Some definitions:
(i) MOMENTS: Nth degree moment generating function of an arbitrary distribution D(X) of i samples x1,x2...x(i), around point c, is defined as: SUM(i=1,...,m)[D(x(i))*(x(i)-c)^n]
If the distribution is continuous, the formula turns into
Intergral[of x][D(x)*(x-c)^n][dx]
(ii) CENTROID: Centroid of a volume is the origin of coordinate axes for which first moments of the volume are zero. It is considered center of a volume. For a homogeneous body in a parallel gravity field, mass center and center of gravity coincide with the centroid.
(iii) FIRST MOMENT OF A FINITE DISTRIBUTION D(X(i)) AROUND POINT 'c':
SUM(i=1,...,m)[D(X(i))*(X(i)-c)]
(iv) THUS FOR C TO BE THE CENTROID OF VOLUME:
SUM(i=1,...,m)[D(X(i))*(X(i)-c)] = 0 <=>
D(x1)*(x1-c) + D(x2)*(x2-c) + ... + D(x(i))*(x(i)-c) = 0 <=>
(X1-C) + (X2-C) + ... + (X(i)-C) = 0 <=>
X1+X2+...+Xi = i*c <=>
C = (X1+X2+...+X(i))/i (points' average!)
where D(Xk)=D(Xj) for all k,j in 1,...,i since the body is homogenous
Quote:
@someusername
I am looking for the center of the volume. If you consider this volume as the infently points masses, the integral over all these point masses will give you the center of gravity.(my comments: For a homogeneous body in a parallel gravity field, mass center and center of gravity coincide with the centroid of volume) You only consider 4 vertices from a solid body - what would be wrong...
You don't want to find the volume; you want the centroid. Even if you manage to define a valid continuous distribution of mass for this volume and solve the integral it would give you the very same result, because:
the convex closure of those 8 points in R^3 is not missing any of the volume if you found a symbolic parametrization for it. Nor is it adding any.
[ edit: I'm missing something here! The original 4vertices polygon needs to be convex for this to hold. ]
(And I din't imply 4 vertices, I implied all 8.)
I am trying to make my point through math here. If my math is wrong then my point is wrong!
This is the last post from me on this subject
[Edited by - someusername on December 1, 2005 12:03:17 PM]