At this point we can parameterize any circle in the plane by transforming the unit circle using the following guide:[br][math]\vec{c}\left(t\right)=\left(R\cos\left(at+b\right)+h,R\sin\left(at+b\right)+k\right),t\in\left[0,2\pi\right][/math][br][list][*][math]R[/math] is the radius of the circle[/*][*][math]\left(h,k\right)[/math] is the center of the circle[/*][*][math]a[/math] changes the [b][color=#ff0000]speed[/color][/b] and the [b][color=#ff0000]orientation[/color][/b] 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 [b][color=#ff0000]positive orientation[/color][/b] (counter-clockwise). When [math]a<0[/math] the resulting path will have the [b][color=#ff0000]negative orientation[/color][/b] (clockwise).[/*][*][math]b[/math] shifts the parameter counterclockwise about the circle by [math]b[/math] radians.[/*][/list][br]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 [math]x[/math] and [math]y[/math] axes. Return to the applet above and attempt to parameterize an ellipse. (Recall that the common rectangular equation for an ellipse is [math]\left(\frac{x-h}{r_1}\right)^2+\left(\frac{y-k}{r_2}\right)^2=1[/math])
[math]\vec{c}\left(t\right)=\left(r_1\cos\left(at+b\right)+h,r_2\sin\left(at+b\right)+k\right),t\in\left[0,2\pi\right][/math] parameterizes an ellipse whose axes are parallel to the [math]x[/math] and [math]y[/math] axis. The center of the ellipse is at [math]\left(h,k\right)[/math]. The horizontal semi-axis has length [math]r_1[/math] and the vertical semi-axis has length [math]r_2[/math].[br][br]As a challenge, you could combine your understanding of rotations of [math]\mathbb{R}^2[/math] with your understanding of the standard parameterization for an ellipse to deduce a parameterization for an oblique ellipse - that is an ellipse whose axes are no longer parallel to the primary axes.