Conic Section Gallery

The following is a gallery of demos for illustrating selected families of conic sections These figures and animations can be used by instructors in a classroom setting or by students to aid in acquiring a visualization background relating to the change of parameters in expressions. Two file formats, gif and mov are available.

1. The gif animations should run on most systems and the file sizes are relatively small. 

2. The mov animations require the QuickTime Player (version 6) which is a free download available by clicking here; these files are also small. (The mov files may not execute properly in older versions of QuickTime.) 

3. The collection of animations in gif and mov format can be downloaded; see the zipped download category at the bottom of the following table.

4. We use abbreviations, GSP for Geometer's Sketch Pad 4, MAT for MATLAB, and EX for Excel spread sheet, to indicate the software used to generate the frames of the animation. If alternate corresponding software is also available there will be more than one abbreviation in that category. Software routines can also be downloaded.

Circles, Ellipses, Parabolas, and Hyperbolas

The conic sections are the non-degenerate curves generated by the intersections of a plane with one or two cones in the double cone as pictured below. For a plane perpendicular to the axis of the cone, a circle is produced. For a plane which is not perpendicular to the axis and which intersects only one cone, the curve produced is either an ellipse or a parabola. The curve
produced by a plane intersecting both cones is a hyperbola.

• To see an animation of the conic sections as a plane is being rotated through a double cone go to http://www.math.odu.edu/cbii/calcanim/index.html The animation includes the three-dimensional image of the cone
  with the plane, as well as the corresponding two-dimensional image of the plane itself. This excellent demo was
  done in Mathcad. The authors granted us permission to use their original file as the basis for the following animation. To download this animation in both gif and mov format click here.
  
  

• At http://math2.org/math/algebra/conics.htm is a discussion of conic sections generated by intersecting a double cone with detailed descriptions of the cases including degenerate cases. Also included are very nice figures.

Gallery of Animations

Circles GSP		Vary r: Gif Animation               Mov Animation Vary h: Gif Animation                Mov Animation Vary k: Gif Animation                Mov Animation
Ellipses centered at (0, 0) GSP		Vary a: Gif Animation                Mov Animation Vary b: Gif Animation                Mov Animation
Ellipses centered at (h, k)) GSP		Vary h: Gif Animation                Mov Animation Vary k: Gif Animation                Mov Animation
Parabolas in the form GSP also an EX file		Vary a: Gif Animation                Mov Animation Vary b: Gif Animation                Mov Animation Vary c: Gif Animation                Mov Animation
Parabolas in the form GSP also an EX file		Vary h: Gif Animation                Mov Animation Vary k: Gif Animation                Mov Animation
Hyperbolas centered at (0, 0) GSP		Vary a: Gif Animation                Mov Animation Vary b: Gif Animation                Mov Animation
Hyperbolas centered at (h,k) GSP		Vary h: Gif Animation                Mov Animation Vary k: Gif Animation                Mov Animation
Conic Sections (polar form) EX		Vary k: Gif Animation                Mov Animation Vary e: Gif Animation                Mov Animation
Zipped downloads ==>	Download (all) Gif and Mov Animations (large file) Download (1) Geometer's Sketch Pad file with 7 programs Download (3) Excel files

Other Selected Resources:

• A demo that provides a visual development for the "locus of points" definitions of the conic sections that can be implemented on a whiteboard (or blackboard) is available at  mathdemos.org. Click on Go to Demo List, then click on Physical Demos, and finally click on 'Constructing the Conic Sections on a Whiteboard'. 
  
  
• Another demo at mathdemos.org shows in an  immediate way that parabolas arise as graphs of natural phenomena. A ball is given a push up an inclined plane. As the ball moves up and then back down the inclined plane its distance from the starting point at the base is recorded by a motion sensor. The time vs. distance data is simultaneously projected on a computer monitor. When the ball returns to the starting point the data points are connected and the resulting graph is parabolic. If we would continuously record the data the resulting curve would indeed be a parabola. At  mathdemos.org. click on Go to Demo List, then click on Physical Demos, and finally click on 'Parabola!'. 
  
  

• A parabola (and more) java applet at http://www-groups.dcs.st-and.ac.uk/~history/Java/Parabola.html
  
  

• The Standard and General Form Equations for Parabolas applet at http://cs.jsu.edu/~leathrum/Mathlets/parabolas.html

• Javasketchpad applet: The Conic Sections As the Locus of Perpendicular Bisectors http://www.keypress.com/sketchpad/javasketchpad/gallery/pages/conics.php
  

• Nine applets for conics (fee involved) at http://www.ies.co.jp/math/java/conics/index.html
  

• Translations and Rotations of Conic Sections at http://cs.jsu.edu/~leathrum/Mathlets/conics.html This applet lets you investigate the general quadratic                  Ax^2 + Bxy+ Cy^2 + Dx + Ey + F = 0.
  

• A Geometer's Sketch Pad routine for a general quadratic  is available for download by clicking here.
  
  

• An Introduction To Conic Sections at http://www.krellinst.org/UCES/archive/resources/conics
  with historical, geometric, and algebraic notes, as well as exercises.
  
  

• Various conic contributions to the Math Archives at http://archives.math.utk.edu/topics/precalculus.html
  
  

• The Ellipse Game at http://johnbanks.maths.latrobe.edu.au/Games/Ellipse/ is an applet to provide
   practice finding the foci.
  
  

• Eric Weisstein's World of Mathematics discussion of conic sections at
   http://mathworld.wolfram.com/ConicSection.html
  

Credits: 

• Special thanks to P. Bogacki and G. Melrose of Old Dominion University for their permission to use the 
  animation of a plane intersecting a double cone. They kindly regenerated the animation to incorporate 
  the labeling of the conic sections.

• The Geometer's Sketchpad routines were constructed by Mark Yates of The McCallie School and are 
  used with his permission.

• The Excel files were written by Xia Zhao, a student at Temple University, and David R. Hill, Temple University.

• Special thanks to Deane Arganbright of University of Tennessee at Martin and to Walter Hunter at Montgomery County Community College for their assistance with the development of the Excel files that accompany this gallery. The book "Mathematical Modeling with Excel", by Erich Neuwrith and Deane Arganbright (Brooks-Cole) provides a rich source of information for developing Excel routines for mathematical instruction.

• This gallery was developed by David R. Hill, Temple University.

8/6/03DRH          Last updated 9/15/2010 DRH

As of 8/29/03