Parallel Transport Approach

to Curve Framing

Andrew J. Hanson and Hui Ma

Department of Computer Science

Lindley Hall 215

Indiana University

Bloomington, IN 47405

January 11, 1995

Abstract

We propose an algorithm for generating a moving coordinate frame on a space curve based on the concept of parallel transport. Such algorithms can be used for creating ribbons, tubes, and camera orientations that are smoothly varying and controlled by the curve geometry itself. The more familiar Frenet frame approach suffers from ambiguity and sudden orientation changes when the curve straightens out momentarily. We compare the properties of alternative framing methods and point out when the parallel transport approach has unique advantages. We discuss a variety of implementation issues and illustrate the application of the algorithm to ribbons and tubes based on open and closed curves, as well as to the generation of moving camera orientations.

1 Introduction

We attack the problem of associating moving coordinate frames to three-dimensional space curves in ways that are well-understood mathematically and that have optimal behavior for certain classes of computer graphics applications. Classical differential geometry typically treats moving frames using the Frenet frame formalism because of its close association with a curve's curvature and torsion, which are coordinate-system independent [2, 9, 4]. The Frenet frame, unfortunately, has the property that it is undefined when the curve is even momentarily straight (has vanishing curvature), and exhibits wild swings in orientation around points where the osculating plane's normal has major changes in direction. We propose an alternative approach, the parallel-transport frame method (see Bishop [1]), which has a mathematically sound foundation and more appropriate behavior for computer graphics; in cases where the Frenet frame has desirable properties, a hybrid strategy is also feasible.