Constructing the Conic Sections on a Whiteboard

  • Objective
  • Level
  • Prerequisites
  • Platform
  • Instructor's Notes
  • Credits
  • Generating an Ellipse

















     

     
    Objective: This demo provides a visual development for the "locus of points" definitions of the conic sections. 

    Level: Precalculus or Calculus I.

    Prerequisites: A discussion of the locus of points definitions of the conic sections. 

    Platform: Any whiteboard (or blackboard).

    Instructor's Notes:
    This is a good activity to get students out of their seats and doing something physical and mathematical.  It illustrates that the "locus definitions" of the conic sections can actually be used to construct the conic sections. 

    The word locus may not be familiar to some students. Hence for the prerequisite mentioned above you can use the following. 

    The word locus is used in geometry to mean the set of all points 
    which satisfy a given geometric condition. It is often convenient to think of the locus as the path traced out by a point which moves in such a way as always to satisfy the given condition.
    A particularly simple locus is the set of all points a fixed distance r > 0  from a fixed point. It has been helpful to have students construct this locus on paper and provide the common names associated with the fixed distance and the fixed point. (Of course this locus is a circle. Click on the thumbnail sketch below to see the animation.)

    Generating a Circle
    Although students may have already seen the construction of an ellipse, they may not have seen how the definitions may be used to construct a parabola and hyperbola. We can also justify that the shapes produced by the constructions are actually the conic sections. This takes some thinking to produce the correct relationships.

    If the construction of the ellipse is well known to the students, then it can be demonstrated at the board with the help of a student. The instructor can provide the justification that the points drawn by the pen actually satisfy the locus definition of the ellipse. 

    The construction and justification of the parabola are a bit more difficult. Most students have not seen this construction, and most are a bit surprised that the construction actually creates a parabola. 

    It is very rare that a student knows how to construct a hyperbola. A survey of colleagues showed that most of them didn't know either.

    All of the constructions can be done by pairs of students, but groups of three work better.

    Materials: 

    *   string
    *   tacks for attaching the string to the wooden bars
    *   small suction cups (available at craft shops for window hangings) 
         make good fixed points or foci
    *   a large "T-square" (to slide along the pen tray) can be easily made 
         by gluing a wooden bar ( about 3 feet long)  to a 6 inch piece of 4x4
    *   a wooden bar (3 to 4 feet long)  for the hyperbola construction
    *   colored dry erase marking pens
    (Note: a blackboard can be used when the suction cups are replaced by tape and chalk replaces the dry erase markers.)

    Demo Directions and Justifications

    1.   Constructing an Ellipse

    Attach suction cups to the whiteboard at two points, F1 and F2. 
    Cut a piece of string and tie the ends to F1 and F2. 
    Place the marking pen at the location P, and, keeping the string taut,
    slowly move the pen. The resulting figure is an ellipse.
     

    (An animation of this construction is available at the beginning of this document.)

    Definition:  An ellipse is the set of points P whose distances from F1 and F2 always add together to give the same number.

    Justification that the construction produces an ellipse:

    Suppose the length of the string is S.  Then the distance from P to F1 plus the distance from P to F2 is simply the length of the string which is constant.  For each point P,
                 dist( P, F1 ) + dist( P, F2 ) = length of the string = S 
    so the collection of the points P is an ellipse.


     2.   Constructing a Parabola

    Cut a string slightly longer than H and attach one end to the whiteboard at F (suction cup) and pin the other end to the top of the T-bar at A.  (The "free"
    length of the string should be approximately equal to H, the height of the T-bar.)
    Put the base of the T-bar on the pen tray of the whiteboard. Place the marking pen at the location P, and, keeping the string taut, slowly move the T-bar. The pen should stay adjacent to the wood bar as the T-bar is moved. The resulting figure is a parabola.

    (To see an animation of this construction click on the thumbnail icon below.)
     
    Generating a Parabola


    Definition:  A parabola is the set of points P whose distance from a fixed point,
    F, equals the distance from a fixed line, L .

    Justification that the construction produces a parabola:

    Suppose the length of the string is S (and S = H).  Then the distance from P to F plus the distance from P to A is simply the length of the string, S:
                  dist( P, F ) + dist( P, A ) = S
    We also know that the distance from the line L to P plus the distance from P to A is just the height of the T-bar, H:
                   dist( P, L ) + dist( P, A ) = H
     
    But  S = H 
    so     dist( P, F ) + dist( P, A ) = dist( P, L ) + dist( P, A )
    and   dist( P, F ) = dist( P, L )
    so the collection of the points P is a parabola.


    3.   Constructing a Hyperbola

    Cut a string slightly shorter than H and attach one end to the whiteboard at F2 (suction cup) and pin the other end to the top of the wood bar at A.  (The "free" length of the string should approximately equal  H, the length of the wood bar.) Put the corner of the wood bar at F1. Place the marking pen at the location P, and, keeping the string taut, slowly rotate the wood bar around the pivot point F1. The pen P should stay adjacent to the wood bar as the bar is rotated. The resulting figure is a hyperbola.

    (To see an animation of this construction click on the thumbnail icon below.)
     
    Generating a Hyperbola
    Definition:  A hyperbola is the set of points P whose difference of distances from two fixed points, F1 and F2, is constant .

    Justification that the construction produces a hyperbola:
     

    Suppose the length of the string is S (and S = H).  Then the distance from P to F2 plus the distance from P to A is simply the length of the string, S:
                  dist( P, F2 ) + dist( P, A ) = S
    We also know that the distance from the line P to F1 plus the distance from P to A is just the length of the wood bar, H:
                   dist( P, F1 ) + dist( P, A ) = H
    When we subtract the first distance equation from the second, we get
      H - S = { dist( P, F1 ) + dist( P, A ) } - { dist( P, F2 ) + dist( P, A ) } 
                = dist( P, F1 ) - dist( P, F2 )

    so dist( P, F1 ) - dist( P, F2 ) is constant  ( = H -S)  for all points P, and the set of points P is a hyperbola.

    Notes:

    1. A natural extension to the geometric constructions is the derivation of the quadratic equations for the conics from the locus definitions.

    2. Each of the animations for constructing the conic sections, parabla, ellipse and hyperbola, is available in a larger form by clicking on big-conic. These are suitable for a classroom monitor or display when using this demo.

    Credits:  This demo was submitted by 

    Dale Hoffman
    Department of Mathematics 
    Bellevue Community College

    and is included in Demos with Positive Impact with his permission.



    DRH 2/14/00   Last updated 5/18/2006

    Since 3/1/2002