This page looks best with JavaScript enabled

Quadratic Approximation of Cubic Curves

 ·  ☕ 2 min read · 👀... views

A simple degree reduction technique for converting piecewise cubic splines into piecewise quadratic splines that maintain parameterization and $C^1$ continuity.

Nghia Truong, Cem Yuksel and Larry Seiler
High-Performance Graphics 2020

Abstract

We present a simple degree reduction technique for piecewise cubic polynomial splines, converting them into piecewise quadratic splines that maintain the parameterization and $C^1$ 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 $C^1$-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 $C^1$ 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.

Download

Paper: (preprint, 10MB pdf)
Code (github)
Windows binary

WebGL Demo

Usage:
* Left mouse drag rotates the camera
* Right mouse drag translates the camera
* Mouse wheel zooms in/out
* S toggles smooth camera navigation

Notes:
* This WebGL demo requires WebAssembly-capable browser with WebGL enabled and may not work correctly in Firefox.
* The blue spheres are the control points of the cubic Bezier curve and red spheres are the control points of the approximated quadratic Bezier curves.
* The blue control points (spheres) can be move around. Just click on them, and some handles will show up that allow translation.

Initialization...

Video



Share on

Nghia Truong
WRITTEN BY
Nghia Truong
(Trương Trọng Nghĩa)


What's on this Page