1.1.8 Circles and Ellipses
At this point we can parameterize any circle in the plane by transforming the unit circle using the following guide:
- is the radius of the circle
- is the center of the circle
- changes the speed and the orientation of the parameterization. I think speed is self-explanatory but orientation roughly refers to the direction of movement as the image curve is traced out. For our purposes, the standard parameterization of the unit circle will be said to have the positive orientation (counter-clockwise). When the resulting path will have the negative orientation (clockwise).
- shifts the parameter counterclockwise about the circle by radians.
A slight adjustment to our understanding of transformed circles allows us to also parameterize an ellipse whose axes are parallel to the and axes. Return to the applet above and attempt to parameterize an ellipse. (Recall that the common rectangular equation for an ellipse is )