Dandelin

Dandelin refers to a geometric construction named after the 19th-century French mathematician Germinal Pierre Dandelin. The construction provides a purely geometric proof that conic sections—the ellipse, parabola, and hyperbola—have focal properties and are obtained as plane sections of a cone.

Construction and idea: Take a right circular cone and a plane that intersects the cone. Inside the cone, place two spheres so that each sphere is tangent to the cone and tangent to the plane. Each sphere touches the cone along a circle and touches the plane along another circle. The plane intersects each sphere in a circle; the centers of those two circles lie on the plane and are the perpendicular projections of the spheres’ centers. These two points on the plane are the foci of the resulting conic section.

Focal property: For any point on the conic, the sum of its distances to the two foci is constant (ellipse), or the absolute difference of the distances is constant (hyperbola). If the cutting plane is parallel to a generating line of the cone, one focus recedes to infinity and the conic becomes a parabola. Thus, the Dandelin spheres give a constructive, purely geometric demonstration of the focus–directrix nature of conics.

Significance: Dandelin’s method provides a classical, visual proof that the conic sections are indeed portions of a cone and clarifies their focal properties. It remains a standard teaching tool in geometry for illustrating why ellipses, parabolas, and hyperbolas have foci.