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.
