Back to . . .  Deposit #99 - b Louis A. Talman Dept. of Mathematical and Computer Sciences Metropolitan State College of Denver ____________ Talman created individual images for the Quick Time Movie  using Wolfram Mathematica®.  ____________ If you do not have Quick Time installed on your computer click here for the latest version. ____________

This section . . . .           Dandelin Constructions ~ Special Representations of the Conic Sections

The Hyperbola

 Ellipse Animation Deposit #99 - a Parabola Animation Deposit #99 - c 2010

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.

 References Gray, Alfred, Modern Differential Geometry of Curves and Surfaces with MATHEMATICA®, CRC Press, 1998. For biography: < http://www-history.mcs.st-and.ac.uk/history/Biographies/Dandelin.html > For Fermat animations: < http://curvebank.calstatela.edu/hyperbolafermat/hyperbolafermat.htm > For the "Father of Conics": < http://curvebank.calstatela.edu/birthdayindex/sep/sep22parabola/sep22parabola.htm > ( Other deposits on the conics are in the National Curve Bank. Use "Google" at the bottom of the Home page.)

 The National Curve Bank thanks Lou Talman for Deposit #99. [email:   talmanl@mscd.edu ]