Projection matrices and related viewing frustums: new ways to create and apply

TL;DR

This paper explores the properties and transformations of arbitrary affine frustums in computer graphics, proposing new methods for their creation, manipulation, and application without relying solely on projection matrices, especially for resource-limited devices.

Contribution

It introduces novel techniques for constructing and transforming arbitrary affine frustums and applies these to practical scenarios without dependence on standard projection matrices.

Findings

Derived dependencies between frustum planes and key points from inverse projection matrices.

Developed formulas for frustum transformation simulating reflection, refraction, and cropping.

Proposed a method to apply arbitrary frustums without a projection matrix to limit visible volume.

Abstract

In computer graphics, the field of view of a camera is represented by a viewing frustum and a corresponding projection matrix, the properties of which, in the absence of restrictions on rectangular shape of the near plane and its parallelism to the far plane are currently not fully explored and structured. This study aims to consider the properties of arbitrary affine frustums, as well as various techniques for their transformation for practical use in devices with limited resources. Additionally, this article explores the methods of working with the visible volume as an arbitrary frustum that is not associated with the projection matrix. To study the properties of affine frustums, the dependencies between its planes and formulas for obtaining key points from the inverse projection matrix were derived. Methods of constructing frustum by key points and given planes were also considered. Moreover, frustum transformation formulas were obtained to simulate the effects of reflection, refraction and cropping in devices with limited resources. In conclusion, a method was proposed for applying an arbitrary frustum, which does not have a corresponding projection matrix, to limit the visible volume and then transform the points into NDC space.