In this paper we apply perturbation methods to the problem of computing specular reflections in curved surfaces. The key idea is to generate families of closely related optical paths by expanding a given path into a high-dimensional Taylor series. Our path perturbation method is based on closed-form expressions for linear and higher-order approximations of ray paths, which are derived using Fermat's Variation Principle and the Implicit Function Theorem (IFT). The perturbation formula presented here holds for general multiple-bounce reflection paths and provides a mathematical foundation for exploiting path coherence in ray tracing acceleration techniques and incremental rendering. To illustrate its use, we describe an algorithm for fast approximation of specular reflections on curved surfaces; the resulting images are highly accurate and nearly indistinguishable from ray traced images.

We describe an approach for interactively approximating specular reflections in arbitrary curved surfaces. The technique is applicable to any smooth implicitly defined reflecting surface that is equipped with a ray intersection procedure; it is also extremely efficient as it employs local perturbations to interpolate point samples analytically. After ray tracing a sparse set of reflection paths with respect to a given vantage point and static reflecting surfaces, the algorithm rapidly approximates reflections of arbitrary points in 3-space by expressing them as perturbations of nearby points with known reflections. The reflection of each new point is approximated to second-order accuracy by applying a closed-form perturbation formula to one or more nearby reflection paths. This formula is derived from the Taylor expansion of a reflection path and is based on first and second-order path derivatives. After preprocessing, the approach is fast enough to compute reflections of tessellated diffuse objects in arbitrary curved surfaces at interactive rates using standard graphics hardware. The resulting images are nearly indistinguishable from ray traced images that take several orders of magnitude longer to generate.