26 Jan 1996 Program creation.
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3 Aug 1996 Corrected bug arising when polyhedron density
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is not 1.0. Changes confined to function main().
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Thanks to Zoran Popovic for catching this one.
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2. POLYHEDRON GEOMETRY FILES
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The program reads a data file specified on the
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command line. This data file describes the
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geometry of a polyhedron, and has the following
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format:
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N
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x_0 y_0 z_0
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x_1 y_1 z_1
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.
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.
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.
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x_{N-1} y_{N-1} z_{N-1}
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M
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k1 v_{1,1} v_{1,2} ... v_{1,k1}
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k2 v_{2,1} v_{2,2} ... v_{2,k2}
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.
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.
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.
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kM v_{M,1} v_{M,2} ... v_{M,kM}
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where:
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N number of vertices on polyhedron
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x_i y_i z_i x, y, and z coordinates of ith vertex
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M number of faces on polyhedron
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ki number of vertices on ith face
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v_{i,j} jth vertex on ith face
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In English:
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First the number of vertices are specified. Next
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the vertices are defined by listing the 3
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coordinates of each one. Next the number of faces
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are specified. Finally, the faces themselves are
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specified. A face is specified by first giving
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the number of vertices around the polygonal face,
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followed by the (integer) indices of these
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vertices. When specifying indices, note that
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they must be given in counter-clockwise order
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(when looking at the face from outside the
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polyhedron), and the vertices are indexed from 0
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to N-1 for a polyhedron with N faces.
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White space can be inserted (or not) as desired.
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Three example polyhedron geometry files are included:
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cube A cube, 20 units on a side, centered at
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the origin and aligned with the coordinate axes.
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tetra A tetrahedron formed by taking the convex
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hull of the origin, and the points (5,0,0),
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(0,4,0), and (0,0,3).
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icosa An icosahedron with vertices lying on the unit
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sphere, centered at the origin.
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3. RUNNING THE PROGRAM
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Simply type,
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% volInt <polyhedron geometry filename>
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The program will read in the geometry of the
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polyhedron, and the print out the ten volume
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integrals.
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The program also computes some of the mass
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properties which may be inferred from the volume
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integrals. A density of 1.0 is assumed, although
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this may be changed in function main(). The
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center of mass is computed as well as the inertia
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tensor relative to a frame with origin at the
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center of mass. Note, however, that this matrix
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may not be diagonal. If not, a diagonalization
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routine must be performed, to determine the
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orientation of the true body frame relative to
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the original model frame. The Jacobi method
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works quite well (see Numerical Recipes in C by
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Press, et. al.).
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4. DISCLAIMERS
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1. The volume integration code has been written
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to match the development and algorithms presented
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in the paper, and not with maximum optimization
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in mind. While inherently very efficient, a few
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more cycles can be squeezed out of the algorithm.
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This is left as an exercise. :)
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2. Don't like global variables? The three
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procedures which evaluate the volume integrals
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can be combined into a single procedure with two
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nested loops. In addition to providing some
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speedup, all of the global variables can then be
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made local.
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3. The polyhedron data structure used by the
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program is admittedly lame; much better schemes
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are possible. The idea here is just to give the
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basic integral evaluation code, which will have
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to be adjusted for other polyhedron data
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structures.
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4. There is no error checking for the input
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files. Be careful. Note the hard limits
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#defined for the number of vertices, number of
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faces, and number of vertices per faces.
