In this paper, we present the closed-form representation of a subset of the allowable motion of an ellipse and an ellipsoid inside a simple elliptical and ellipsoidal environment based on the idea of kinematics of containment. Identifying an object’s free motion space, or equivalently the configuration space (C-space), is of great interest and practical use in computer-aided design and path planning problems. A special case exists when the object is constrained in an environment that is slightly larger than itself. We motivate our research by looking at the amount of free space in a cell for motion of the nucleus relative to the cell membrane. Therefore, we focus on elliptical and ellipsoidal shapes because of their similarity to the actual shape of a cell and its nucleus.

The subset of C-space defined by our closed-form representation is in spherical shape and the convexity of sphere makes it convenient to check whether two nodes in the C-space can be connected without collision. We derive the explicit expressions of the equation for both the single ellipse and single ellipsoid case. Existing methods are implemented and verified while a new method is proposed that shows better performance at enclosing a sub C-space with a larger volume.

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