Quadratic Approximation of Cubic Curves

PubDate: August, 2020

Teams: University of Utah, Facebook Reality Labs

Writers: Nghia Truong, Cem Yuksel, Larry Seiler

PDF: Quadratic Approximation of Cubic Curves

Abstract

We present a simple degree reduction technique for piecewise cubic polynomial splines, converting them into piecewise quadratic splines that maintain the parameterization and ?1 continuity. Our method forms identical tangent directions at the interpolated data points of the piecewise cubic spline by replacing each cubic piece with a pair of quadratic pieces. The resulting representation can lead to substantial performance improvements for rendering geometrically complex spline models like hair and fiber-level cloth. Such models are typically represented using cubic splines that are ?1 -continuous, a property that is preserved with our degree reduction. Therefore, our method can also be considered a new quadratic curve construction approach for high-performance rendering. We prove that it is possible to construct a pair of quadratic curves with ?1 continuity that passes through any desired point on the input cubic curve. Moreover, we prove that when the pair of quadratic pieces corresponding to a cubic piece have equal parametric lengths, they join exactly at the parametric center of the cubic piece, and the deviation in positions due to degree reduction is minimized.