This applet describes stereographic projection from the two-sphere minus the North Pole to the plane, in particular the images of circles on the sphere, that are all sent to circles or straight lines in the plane. In this illustration we demonstrate stereographic projection from the North Pole on the unit sphere in three-space to the Equatorial plane. Any circle on the unit sphere is projected to a circle in the plane, with a circle through the origin being projected to an infinitely large circle, i.e. a straight line.
When the angle c = 0, changing the value a gives a family of circles in parallel planes at heights z = sin(a). Changing the value of c rotates the circle so that when the circle passes through the North Pole, the image is a straight line in the Equatorial plane.