What are Dandelin
constructions?

Mastery of the conic sections hones a
student's skills for Calculus. In particular, the circle,
ellipse, parabola and hyperbola present a clear opportunity for
students to learn algebra, graphing, vocabulary and definitions.

The Dandelin definitions and constructions are an enrichment, or
refinement, of the conics that involve first placing a sphere inside a
cone. [Look above at the animations.] If a cone is
intersected by a plane, then the foci of the conics are all points
where the plane touches the inscribed sphere. This is true for
all four conics.

Germinal Pierre Dandelin (1794 - 1847) published this discovery when he
was only 28 years old and having already lived through extremely
difficult times. As an 18 year old Belgium student studying in
the elite Ecole Polytechnic in Paris, he volunteered to serve in
Napoleon's army. When the advancing armies of Britain, Russia,
Austria and Prussia forced Napoleon to retreat to Paris, Dandelin was
wounded. He was one month short of this 20th birthday. He
would spend the next eight years working as an engineer in the Ministry
of Interior, returning to Belgium and becoming a citizen of the
Netherlands. While investigating the conics launched his career
as a mathematician, he would later publish in stereographic
projections, statics, algebra and probability. In particular, his
method of approximating roots of an equation is now known as the
Dandelin-Graffe method.