In 1609, Johannes Kepler discovered that the planets traveled in elliptical orbits, with the sun at one focus.
Kepler didn’t have a rectangular coordinate system to help follow the courses of the planets in the sky. But he did have system very similar to polar coordinates.
And even though he didn’t have a formal notion of a pole, he placed the focus, with the sun, at the pole
With the other focus falling on one of the rectangular coordinate axes, the polar equation of an ellipse, and of the other conic sections, takes an especially simple form.
This program allows you to experiment with this simple form by adjusting the values of the two parameters in the polar equation.
First, choose the conic you want to investigate.
Then, use the sliders to adjust the values of the eccentricity e and the latus rectum l.
The eccentricity controls the shape of the conic section ...
... while the latus rectum controls the size. You can also type numbers directly into the boxes.
Radio buttons allow you to control the orientation of the conic section by choosing the sign and the trigonometric function used in the denominator.
The conic section will be shown in the graph area, ...
... and various measurements and other data about the conic section will be displayed below the control area.
The points and line segments being measured will also appear in the graph, and can be turned off by de-selecting the checkboxes.
The buttons at bottom will allow you to investigate other conic sections. Try them all! Keep an eye on the data as well as the graph. Look for patterns!