The Conics

By: Diana Brown


Day Four:

Ellipse Introduction


Construction of an ellipse with wax paper

Start out with a piece of uncrumbled wax paper about the size of half a sheet of standard notebook paper (8.5 x 11).  Draw a circle in the center of the wax paper without writing off the paper.  Then draw a point anywhere inside the circle excluding on the circle and the center.  The next step is to fold up the circle so it touches the point inside.  When this portion of the circle is aligned with the point, crease the paper and fold it accordingly.  Choose another part of the circle and align this with the point, creasing the wax paper.  Repeat this step several times until the majority of the circle has touched the point inside or until an ellipse is visible.

 

Construction of an ellipse on Geometers Sketchpad

Draw a circle

 

 

Place a point anywhere inside the circle (point A) and one on the circle (Point B)

 

 

Construct a segment from point A to B and construct line segment AB’s midpoint (Point C)

 

 

Construct the perpendicular bisector of line segment AB and while this line is still selected, choose Trace Line from the Display menu. 

 

To simulate the process of folding, point B and under Display click animate point.
Watch the ellipse form!

 


Proof

Statement: The sum of the distances from two points (the foci) inside an ellipse to any point on the ellipse is constant.
Proof: The foci on our sketch are the points A (the center of the circle) and B (the point inside the circle).  We will again use Geometer's Sketchpad to visualize our proof.  Construct a line segment from A to C and place a point of intersection E where this new line segment intersects the perpendicular line passing through point D.  This point E traces out the ellipse.  Construct another line segment between points E and B.  You should now have a triangle BCE with a perpendicular bisector ED.  Because ED is the perpendicular bisector, BD is congruent to DC and the angles BDE and CDE are both 90 degrees.  Of course, DE is congruent to itself, and thus we have two triangles with two congruent sides with an included congruent angle.  By SAS, the triangles are congruent and therefore EB is congruent to EC.  The radius of the circle is AC = AE+EC = AE+EB.  So since the radius is constant, the sum AE+EB is always constant.

 

 

 


Ellipses in real life:

 


Example 1) A portion of the white house lawn is called The Ellipse.  It is 1060 feet long and 890 feet wide.

 

White House and Ellipse, View from atop the Washington Monument

 

 

Example 2)  The orbits of the planets create an ellipse.

 

 

 

 

 


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