3.5 Other Quadrics

pbrt supports three more quadrics: cones, paraboloids, and hyperboloids. They are implemented in the source files shapes/cone.h, shapes/cone.cpp, shapes/paraboloid.h, shapes/paraboloid.cpp, shapes/hyperboloid.h, and shapes/hyperboloid.cpp. We won’t include their full implementations here, since the techniques used to derive their quadratic intersection coefficients, parametric coordinates, and partial derivatives should now be familiar. However, we will briefly summarize the implicit and parametric forms of these shapes. A rendered image of the three of them is in Figure 3.10.

3.5.1 Cones

The implicit equation of a cone centered on the axis with radius and height is

Cones are also described parametrically:

The partial derivatives at a point on a cone are

and the second partial derivatives are

3.5.2 Paraboloids

The implicit equation of a paraboloid centered on the axis with radius and height  is

and its parametric form is

The partial derivatives are

and

3.5.3 Hyperboloids

Finally, the implicit form of the hyperboloid is

and the parametric form is

The partial derivatives are

and