The Art of the Ellipse

On this page we recall the basic theory of the ellipse, and we develop Kepler's image of an ellipse inscribed in a unit circle.

The standard ellipse, Fig. 01

By definition, the ellipse in standard position is the locus of points R = (X, Y) such that
(X/A)2 + (Y/B)2 = 1
As shown in the figure, we visualize the ellipse with the major axis vertical, along the X axis. Let Q denote the upper end of the major axis, and C its center. Let CQ denote the distance from point C to point Q. Then the parameters of the ellipse are:
  • semi-major axis, A = CQ,
  • focal distance, C = CS = CS',
  • eccentricity, C/A, and
  • semi-minor axis, B = DC.
In this example, we have used A=5/2, B=3/2, and C=4/2=2 to draw the figure, so the eccentricity is 0.8.

The ellipse may be drawn by attaching a string of length L at the two foci, S and S', and using it to guide a pencil around the ellipse. When the pencil is at Q, the string covers S'Q twice and SS' once, so 2*(A-C) + 2*C = L, or L = 2*A. Considering the pencil at (0, B), the end of the semi-minor axis, the Pythagorean theorem gives us the equations,

B = sqrt(A2 - C2)
= A*sqrt(1 - e2).
relating A, B, C, and e.

Kepler's ellipse, Fig. 02

Here we follow Kepler, as described in [Katz, 1993; p. 378]. We normalize the figure by rescaling both X and Y axes by 1/A. We now have another ellipse of the same eccentricity, but having a=A/A=1, b=B/A=0.6, and c=C/A=e=0.8.

Now enclose the ellipse (no matter the eccentricity) in a circle of radius one. Note that the focal distance is now c = e, and the semi-minor axis of the ellipse is now b = sqrt(1 - e2) < 1.

If the eccentricity is small, as it is for all planets, then we have the approximation, b ~= 1 - e2/2.

Kepler's ellipse and circle, Fig. 03a

Again following Kepler, we choose a point R on the ellipse, and draw a horizontal line through it, meeting the unit circle at W and the X (vertical) axis at V. If W = (X0, Y0), R = (X1, Y1), and V = (X2, Y2), then:
  • X0 = X1 = X2 as all three points are on the same horizontal line (X = X0),
  • Y2 = 0 as V is on the X axis (Y = 0),
  • X02 + Y02 = 1 as W is on the unit circle, and
  • (X1/a)2 + (Y1/b)2 = 1 as R is on our ellipse.
But a=1, and X0=X1, so subtracting these two equations we have,
(Y0)2 = (Y1/b)2
so Y1 = b*Y0. Thus the ellipse within the circumscribed unit circle may be understood as the circle uniformly compressed in the horizontal dimension by the factor b < 1.
Rev'd 29 dec 2001 by ralph abraham