Rational frames of minimal twist along space curves under specified boundary conditions (Q1633017)

To describe the spacial motion of a rigid body, one must specify the variation of its position and orientation with respect to time. The path of a point is employed to specify the variation of position as a parametric curve, to describe the variation of orientation one can use an orthonormal frame embedded within the body. In many contexts, a frame whose first component is the curve tangent and the second and third components span the curve normal plane at each point is desirable. An example is the Frenet frame and the rotation minimizing adapted frame called the Bishop frame, useful in computer animation, robotics, spatial motion planning, the construction of swept surfaces, and related applications. In this article, the authors address the problem of constructing an adapted orthonormal frame on a space curve that exhibits minimal ``twisting'' between prescribed initial and final instances. Their construction uses the Euler-Rodrigues frame (ERF) on spatial Pythagorean-hodograph (PH) curves. One of the advantages of this initial frame is that it is inherently rational and non-singular. In order to ensure monotonicity of the rational minimal twist frame, or of further suppressing variations of the tangent angular velocity component, the authors propose a strategy of subdividing the curve at the points that correspond to inflections and/or extrema of the ERF tangent angular velocity component. The article includes an algorithm that yields (rational) solutions to the problem of smoothly varying the orientation of a rigid body along a given spatial path, compatible with prescribed initial and final orientations.