Dandelin Spheres and Conic Sections

This example demonstrates the geometric construction of Dandelin spheres, used to prove the focal properties of conic sections such as ellipses, parabolas, or hyperbolas obtained by intersecting a plane with a cone. Through 3D solid geometry construction, it shows the tangency relationships between the spheres, the cone, and the intersecting plane, intuitively revealing the geometric nature of foci and directrices in conic sections. The applet allows users to manipulate the angle and position of the cutting plane, dynamically observe how the resulting conic section changes from ellipse to parabola to hyperbola, and explore how one or two Dandelin spheres are inscribed within the cone and tangent to both the cone surface and the cutting plane. The points of tangency between each sphere and the cutting plane correspond exactly to the foci of the conic section, while the circles of tangency between each sphere and the cone project onto the cutting plane as the corresponding directrices. This interactive visualization helps students develop a deep geometric understanding of why conic sections satisfy their defining focal properties, connecting classical Euclidean geometry with analytic concepts in an intuitive and engaging way.