Introduction

I had heard about 'dual-paraboloid mapping' some time ago and always wanted to look into how the algorithm worked. The name itself begs to be investigated, and the paraboloid mapping could be applied to environment maps as well as shadow maps so it seemed like a good idea to look into it.

When I recently found some spare time to read up on it I found several different academic papers on the topic explaining the theory behind the algorithm. However, after several rounds with Google I realized that there weren't many good resources on how to implement the mapping scheme. So I decided that I would investigate the topic and try to provide an easy to understand guide to the theory as well as a reference to the implementation of paraboloid mapping in the modern programmable rendering pipeline.

The implementation is written in DirectX 9 HLSL. The effect files that are developed in this exercise are provided with the article and may be used in whatever manner you desire.

Algorithm History

Before diving into the algorithm, I want to point out where the idea came from and why it was developed. The original motivation behind paraboloid mapping was presented in the paper "View-Independent Environment Maps" by Heidrich and Seidel. This new parameterization was intended to provide an alternative to spherical and cubical parameterizations used in environment mapping.

Spherical mapping suffers from wildly non-uniform sampling rates across its surface, making it difficult to use as a general-purpose environment mapping strategy. On the other hand, cube mapping provides a much more uniform sampling rate, but also requires six rendering passes to generate a complete environment of the current scene. This is quite an expensive algorithm to implement in a real-time rendering scenario.

Paraboloid mapping provides a good balance between these two methods. It produces a representation of the current scene about a single point with two surfaces, one for the forward facing hemisphere and one for the backward facing hemisphere. Its sampling rate is also more uniform than spherical mapping.

Another paper introduced the paraboloid parameterization to shadow maps: "Shadow Mapping for Hemispherical and Omnidirectional Light Sources" by Stefan Brabec. This paper provided the framework for building shadow maps that could cover an entire half of a scene with a single map or the entire scene with two maps. As opposed to standard cubical shadow mapping, the paraboloid parameterization saves a significant amount of work.

Algorithm Overview

So what exactly is paraboloid mapping? First let's look at the shape of a paraboloid to understand how it works. Figure 1 shows a paraboloid.

Figure 1:  Surface plot of a paraboloid (image taken from Brabec)

The basic idea behind it is relatively simple. For a given point P in a 3D scene we can divide the scene into two hemispheres, we'll call them forward and backward. For each of these hemispheres there exists a paraboloid surface that will focus any ray traveling toward point P into the direction of that hemisphere. Here is a 2D image that shows this idea.

Figure 2:  Dual paraboloids reflecting rays into parallel directions (image taken from Brabec)

Figure 2 shows both hemispheres reflecting rays into a parallel stream away from the point P. This is the key to the paraboloid mapping – environment mapping encodes the light surrounding a point into a grid of values to be stored in a map while shadow mapping does the same for depth values.

This encoding is based on the incoming rays being reflected about the paraboloid surface's normal vector. As we will see in the Mathematic Background section, this idea is the basis of the implementation of paraboloid mapping.

Mathematic Background