The general formula for a circle centered at point (
) is given as:
(x-h)^2 + (y-k)^2 = r^2.
Although Ptolemy (around 100 AD) probably studied, and was familiar with, conic sections such as the ellipse, they were not part of his astronomy. It would be nearly two millennia from the inception of conic sections before conic sections, other than the circle, would play a role in astronomy.
Early observations led to the belief that the Earth did not move, and consequently must be the center of the universe. It is a natural conclusion since it is not readily apparent that the Earth is actually moving. After all, on a still day there is no wind or any other indication of motion.
The problem remained, however, of how to explain the motion of the heavens. As idealists, the Greeks were certain that the natural world must represent perfection. The perfect motion, or shape, about something is a circle. Unfortunately, observations of the heavens did not support the idea of the planets and stars rotating in perfect circles around the Earth. Problems, such as retrograde motion, varying speeds and apparent distance, could not be explained with this model.
In order to maintain the inherent perfection of this geocentric world view, with perfect circular motion, Ptolemy developed a system where the planets would travel in small circles (called epicycles) within a larger circle (called a deferent) around the earth. As complicated as it was, it was a better description than had previously been developed.
Copernicus (around 1500 AD) had the revolutionary idea that rather than the Earth being the center around which all heavenly things revolved it was actually the Sun, which was at the center. Instead of a geocentric universe, it was a heliocentric universe. According to the Copernican view, planets still moved in circular orbits, which meant that epicycles were still necessary to explain the observed motions of the heavens.
Although analytic geometry, as we know it, was not yet formally born we will introduce the definition and general equation of the circle.
A circle is the set of all points in a plane such that the distance (radius) from a given point (center of the circle) is constant. The general equation of the circle with center (
) and radius r is: (x-h)^2 + (y-k)^2 = r^2. Students should be able to write the equation of circles given identifying characteristics and graph the equations of circles.