Linear interpolation of three 3D points in 3D space

Question

I have three 3D points like p1(x1,y1,z1), p2(x2,y2,z2), p3(x3,y3,z3). I have another point, but I know only x, y

Answer 1

This is to support both MBo's and Konstantin's answers. Please don't accept this question, but one of the others.

This is how you would implement a solution in MATLAB:

%// Your known 3 points
p1 = [ 1 10  0]';
p2 = [-1 10 10]';
p3 = [ 0  0 10]';

%// your 4th target point
p4 = [0 5  NaN]';

%// Difference matrix/vector
A = [p2-p1  p3-p1];
b = p4-p1;

%// Compute solution
p4(end) = p1(end) + A(3,:)*(A(1:2,:)\b(1:2));

Now, in C++, the mere fact of including the relevant eigen libraries blows up the executable size rather spectacularly. What eigen is capable of is complete overkill for this simple 2x2 system.

So I wouldn't go as far as resort to eigen, unless you have tons of other linear algebra things to do. It is a simple 2x2 system, which is easy enough to solve by hand.

Just KISS it; see DanielKO's answer :)