Ellipse with a Cake Pan "Spirograph"

create an ellipse by rolling a disk inside a cake pan

The path a point on a disk takes as the disk rolls inside a circle can be an elaborate Spirograph®-like design. However, if the inner disk is one half of the diameter of the circle that it rolls inside, the path of a point on the disk is an ellipse.


Materials

  • Round cake pan
  • Cardboard
  • Wide rubber band
  • Paper and pencil
  • Tape

Steps

  1. Cut out a circle from cardboard one half the diameter of the cake pan.
  2. Place a rubber band around the edge of the cardboard disk (to provide traction).
  3. Tape a piece of paper to the bottom of the cake pan.
  4. Poke a hole in the cardboard. The closer the hole is to the edge of the cardboard disk, the greater the eccentricity of the ellipse, i.e., the longer and skinnier it will become. (If the pencil is on the edge of the circle, it will draw a line!)
  5. Put a pencil in the hole and trace the path of the hole as the cardboard disk rolls around the inside of the cake pan. Proceed slowly around the cake pan holding the pencil in one hand and using the other hand to roll the disk and prevent it from slipping. The path is an ellipse.

Notes

  • A curve created by the path of a point attached to a circle rotating inside another circle is called a hypotrochoid. The ellipse in this project is a special case of a hypotrochoid. The Spirograph toy creates hypotrochoids and epitrochoids (paths formed by a circle travelling around the outside another circle).

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