Solving tri-linear interpolation parameters

[aadster]

I'm trying to find the tri-linear interpolation parameters of a point C within a hexahedron of 3d vectors (C000, C100, C010, C011 etc)

You could call this "inverse tri-linear interpolation"

Ive used the same variable names as this wikipedia article: http://en.wikipedia.org/wiki/Trilinear_interpolation

if Xd, Yd & Zd are the 0..1 trilinear parameters for x,y & z respectively, how can I solve the equations for these parameters? The interpolated point C is known.

Many thanks

 

[aadster]

to break this down further, this is the tri-linear equation.

R = ((1-Yd) * (p000 * (1-Xd) + p100 * Xd ) + (p010 * (1-Xd) + p110 * Xd ) * Yd) * (1-Zd) +
((1-Yd) * (p001 * (1-Xd) + p101 * Xd ) + (p011 * (1-Xd) + p111 * Xd ) * Yd) * Zd;

where R is tri-linear interpolated result,
P000-P111 are the 8 points of a hexahedron that define the 3D interpolation space
R and P can be either scalar or vector

Xd, Yd, and Zd are scalars and are the tri-lin parameters I am trying to find


Known:
When Xd,Yd & Zd = 0, R = p000 and when Xd,Yd & Zd = 1, R = p111 etc

Clearly there are some degenerate cases here, but my maths is a little rusty and I am finding tricky to solve for R... any ideas? Thanks!

 

[aadster]

I'll add to this as I progress, but interestingly Wolfram Alpha could only solve this for the simplest factor Z: (only 2 occurances)

Z = (a X Y+a (-X)-a Y+a-b X Y+b X-c X Y+c Y+d X Y-R)/(a X Y+a (-X)-a Y+a-b X Y+b X-c X Y+c Y+d X Y-e X Y+e X+e Y-e+f X Y-f X+g X Y-g Y-h X Y)

Since factors a-h form a cube of values, I can swap them around to create two other equations in the exact form above wrt X and Y.

Assuming I have all 3 equations, what is the safest way to combine all 3 equations to calculate X, Y & Z?