極曲線

Picture a spirograph toy placed flat on a table. As a wheel rotates inside a fixed ring, a pen placed in one of the pen holes traces a repeating path on the paper.

Even though these patterns look complex, the pen's position at any moment is defined by its distance from the center and its angle of rotation.

This is the basis of a polar curve, where points are defined by a radius r and an angle theta.

To see how this works mathematically, let’s look at a specific function. Suppose the radius changes with angle, defined by the equation r equals one plus sine theta.

As the angle moves from 0 to 2 pi, the value of the radius r increases and decreases smoothly. This creates a specific path that forms an outer arc and a sharp inward loop, symmetrically closing the curve at the origin.

This specific shape is called a cardioid—a smooth, heart-like curve formed by a simple variation in radius.

By changing the function, one can move from simple shapes like this to the intricate, multi-layered patterns traced by a spirograph.