Three-Dimensional Compensation to the Paraboloid | Steve Wait – April 5, 2019

z    = ax² (xz plane)

a    = coefficient
F    = .25a = focus (focus to vertex)
D   = ball diameter
x    = x coordinate on parabola (in xz plane)
X_c = x compensated, rotationally transformed value (ball center)
Y_c = y compensated, rotationally transformed value (ball center)
z    = z coordinate on parabola (in xz plane)
Z_c = z compensated, (ball center)
θ   = angular displacement (in xy plane)

Paraboloid 3D Compensation Equations (Ref. Fig. 1)

X_c = cos θ (x ± (xD / (2√(4F²+x²))

Y_c = sin θ (x ± (xD / (2√(4F²+x²))

Z_c = z ± (FD / √(4F²+x²))

Fig. 1

Tags: geometry