Question

I’m developing a 3D chess game in which I need to be able to calculate positions X,Y,Z of a trajectory in a way that this trajectory will describe a parabola (for the pieces animation).

Thus I need the formulas below (or the general formula) for the given equidistant points p1=(x1,y1,z1), p2=(x2,y2,z2) and p3(x3,y3,z3):

x=f(y,z)=???  
y=f(x,z)=???   
z=f(x,y)=???

Answer 1

For each component x, y and z consider a separate parabola defined by

x(t) = x1 - t*(3*x1-4*x2+x3) + 2*t^2*(x1-2*x2+x3)     //t=0..1
y(t) = y1 - t*(3*y1-4*y2+y3) + 2*t^2*(y1-2*y2+y3)     //t=0..1
z(t) = z1 - t*(3*z1-4*z2+z3) + 2*t^2*(z1-2*z2+z3)     //t=0..1

at t=0 then x=x1, at t=0.5 then x=x2 and at t=1 then x=x2. Similarly for y(t) and z(t).