The Circle

Introduction:

Perhaps the most well-known curve in mathematics and in popular culture, the circle was believed by the Greeks to be a perfect form. It appears in virtually all areas of mathematics, from elementary geometry to complex analysis and differential geometry. It can be viewed from many different perspectives, and defined in many different ways. Some of the more common definitions are given below (see equations).

Another well-known curve that results from the circle is its involute:

The involute is defined geometrically as the curve that results from unwinding a taut string from a curve, tracing the end of the string.

Definition:

Implicitly, the circle is defined by

Geometrically, the circle is the set of points at a given distance (called the radius) from a certain point (called the center) in a plane.

Properties:

Velocity and Acceleration:

The Tangent Vector:

Arc Length:

Curvature: