MODELING OF FUNDAMENTAL SPLINES IN THE FORM OF QUATERNION CURVES

Abstract

Spherical curves are essential for geometric modeling, computer science and for various branches of engineering, and the definition of conditions for their construction has been studied in many works and has a large number of methods. The main field of application is computer animation and geometric modeling of motion trajectories in three-dimensional space. To model such curves, unit quaternions are used. However, this topic is not yet exhausted when using quaternions to construct various splines that retain their properties in accordance with those existing in three-dimensional space.

The paper highlights the experience of previous research in the field of constructing spherical quaternion curves. Spherical curve modeling has already been explored for Bezier curves, and some other interpolation curves based on quaternion theory. The authors of the work present the construction of fundamental splines (in particular, Catmull-Rom and Kochanek-Bartels) with verification of their properties after transferring to the surface of the sphere. These splines are important in modeling because of their property of being loopable, which allows to use them for continuous looping motion in animation.

The authors conduct research based on the method of representing functions in a cumulative form. This requires a preliminary representation of the spatial curve function as the sum of the products of some basis functions by key points. As an example, authors investigate the behavior of spherical splines when changing the parameters of tension, bias and continuity. The paper shows how the resulting spherical curves inherit the properties of their three-dimensional versions.

Further research is related to the application of simulated quaternion curves to construct surfaces.

Key words: isotropic curve, Bézier curve, Pythagorean-Hodograph curve.