ellipseacross

Ellipseacross is a geometric construct that refers to a family of ellipses used to model cross-boundary traversal and coverage across a planar region. The concept is commonly described in the context of computational geometry, computer graphics, and robotic path planning, where a sequence of oriented ellipses is generated to sweep across a domain from one side to the opposite side, maintaining a predefined orientation and aspect ratio.

Formally, an ellipseacross is defined by a fixed orientation theta and axis lengths a and b, together with a parametric center path c(t) in the plane. Each ellipse in the family has equation (R(theta) · (x − c(t)))^T diag(1/a^2, 1/b^2) (R(theta) · (x − c(t))) = 1, where R(theta) is the rotation matrix. As t varies, the centers c(t) trace a path that causes successive ellipses to intersect a chosen boundary or line segment, producing an acrossing effect. The envelope of the family can form a boundary curve that highlights the region covered by the sweep.

Key properties include smoothness of the center path, controllable overlap between consecutive ellipses by adjusting a and b, and the ability to tailor the sweep to specific boundaries. Ellipseacross supports variations such as time-dependent orientation, adaptive axis lengths, and non-linear center trajectories, expanding its applicability to dynamic environments.

Applications span shape morphing, sensor coverage optimization, and inspection or painting tasks in robotics and graphics. Related concepts include ellipse envelopes, coverage paths, and Minkowski combinations in geometric modeling. See also: ellipse, envelope, coverage path planning, Minkowski sum.