Principles and applications of pencil tracing

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Abstract
Pencil tracing, a new approach to ray tracing, is introduced for faster image synthesis with more physical fidelity. The paraxial approximation theory for efficiently tracing a pencil of rays is described and analysis of its errors is conducted to insure the accuracy required for pencil tracing. The paraxial approimation is formulated from a 4x4 matrix (a system matrix) that provides the basis for pencil tracing and a variety of ray tracing techniques, such as beam tracing, ray tracing with cones, ray-object intersection tolerance, and a lighting model for reflection and refraction. In the error analysis, functions that estimate approximation errors and determine a constraint on the spread angle of a pencil are given.The theory results in the following fast ray tracing algorithms; ray tracing using a system matrix, ray interpolation, and extended 'beam tracing' using a 'generalized perspective transform'. Some experiments are described to show their advantages. A lighting model is also developed to calculate the illuminance for refracted and reflected light.

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