Construction of the Quadrivium logo

Focus and directrix construction of an ellipse

An ellipse is the set of points the same distance from a point C and a circle.
For the point H on the ellipse, segments CH and CD have the same length, and the tangent line to the ellipse is the perpendicular bisector of CD.
C is one focus of the ellipse, and the center of the circle, A, is the other focus.

One of the main claims to fame of Newton's calculus was a derivation of the motion of planets around the sun. Their orbits are elliptical, with the sun at one focus. The speed is not constant. The same results can be derived without calculus, as this diagram shows. Here, the direction and size of the velocity are shown by the green vector. Its length is the same as CI.

About the diagram:

To animate the diagram, click the triangle in the lower left. Or use the slider to manually move point D around the circle, causing H to move around the ellipse.
Some points can be dragged; try it. What changes?
To reset the diagram, click the arrows in the upper right corner.

Susan Addington, Created with GeoGebra

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