Two Linear Motions Generate 8 Ellipses

This example demonstrates how combining two linear motions/linear constructions can systematically produce and display eight ellipses with clear symmetry and structural relationships. The key mathematics includes conic sections—especially ellipses—as well as geometric transformations such as reflection (mirror) and rotation. The construction relies on how intersections and constraints defined by circles and segments determine ellipse-related curves. In GeoGebra, the build uses Midpoint to create key reference points, Mirror to reflect configurations, Segment and Point to form the foundational geometry, Rotate to move/rotate the configuration to new symmetric positions, Circle to define circular constraints, and Intersect to compute crucial intersection points. The workflow builds multiple intermediate objects (e.g., eq1/eq2, control points like O, A, B, C and their transformed variants such as C′, D′, I′, and curve-defining objects like c and f′/f″) that collectively yield the eight-ellipse set. Suitable for high school through early undergraduate instruction in geometry, conic sections, and dynamic geometric construction.