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.
In the applet below you can type in the rectangular equation of a transformed circle and practice parameterizing. Get comfortable shifting the parameter to start your parameterization at various points along the circle. Change the speed to achieve a parameterization that wraps around the circle multiple times, creates an image curve that includes just a portion of the circle, or has a negative orientation.

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 )