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src/meshTools/momentOfInertia/volumeIntegration/README
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1. OVERVIEW
This code accompanies the paper:
This code accompanies the paper:
Brian Mirtich, "Fast and Accurate Computation of
Polyhedral Mass Properties," journal of graphics
tools, volume 1, number 2, 1996.
Brian Mirtich, "Fast and Accurate Computation of
Polyhedral Mass Properties," journal of graphics
tools, volume 1, number 2, 1996.
It computes the ten volume integrals needed for
determining the center of mass, moments of
inertia, and products of inertia for a uniform
density polyhedron. From this information, a
body frame can be computed.
It computes the ten volume integrals needed for
determining the center of mass, moments of
inertia, and products of inertia for a uniform
density polyhedron. From this information, a
body frame can be computed.
To compile the program, use an ANSI compiler, and
type something like
% cc volInt.c -O2 -lm -o volInt
To compile the program, use an ANSI compiler, and
type something like
% cc volInt.c -O2 -lm -o volInt
Revision history
Revision history
26 Jan 1996 Program creation.
26 Jan 1996 Program creation.
3 Aug 1996 Corrected bug arising when polyhedron density
is not 1.0. Changes confined to function main().
Thanks to Zoran Popovic for catching this one.
3 Aug 1996 Corrected bug arising when polyhedron density
is not 1.0. Changes confined to function main().
Thanks to Zoran Popovic for catching this one.
2. POLYHEDRON GEOMETRY FILES
The program reads a data file specified on the
command line. This data file describes the
geometry of a polyhedron, and has the following
format:
The program reads a data file specified on the
command line. This data file describes the
geometry of a polyhedron, and has the following
format:
N
N
x_0 y_0 z_0
x_1 y_1 z_1
.
.
.
x_{N-1} y_{N-1} z_{N-1}
x_0 y_0 z_0
x_1 y_1 z_1
.
.
.
x_{N-1} y_{N-1} z_{N-1}
M
M
k1 v_{1,1} v_{1,2} ... v_{1,k1}
k2 v_{2,1} v_{2,2} ... v_{2,k2}
.
.
.
kM v_{M,1} v_{M,2} ... v_{M,kM}
k1 v_{1,1} v_{1,2} ... v_{1,k1}
k2 v_{2,1} v_{2,2} ... v_{2,k2}
.
.
.
kM v_{M,1} v_{M,2} ... v_{M,kM}
where:
N number of vertices on polyhedron
x_i y_i z_i x, y, and z coordinates of ith vertex
M number of faces on polyhedron
ki number of vertices on ith face
v_{i,j} jth vertex on ith face
where:
N number of vertices on polyhedron
x_i y_i z_i x, y, and z coordinates of ith vertex
M number of faces on polyhedron
ki number of vertices on ith face
v_{i,j} jth vertex on ith face
In English:
In English:
First the number of vertices are specified. Next
the vertices are defined by listing the 3
coordinates of each one. Next the number of faces
are specified. Finally, the faces themselves are
specified. A face is specified by first giving
the number of vertices around the polygonal face,
followed by the (integer) indices of these
vertices. When specifying indices, note that
they must be given in counter-clockwise order
(when looking at the face from outside the
polyhedron), and the vertices are indexed from 0
to N-1 for a polyhedron with N faces.
First the number of vertices are specified. Next
the vertices are defined by listing the 3
coordinates of each one. Next the number of faces
are specified. Finally, the faces themselves are
specified. A face is specified by first giving
the number of vertices around the polygonal face,
followed by the (integer) indices of these
vertices. When specifying indices, note that
they must be given in counter-clockwise order
(when looking at the face from outside the
polyhedron), and the vertices are indexed from 0
to N-1 for a polyhedron with N faces.
White space can be inserted (or not) as desired.
Three example polyhedron geometry files are included:
White space can be inserted (or not) as desired.
Three example polyhedron geometry files are included:
cube A cube, 20 units on a side, centered at
the origin and aligned with the coordinate axes.
cube A cube, 20 units on a side, centered at
the origin and aligned with the coordinate axes.
tetra A tetrahedron formed by taking the convex
hull of the origin, and the points (5,0,0),
(0,4,0), and (0,0,3).
tetra A tetrahedron formed by taking the convex
hull of the origin, and the points (5,0,0),
(0,4,0), and (0,0,3).
icosa An icosahedron with vertices lying on the unit
sphere, centered at the origin.
icosa An icosahedron with vertices lying on the unit
sphere, centered at the origin.
3. RUNNING THE PROGRAM
Simply type,
% volInt <polyhedron geometry filename>
Simply type,
The program will read in the geometry of the
polyhedron, and the print out the ten volume
integrals.
% volInt <polyhedron geometry filename>
The program also computes some of the mass
properties which may be inferred from the volume
integrals. A density of 1.0 is assumed, although
this may be changed in function main(). The
center of mass is computed as well as the inertia
tensor relative to a frame with origin at the
center of mass. Note, however, that this matrix
may not be diagonal. If not, a diagonalization
routine must be performed, to determine the
orientation of the true body frame relative to
the original model frame. The Jacobi method
works quite well (see Numerical Recipes in C by
Press, et. al.).
The program will read in the geometry of the
polyhedron, and the print out the ten volume
integrals.
The program also computes some of the mass
properties which may be inferred from the volume
integrals. A density of 1.0 is assumed, although
this may be changed in function main(). The
center of mass is computed as well as the inertia
tensor relative to a frame with origin at the
center of mass. Note, however, that this matrix
may not be diagonal. If not, a diagonalization
routine must be performed, to determine the
orientation of the true body frame relative to
the original model frame. The Jacobi method
works quite well (see Numerical Recipes in C by
Press, et. al.).
4. DISCLAIMERS
1. The volume integration code has been written
to match the development and algorithms presented
in the paper, and not with maximum optimization
in mind. While inherently very efficient, a few
more cycles can be squeezed out of the algorithm.
This is left as an exercise. :)
1. The volume integration code has been written
to match the development and algorithms presented
in the paper, and not with maximum optimization
in mind. While inherently very efficient, a few
more cycles can be squeezed out of the algorithm.
This is left as an exercise. :)
2. Don't like global variables? The three
procedures which evaluate the volume integrals
can be combined into a single procedure with two
nested loops. In addition to providing some
speedup, all of the global variables can then be
made local.
2. Don't like global variables? The three
procedures which evaluate the volume integrals
can be combined into a single procedure with two
nested loops. In addition to providing some
speedup, all of the global variables can then be
made local.
3. The polyhedron data structure used by the
program is admittedly lame; much better schemes
are possible. The idea here is just to give the
basic integral evaluation code, which will have
to be adjusted for other polyhedron data
structures.
3. The polyhedron data structure used by the
program is admittedly lame; much better schemes
are possible. The idea here is just to give the
basic integral evaluation code, which will have
to be adjusted for other polyhedron data
structures.
4. There is no error checking for the input
files. Be careful. Note the hard limits
#defined for the number of vertices, number of
faces, and number of vertices per faces.
4. There is no error checking for the input
files. Be careful. Note the hard limits
#defined for the number of vertices, number of
faces, and number of vertices per faces.