Visualizing flow with quaternion frames

Flow fields, geodesics, and deformed volumes are natural sources of families of space curves that can be characterized by intrinsic geometric properties such as curvature, torsion, and Frenet frames. By expressing a curve's moving Frenet coordinate frame as an equivalent unit quaternion, we reduce the number of components that must be displayed from nine with six constraints to four with one constraint. We can then assign a color to each curve point by dotting its quaternion frame with a 4D light vector, or we can plot the frame values separately as a curve in the three-sphere. As examples, we examine twisted volumes used in topology to construct knots and tangles, a spherical volume deformation known as the Dirac string trick, and streamlines of 3D vector flow fields.<>