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@ -1,21 +0,0 @@
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8
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-10 -10 -10
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+10 -10 -10
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+10 +10 -10
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-10 +10 -10
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-10 -10 +10
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+10 -10 +10
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+10 +10 +10
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-10 +10 +10
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6
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4 0 3 2 1
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4 4 5 6 7
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4 0 1 5 4
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4 6 2 3 7
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4 1 2 6 5
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4 0 4 7 3
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@ -1,38 +0,0 @@
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12
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+0.00000000 +0.00000000 +1.00000000
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+0.00000000 +0.00000000 -1.00000000
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+0.89442719 +0.00000000 +0.44721360
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+0.27639320 +0.85065081 +0.44721360
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-0.72360680 +0.52573111 +0.44721360
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-0.72360680 -0.52573111 +0.44721360
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+0.27639320 -0.85065081 +0.44721360
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+0.72360680 +0.52573111 -0.44721360
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-0.27639320 +0.85065081 -0.44721360
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-0.89442719 +0.00000000 -0.44721360
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-0.27639320 -0.85065081 -0.44721360
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+0.72360680 -0.52573111 -0.44721360
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20
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3 6 11 2
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3 3 2 7
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3 7 2 11
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3 0 2 3
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3 0 6 2
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3 10 1 11
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3 1 7 11
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3 10 11 6
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3 8 7 1
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3 8 3 7
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3 5 10 6
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3 5 6 0
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3 4 3 8
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3 4 0 3
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3 9 8 1
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3 9 1 10
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3 4 5 0
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3 9 10 5
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3 9 5 4
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3 9 4 8
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@ -1,16 +0,0 @@
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4
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0 0 0
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5 0 0
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0 4 0
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0 0 3
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4
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3 0 3 2
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3 3 0 1
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3 2 1 0
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3 1 2 3
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@ -1,380 +0,0 @@
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/*******************************************************
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* *
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* volInt.c *
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* *
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* This code computes volume integrals needed for *
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* determining mass properties of polyhedral bodies. *
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* *
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* For more information, see the accompanying README *
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* file, and the paper *
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* *
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* Brian Mirtich, "Fast and Accurate Computation of *
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* Polyhedral Mass Properties," journal of graphics *
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* tools, volume 1, number 1, 1996. *
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* *
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* This source code is public domain, and may be used *
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* in any way, shape or form, free of charge. *
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* *
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* Copyright 1995 by Brian Mirtich *
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* *
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* mirtich@cs.berkeley.edu *
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* http://www.cs.berkeley.edu/~mirtich *
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* *
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*******************************************************/
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/*
