Computation of rotation minimizing frames

Abstract

Due to its minimal twist, the rotation minimizing frame (RMF) is widely used in computer graphics, including sweep or blending surface modeling, motion design and control in computer animation and robotics, streamline visualization, and tool path planning in CAD/CAM. We present a novel simple and efficient method for accurate and stable computation of RMF of a curve in 3D. This method, called the double reflection method, uses two reflections to compute each frame from its preceding one to yield a sequence of frames to approximate an exact RMF. The double reflection method has the fourth order global approximation error, thus it is much more accurate than the two currently prevailing methods with the second order approximation error-the projection method by Klok and the rotation method by Bloomenthal, while all these methods have nearly the same per-frame computational cost. Furthermore, the double reflection method is much simpler and faster than using the standard fourth order Runge-Kutta method to integrate the defining ODE of the RMF, though they have the same accuracy. We also investigate further properties and extensions of the double reflection method, and discuss the variational principles in design moving frames with boundary conditions, based on RMF.

Construction of Rational Curves with Rational Rotation-Minimizing Frames via Möbius Transformations

“…4. The numbers indicate that the average error decreases as h 3 , where h = 2 −k is the stepsize, while the error at the joints -and in particular at the end point of the curve -decreases faster with order 4 (in accordance with the phenomenon which is described in [22,23]). …”

“…The first publication discussing this frame was probably [4]. Since then, the computation of this frame and of the resulting sweep surfaces has been discussed in a substantial number of publications, see [23] and the references cited therein. In particular, the construction of rational parameterizations of these sweep surfaces is of interest, due to the importance of NURBS representations for Computer-Aided Design.…”

Autorotation

“…For descriptions of the mathematics of the special case of so-called rotation-minimising-frames see refs. [31,32,33]. Now we introduce and define autorotation.…”

Swept Volume Parameterization for Isogeometric Analysis

“…As another example, which is motivated by a figure in [21], we consider the space curve c(t) = (r cos(t), r sin(t), cos(αt)), t ∈ [0, 2π] which we use as the guiding curve for a swept volume. Figure 8 shows this curve.…”