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# Determining Center of Gravity, Principal Moments, Volume and Surface Area of a Polyhedron

IP.com Disclosure Number: IPCOM000085347D
Original Publication Date: 1976-Mar-01
Included in the Prior Art Database: 2005-Mar-02
Document File: 3 page(s) / 28K

IBM

Appel, A: AUTHOR

## Abstract

A technique to find a three-dimensional center of gravity is as follows: 1. Read in the vertex list and the topological map which describes the polyhedron. For a typical object as shown in the figure, a typical vertex list is: POINT X Y Z 1 1. .5 3. 2 -1. .5 3. 3 01. 1.3 3. . . . . . . . . . . . . . . . .

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Determining Center of Gravity, Principal Moments, Volume and Surface Area of a Polyhedron

A technique to find a three-dimensional center of gravity is as follows: 1. Read in the vertex list and the topological map which

describes the polyhedron. For a typical object as shown

in the figure, a typical vertex list is:

POINT X Y Z

1 1. .5 3.

2 -1. .5 3.

3 01. 1.3 3.

. . . .

. . . .

. . . .

. . . .

A typical topological map is:

SURFACE CONNECTIVITY OF POINTS

1 1,12,10,8,5,4,0

2 1,2,11,12,0

. .

. .

. .

. .

2. Calculate the equation of each face of the polyhedron from

three points on the surface:

X(1) Y(1) Z(1)

A X(2) +B Y(2) +C Z(2) +D = 0

X(3) Y(3) Z(3)

3. Check to make certain that the vector [A,B,C] points into

the volume of the polyhedron. If not, reverse the signs of

A,B,C,D. 0r make certain that all surface connectivity

data is definitely clockwise or counterclockwise.

4. Take each face of the polyhedron, one at a time:

4.1 Take each line on the boundary of the face, one at

a time;

4.2 Project this line onto the coordinate plane which

forms the smallest angle with the face the line

lies on;

4.3 Consider this line projection and the origin as a

triangle on the coordinate plane. Find the area

and center of gravity of this triangle;

4.4 Calculate the moment of the triangle about the

three-coordinate axis:

M(x) = (C.G)(x) X AREA

M(y) = (C.G)(y) X AREA

4.5 Sum up the areas due to each line of the face:

AREA TOTAL = Sigma area each triangle.

4.6 Sum up each area moment:

TOTAL M(x) = Sigma M(x) each triangle.

1

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TOTAL M(y) = Sigma M(y) each triangle.

4.7 Calculate the area of the face by:

AREA(F) = (AREA TOTAL)/C if projected onto XY

plane.

AREA(F) = (AREA TOTAL)/B if projected onto XZ

plane.

AREA(F) = (AREA TOTAL)/A if projected onto YZ

plane.

4.8 Calculate C.G. of the face in three dimensions by:

If projected onto XY plane;

CGX = AREA(F)/TOTAL M(x)

CGY = AREA(F)/TOTAL M(y)

CGZ =-(B*CGY+A*CGX+D)/C

If projected onto XZ plane;

CGX = AREA(F)/TOTAL M(x)

CGZ = AREA(F)/TOTAL M(z)

CGY =-(A*CGX+C*CGZ+D)/B

If projected onto YZ plane;

CGY = AREA(F)/TOTAL M(y)

CGZ = AREA(F)/TOTAL M(z)

CGX =-(B*CGY+C*CGZ+D)/A.

The area of a face is now known and its three-dimensional center of gravity. 5. Consider the face as a base of a pyramid with the origin

as the...