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Revision history
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26 Jan 1996 Program creation.
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3 Aug 1996 Corrected bug arising when polyhedron density
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is not 1.0. Changes confined to function main().
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Thanks to Zoran Popovic for catching this one.
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27 May 1997 Corrected sign error in translation of inertia
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product terms to center of mass frame. Changes
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confined to function main(). Thanks to
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Chris Hecker.
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*/
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#include <stdio.h>
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#include <string.h>
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#include <math.h>
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/*
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============================================================================
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constants
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============================================================================
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*/
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#define MAX_VERTS 100 /* maximum number of polyhedral vertices */
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#define MAX_FACES 100 /* maximum number of polyhedral faces */
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#define MAX_POLYGON_SZ 10 /* maximum number of verts per polygonal face */
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#define X 0
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#define Y 1
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#define Z 2
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/*
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============================================================================
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macros
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============================================================================
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*/
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#define SQR(x) ((x)*(x))
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#define CUBE(x) ((x)*(x)*(x))
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/*
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============================================================================
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data structures
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============================================================================
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*/
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typedef struct {
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int numVerts;
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double norm[3];
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double w;
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int verts[MAX_POLYGON_SZ];
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struct polyhedron *poly;
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} FACE;
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typedef struct polyhedron {
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int numVerts, numFaces;
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double verts[MAX_VERTS][3];
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FACE faces[MAX_FACES];
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} POLYHEDRON;
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/*
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============================================================================
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globals
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============================================================================
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*/
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static int A; /* alpha */
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static int B; /* beta */
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static int C; /* gamma */
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/* projection integrals */
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static double P1, Pa, Pb, Paa, Pab, Pbb, Paaa, Paab, Pabb, Pbbb;
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/* face integrals */
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static double Fa, Fb, Fc, Faa, Fbb, Fcc, Faaa, Fbbb, Fccc, Faab, Fbbc, Fcca;
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/* volume integrals */
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static double T0, T1[3], T2[3], TP[3];
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/*
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============================================================================
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read in a polyhedron
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============================================================================
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*/
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void readPolyhedron(char *name, POLYHEDRON *p)
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{
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FILE *fp;
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char line[200], *c;
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int i, j, n;
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double dx1, dy1, dz1, dx2, dy2, dz2, nx, ny, nz, len;
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FACE *f;
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if (!(fp = fopen(name, "r"))) {
