Hi everyone,
I have the following situation: Let T be a two-dimensional manifold in R^3 (in particular I am working with spherical triangles on the unit sphere) and C = conv(T) be its convex hull (in the spherical triangle case this is just the union of the spherical triangle and its flat triangle, i.e. the standard triangle spanned by the vertices A,B and C that also define the spherical triangle, plus the parts of the planes E_1 = {lambda*A+mu*B}, E_2 = {lambda*A+mu*C}, E_3 = {lambda*B+mu*C} which are between the line segment AB (AC, BC, respectively) and the corresponding circular arc through AB).
Now I am searching for a function which is nonnegative in this convex hull, has a maximum at the center of mass of T, i.e. at int_T (x,y,z) dx dy dz / |T| which is inside of C and is zero on the boundary of the convex hull.
An example of this is the octant spanned by A = (1,0,0), B = (0,1,0) and C = (0,0,1). Then, the center of mass is given by (0.5,0.5,0.5) and a function that satisfies the above constraints is
f(x,y,z) = x*y*z*(x+y+z-1)*(1-x^2-y^2-z^2).
Unfortunately, the structure dot(n1,X)*dot(n2,X)*dot(n3,X)*(dot(n4,X)-dot(n4,A))*(1-x^2-y^2-z^2) (where X = (x,y,z) and n_i are the normals of the planes E_i and n_4 is the normal of the plane through the flat triangle) does not have a maximum at the center of mass (although it had in the above example).
Any suggestions how to fix this?
Maybe you are looking for a function with additional
properties, that you do not state explicitly.
As it is formulated now, the problem has an easy
solution: you can take a ball B centered in the center
of mass and contained in the convex hall of T,
and then to build a function that has maximum in the
center, vanishes on the sphere of B and vanishes
outside of B (and hence vanishes on the boundary
of convT as well).
If you are looking for a function that vanishes ONLY
on the boundary of convT, this is also not a problem.
Denote x the center of mass. For every other point y
of convT denote f(y) such a positive coefficient that
x + f(y)(y-x) belongs to the boundary of convT. Now, put
$g(y) = 1 - \frac{1}{f(y)}$ for y not equal to x,
and g(x) = 1. This will be the function you are looking for.
The most complicated question is, how to build an analytic
expression analogous to
dot(n1,X)*dot(n2,X)*dot(n3,X)*(dot(n4,X)-dot(n4,A))*(1-x^2-y^2-z^2)
but having the properties in question.
I don't know the answer, but it looks quite probable, that elevating
each muliplier in some power a1, a2, etc. helps. This new expression
will also be zero on the boundary and positive inside.
It looks like selecting appropriate values of powers
a1, a2, etc. one can move the point where the function attains its
maximum to the location that one wants.
I think you should solve gradient F = 0 evaluated at center of mass, to find the k1, k2 and k4 scalars.
with F = dot(k1 n1,X - A)*dot(k2 n2,X - B)*dot(n3,X - C)*dot(k4 n4,X - A)*(1 - x^2 - y^2 - z^2)
I was wrong, F may be as follows:
F = g (x,y,z,a,b,c) * dot (n1,X - A) * dot (n2,X - B) * dot (n3,X - C) * dot (n4,X - A) * (1 - x^2 - y^2 - z^2)
and g could be, for example, a Gaussian.
g (x,y,z,a,b,c) = exp (-((x - a)^2 + (y - b)^2 + (z - c)^2))
The parameters a, b and c are obtained by solving gradient F = 0 (evaluated at center of mass).
Also you can use a simpler function, but it must have a maximum and three parameters.
Thank you! I will try your solutions but both sound very promising!
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