Nonlinear Projections - Transformations and Projections in Computer Graphics - page 182

Figure 4.36: A Panoramic Projection on a Dodecahedron (courtesy of Dick Termes).

Reference [lampshade 05] shows how to apply the conic panoramic projection to
create original lampshades.
The derivation presented here starts with a point P =( x, y, z ) that is projected
onto the surface of the cone. Once the surface is opened, the coordinates of the projected
point P ∗ are given (Figure 4.37) in terms of the angle θ itmakeswiththetopofthe
cone, and the distance r from the top. These are polar coordinates on the open cone.

z

α

P

r

S

θ

H

P *

S S

z

S
Open (flat) cone

x

R 2R 2R

β

γ

√ x 2 + y 2

y

Figure 4.37: Conic Panoramic Projection.