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printf("i/o error\n");
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exit(1);
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}
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fscanf(fp, "%d", &p->numVerts);
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printf("Reading in %d vertices\n", p->numVerts);
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for (i = 0; i < p->numVerts; i++)
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fscanf(fp, "%lf %lf %lf",
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&p->verts[i][X], &p->verts[i][Y], &p->verts[i][Z]);
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fscanf(fp, "%d", &p->numFaces);
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printf("Reading in %d faces\n", p->numFaces);
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for (i = 0; i < p->numFaces; i++) {
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f = &p->faces[i];
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f->poly = p;
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fscanf(fp, "%d", &f->numVerts);
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for (j = 0; j < f->numVerts; j++) fscanf(fp, "%d", &f->verts[j]);
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/* compute face normal and offset w from first 3 vertices */
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dx1 = p->verts[f->verts[1]][X] - p->verts[f->verts[0]][X];
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dy1 = p->verts[f->verts[1]][Y] - p->verts[f->verts[0]][Y];
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dz1 = p->verts[f->verts[1]][Z] - p->verts[f->verts[0]][Z];
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dx2 = p->verts[f->verts[2]][X] - p->verts[f->verts[1]][X];
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dy2 = p->verts[f->verts[2]][Y] - p->verts[f->verts[1]][Y];
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dz2 = p->verts[f->verts[2]][Z] - p->verts[f->verts[1]][Z];
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nx = dy1 * dz2 - dy2 * dz1;
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ny = dz1 * dx2 - dz2 * dx1;
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nz = dx1 * dy2 - dx2 * dy1;
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len = sqrt(nx * nx + ny * ny + nz * nz);
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f->norm[X] = nx / len;
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f->norm[Y] = ny / len;
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f->norm[Z] = nz / len;
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f->w = - f->norm[X] * p->verts[f->verts[0]][X]
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- f->norm[Y] * p->verts[f->verts[0]][Y]
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- f->norm[Z] * p->verts[f->verts[0]][Z];
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}
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fclose(fp);
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|
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}
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|
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/*
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============================================================================
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compute mass properties
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============================================================================
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*/
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/* compute various integrations over projection of face */
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void compProjectionIntegrals(FACE *f)
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{
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double a0, a1, da;
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double b0, b1, db;
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double a0_2, a0_3, a0_4, b0_2, b0_3, b0_4;
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double a1_2, a1_3, b1_2, b1_3;
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double C1, Ca, Caa, Caaa, Cb, Cbb, Cbbb;
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double Cab, Kab, Caab, Kaab, Cabb, Kabb;
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int i;
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|
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P1 = Pa = Pb = Paa = Pab = Pbb = Paaa = Paab = Pabb = Pbbb = 0.0;
|
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|
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for (i = 0; i < f->numVerts; i++) {
|
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a0 = f->poly->verts[f->verts[i]][A];
|
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b0 = f->poly->verts[f->verts[i]][B];
|
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a1 = f->poly->verts[f->verts[(i+1) % f->numVerts]][A];
|
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b1 = f->poly->verts[f->verts[(i+1) % f->numVerts]][B];
|
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da = a1 - a0;
|
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db = b1 - b0;
|
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a0_2 = a0 * a0; a0_3 = a0_2 * a0; a0_4 = a0_3 * a0;
|
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b0_2 = b0 * b0; b0_3 = b0_2 * b0; b0_4 = b0_3 * b0;
|
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a1_2 = a1 * a1; a1_3 = a1_2 * a1;
|
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b1_2 = b1 * b1; b1_3 = b1_2 * b1;
|
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|
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C1 = a1 + a0;
|
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Ca = a1*C1 + a0_2; Caa = a1*Ca + a0_3; Caaa = a1*Caa + a0_4;
|
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Cb = b1*(b1 + b0) + b0_2; Cbb = b1*Cb + b0_3; Cbbb = b1*Cbb + b0_4;
|
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Cab = 3*a1_2 + 2*a1*a0 + a0_2; Kab = a1_2 + 2*a1*a0 + 3*a0_2;
|
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Caab = a0*Cab + 4*a1_3; Kaab = a1*Kab + 4*a0_3;
