Curves in space are difficult to perceive and analyze, especially when they form dense sets as in typical 3D flow 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 field. The original curves in 3-space are associated with piecewise continuous 4-vector quaternion fields, 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 flexible approaches for interacting with and exploiting the properties of the 4D quaternion fields.
展开▼
