Frames of spatial parametric curves

Marko Černe (2013) Frames of spatial parametric curves. EngD thesis.

Preview

PDF Download (1739Kb)

Abstract

The diploma thesis describes parametric space curves and their properties. Among them torsion and curvature are of special concern because they entirely define the curve up to translation and rotation. That is why we call them invariant properties. Differential geometry is a mathematical discipline which studies the properties of space curves. Differential geometry is in continuous development since many applications in kinematics, robotics and animations use rational curves while designing a rigid body motion. To achieve the motion we need to specify its position and orientation at a given time. Position is given as a point on the parametrized curve. On the other side the orientation is not given precisely therefore an algorithm needs to be designed to determine a natural variation of orientation. Nowadays, there are various frames in use among which a Frenet frame is the most appropriate for analyzing. Modern computer aided design systems can present a space curve only in rational form. A Frenet frame is not as suitable for practical use as it is a rotation minimizing frame that ensures the frame's minimal rotation. Results are supported by implementations of a Frenet frame and a rational minimizing frame. The latter is produced using double reflection method which yields an exact rotation minimizing frame on spherical curves. Moreover, the method gives fruitful approximation as well for general space curves.

Actions (login required)