I want to compute 3-d cross-sections of a pentatope (4-dimensional tetrahedron).
The 3-d cross-sections will be calculated as:
x+y+z+w=c
C is a constant that I will vary to get different cross-sections.
If I view the pentatope as 10 planes, I can calculate the lines that represent the intersection of each plane with my 3-d cross-section. My problem is: how do I calculate the equation for each plane given the three 4-dimensional points that are its corners?
In other words if I have 3 points in 4 dimensional space: (x0,y0,z0,w0) (x1,y1,z1,w1) (x2,y2,z2,w2) What is the equation for the plane?
I don't know how to generalize the method for doing this with 3 points in 3 dimensional space, not least because the cross-product will be a 3-d space, not a 2-d plane.
Feel free to offer broader comments on my approach to the pentatope cross-section problem.
The 2-plane in 4-space can only be expressed in parametric form; in general, only $d-1$-hyperplanes can be expressed without parametrization in $d$-dimensional spaces.
$$f(t,u) = t\left(\mathbf v_1 - \mathbf v_0\right) + u\left(\mathbf v_2 - \mathbf v_0\right)$$
Compare this to the parametric form of the generic line, $f(t) = t\left(\mathbf v_1 - \mathbf v_0\right)$, and the method for generalization is clear.