compute volume of transformed bodies - math

Question

So I want to compute the volume of spheres (unit balls), cuboids (cubes) that are transformed using arbitrary transformation matrices.
e.g.: I got my Sphere with a radius of 1 in the center of my 3D-Space. Then i got a Transformation to apply to that Sphere. What would the volume (resp. radius) be after that? How can i extract that information from the Transformation Matrix? I know that translation and rotation matrices wont affect that, but scaling matrices will.
Thanks in Advance!

Answer 1

The transformation's determinant specifies how the volume of any object changes. If the determinant is 1, the volume is preserved. If it is negative, there is a mirroring included which results in a reversed face order.
Only use the linear part of the matrix to calculate the determinant (disregarding translations and perspective transformations).
Another measure that you might be interested in are the matrice's eigen values and eigen vectors (or singular values). They specify the orientation of the ellipsoid that a matrix transforms a unit sphere into.

Answer 2

If the transformation is a rigid motion, then the mass is unaffected by the transform. If the transform is the composition of a rigid motion and a uniform scaling, then the volume is multiplied by the cube of the scale factor.
That is, if S is a homogeneous scaling transform in the form:
[ s 0 0 0 ]
[ 0 s 0 0 ]
S = [ 0 0 s 0 ]
[ 0 0 0 1 ]
and R is a rigid homogeneous transform, then the mass m of a body transformed by S.R or R.S is m*s^3.
Other than these two special cases (rigid motion and uniform scaled motion), and ignoring trivial cases such as the case where the transform maps all points to a plane, there is no simple formula for the transformed volumes.

Answer 3

OpenGL transform matrix is stored like this:
double m[16]; // it is 4x4 matrix stored as 1 dimensional array for speed
m[0]=xx; m[4]=yx; m[ 8]=zx; m[12]=x0;
m[1]=xy; m[5]=yy; m[ 9]=zy; m[13]=y0;
m[2]=xz; m[6]=yz; m[10]=zz; m[14]=z0;
m[3]= 0; m[7]= 0; m[11]= 0; m[15]= 1;
where:
X(xx,xy,xz) is unit vector of X axis in GCS (global coordinate system)
Y(yx,yy,yz) is unit vector of Y axis in GCS
Z(zx,zy,zz) is unit vector of Z axis in GCS
P(x0,y0,z0) is origin of represented coordinate system in GCS
when orthogonal transforms are applied (rotation,translation,scale...)
then shape is 'preserved'
cube became cuboid and sphere became ellipsoid
then the size of axises vector is the scale in that axis
so if Volume of object before transformation is V0
then Volume after transform is:
V1=V0*|X|*|Y|*|Z|
when non orthogonal transforms are applied (skew,projections,...)
skew is one of the non-orthogonal but linear transforms
Volume can be computed as for orthogonal transforms
but the shape will be broken.
because non orthogonal transforms change
not only size but the shape itself !!!
non-linear transforms like projections always broke the Volume nonlinearly
in this case you have to compute the Volume the hard way
transform object surface and compute the Volume via integration ...
How to determine orthogonality of transform?
easy transforms are orthogonal if X,Y,Z vectors are perpendicular
to each other then transform is linear so:
XY = X x Y; XY /= |XY|; Z0=Z/|Z|;
XZ = X x Z; XZ /= |XZ|; Y0=Y/|Y|;
YZ = Y x Z; YZ /= |YZ|; X0=X/|X|;
if ((XY!=Z0)&&(XY!=-Z0)) non linear;
if ((XZ!=Y0)&&(XZ!=-Y0)) non linear;
if ((YZ!=X0)&&(YZ!=-X0)) non linear;
do not forget to do the comparisons with some accuracy
to avoid FP problems
How to detect non linearity of transform?
create a set of points evenly placed inside some cube (same distance between them)
transform them
now if the point distances are not the same along new axises
but change with distance from origin then transform is non-linear