|
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Cabb = 4*b1_3 + 3*b1_2*b0 + 2*b1*b0_2 + b0_3;
|
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Kabb = b1_3 + 2*b1_2*b0 + 3*b1*b0_2 + 4*b0_3;
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|
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P1 += db*C1;
|
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Pa += db*Ca;
|
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Paa += db*Caa;
|
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Paaa += db*Caaa;
|
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Pb += da*Cb;
|
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Pbb += da*Cbb;
|
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Pbbb += da*Cbbb;
|
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Pab += db*(b1*Cab + b0*Kab);
|
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Paab += db*(b1*Caab + b0*Kaab);
|
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Pabb += da*(a1*Cabb + a0*Kabb);
|
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}
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|
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P1 /= 2.0;
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Pa /= 6.0;
|
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Paa /= 12.0;
|
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Paaa /= 20.0;
|
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Pb /= -6.0;
|
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Pbb /= -12.0;
|
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Pbbb /= -20.0;
|
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Pab /= 24.0;
|
||||
Paab /= 60.0;
|
||||
Pabb /= -60.0;
|
||||
}
|
||||
|
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compFaceIntegrals(FACE *f)
|
||||
{
|
||||
double *n, w;
|
||||
double k1, k2, k3, k4;
|
||||
|
||||
compProjectionIntegrals(f);
|
||||
|
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w = f->w;
|
||||
n = f->norm;
|
||||
k1 = 1 / n[C]; k2 = k1 * k1; k3 = k2 * k1; k4 = k3 * k1;
|
||||
|
||||
Fa = k1 * Pa;
|
||||
Fb = k1 * Pb;
|
||||
Fc = -k2 * (n[A]*Pa + n[B]*Pb + w*P1);
|
||||
|
||||
Faa = k1 * Paa;
|
||||
Fbb = k1 * Pbb;
|
||||
Fcc = k3 * (SQR(n[A])*Paa + 2*n[A]*n[B]*Pab + SQR(n[B])*Pbb
|
||||
+ w*(2*(n[A]*Pa + n[B]*Pb) + w*P1));
|
||||
|
||||
Faaa = k1 * Paaa;
|
||||
Fbbb = k1 * Pbbb;
|
||||
Fccc = -k4 * (CUBE(n[A])*Paaa + 3*SQR(n[A])*n[B]*Paab
|
||||
+ 3*n[A]*SQR(n[B])*Pabb + CUBE(n[B])*Pbbb
|
||||
+ 3*w*(SQR(n[A])*Paa + 2*n[A]*n[B]*Pab + SQR(n[B])*Pbb)
|
||||
+ w*w*(3*(n[A]*Pa + n[B]*Pb) + w*P1));
|
||||
|
||||
Faab = k1 * Paab;
|
||||
Fbbc = -k2 * (n[A]*Pabb + n[B]*Pbbb + w*Pbb);
|
||||
Fcca = k3 * (SQR(n[A])*Paaa + 2*n[A]*n[B]*Paab + SQR(n[B])*Pabb
|
||||
+ w*(2*(n[A]*Paa + n[B]*Pab) + w*Pa));
|
||||
}
|
||||
|
||||
void compVolumeIntegrals(POLYHEDRON *p)
|
||||
{
|
||||
FACE *f;
|
||||
double nx, ny, nz;
|
||||
int i;
|
||||
|
||||
T0 = T1[X] = T1[Y] = T1[Z]
|
||||
= T2[X] = T2[Y] = T2[Z]
|
||||
= TP[X] = TP[Y] = TP[Z] = 0;
|
||||
|
||||
for (i = 0; i < p->numFaces; i++) {
|
||||
|
||||
f = &p->faces[i];
|
||||
|
||||
nx = fabs(f->norm[X]);
|
||||
ny = fabs(f->norm[Y]);
|
||||
nz = fabs(f->norm[Z]);
|
||||
if (nx > ny && nx > nz) C = X;
|
||||
else C = (ny > nz) ? Y : Z;
|
||||
A = (C + 1) % 3;
|
||||
B = (A + 1) % 3;
|
||||
|
||||
compFaceIntegrals(f);
|
||||
|
||||
T0 += f->norm[X] * ((A == X) ? Fa : ((B == X) ? Fb : Fc));
|
||||
|
||||
T1[A] += f->norm[A] * Faa;
|
||||
T1[B] += f->norm[B] * Fbb;
|
||||
T1[C] += f->norm[C] * Fcc;
|
||||
T2[A] += f->norm[A] * Faaa;
|
||||
T2[B] += f->norm[B] * Fbbb;
|
||||
T2[C] += f->norm[C] * Fccc;
|
||||
TP[A] += f->norm[A] * Faab;
|
||||
TP[B] += f->norm[B] * Fbbc;
|
||||
TP[C] += f->norm[C] * Fcca;
|
||||
}
|
||||
|
||||
T1[X] /= 2; T1[Y] /= 2; T1[Z] /= 2;
|
||||
T2[X] /= 3; T2[Y] /= 3; T2[Z] /= 3;
|
||||
TP[X] /= 2; TP[Y] /= 2; TP[Z] /= 2;
|
||||
}
|
||||
|
||||
|
||||
/*
|
||||
============================================================================
|
||||
main
|
||||
============================================================================
|
||||
*/
|
||||
|
||||
|
||||
int main(int argc, char *argv[])
|
||||
{
|
||||
POLYHEDRON p;
|
||||
double density, mass;
|
||||
double r[3]; /* center of mass */
|
||||
double J[3][3]; /* inertia tensor */
|
||||
|
||||
if (argc != 2) {
|
||||
printf("usage: %s <polyhedron geometry filename>\n", argv[0]);
|
||||
exit(0);
|
||||
}
|
||||
|
||||
readPolyhedron(argv[1], &p);
|
||||
|
||||
compVolumeIntegrals(&p);
|
||||
|
||||
|
||||
printf("\nT1 = %+20.6f\n\n", T0);
|
||||
|
||||
printf("Tx = %+20.6f\n", T1[X]);
|
||||
printf("Ty = %+20.6f\n", T1[Y]);
|
||||
printf("Tz = %+20.6f\n\n", T1[Z]);
|
||||
|
||||
printf("Txx = %+20.6f\n", T2[X]);
|
||||
printf("Tyy = %+20.6f\n", T2[Y]);
|
||||
printf("Tzz = %+20.6f\n\n", T2[Z]);
|
||||
|
||||
printf("Txy = %+20.6f\n", TP[X]);
|
||||
printf("Tyz = %+20.6f\n", TP[Y]);
|
||||
printf("Tzx = %+20.6f\n\n", TP[Z]);
|
||||
|
||||
density = 1.0; /* assume unit density */
|
||||
|
||||
mass = density * T0;
|
||||
|
||||
/* compute center of mass */
|
||||
r[X] = T1[X] / T0;
|
||||
r[Y] = T1[Y] / T0;
|
||||
r[Z] = T1[Z] / T0;
|
||||
|
||||
/* compute inertia tensor */
|
||||
J[X][X] = density * (T2[Y] + T2[Z]);
|
||||
J[Y][Y] = density * (T2[Z] + T2[X]);
|
||||
J[Z][Z] = density * (T2[X] + T2[Y]);
|
||||
J[X][Y] = J[Y][X] = - density * TP[X];
|
||||
J[Y][Z] = J[Z][Y] = - density * TP[Y];
|
||||
J[Z][X] = J[X][Z] = - density * TP[Z];
|
||||
|
||||
/* translate inertia tensor to center of mass */
|
||||
J[X][X] -= mass * (r[Y]*r[Y] + r[Z]*r[Z]);
|
||||
J[Y][Y] -= mass * (r[Z]*r[Z] + r[X]*r[X]);
|
||||
J[Z][Z] -= mass * (r[X]*r[X] + r[Y]*r[Y]);
|
||||
J[X][Y] = J[Y][X] += mass * r[X] * r[Y];
|
||||
J[Y][Z] = J[Z][Y] += mass * r[Y] * r[Z];
|
||||
J[Z][X] = J[X][Z] += mass * r[Z] * r[X];
|
||||
|
||||
printf("center of mass: (%+12.6f,%+12.6f,%+12.6f)\n\n", r[X], r[Y], r[Z]);
|
||||
|
||||
printf("inertia tensor with origin at c.o.m. :\n");
|
||||
printf("%+15.6f %+15.6f %+15.6f\n", J[X][X], J[X][Y], J[X][Z]);
|
||||
printf("%+15.6f %+15.6f %+15.6f\n", J[Y][X], J[Y][Y], J[Y][Z]);
|
||||
printf("%+15.6f %+15.6f %+15.6f\n\n", J[Z][X], J[Z][Y], J[Z][Z]);
|
||||
|
||||
}
|
||||
Reference in New Issue
Block a user