Quaternion Frame Approach to Streamline Visualization

Curves in space are diicult to perceive and analyze, especially when they form dense sets as in typical 3D ow and volume deformation applications. We propose a technique that exposes essential properties of space curves by attaching an appropriate moving coordinate frame to each point, reexpressing that moving frame as a unit quaternion, and supporting interaction with the resulting quaternion eld. The original curves in 3-space are associated with piecewise continuous 4-vector quaternion elds, which map into new curves lying in the unit 3-sphere in 4-space. Since 4-space clusters of curves with similar moving frames occur independently of the curves' original proximity in 3-space, a powerful analysis tool results. We treat two separate moving-frame formalisms, the Frenet frame and the parallel-transport frame, and compare their properties. We describe several exible approaches for interacting with and exploiting the properties of the four-dimensional quaternion elds.