Construction of periodic adapted orthonormal frames on closed space curves
- Author(s): Farouki, RT
- Kim, SH
- Moon, HP
- et al.
Published Web Locationhttps://doi.org/10.1016/j.cagd.2019.101802
The construction of continuous adapted orthonormal frames along C closed–loop spatial curves is addressed. Such frames are important in the design of periodic spatial rigid–body motions along smooth closed paths. The construction is illustrated through the simplest non–trivial context — namely, C closed loops defined by a single Pythagorean–hodograph (PH) quintic space curve of a prescribed total arc length. It is shown that such curves comprise a two–parameter family, dependent on two angular variables, and they degenerate to planar curves when these parameters differ by an integer multiple of π. The desired frame is constructed through a rotation applied to the normal–plane vectors of the Euler–Rodrigues frame, so as to interpolate a given initial/final frame orientation. A general solution for periodic adapted frames of minimal twist on C closed–loop PH curves is possible, although this incurs transcendental terms. However, the C closed–loop PH quintics admit particularly simple rational periodic adapted frames. 1 1 1 1