#GRAPHMATICA EQUATION FOR HYPOTROCHOIDS FULL#

By revolving the interior circle once, the resulting image looks as follows:

For example, let the radius of the interior circle b be 3, the radius of the fixed circle a be 5, and the distance from P to the center of the interior circle h be 5.

#GRAPHMATICA EQUATION FOR HYPOTROCHOIDS FULL#

In some cases this will not result in a full pattern, meaning that the point P will not always return to its starting place. If we allow t to vary from 0 to 2, then the interior circle will only make one revolution around the fixed circle. The domain of t has a large effect on what the hypotrochoid will look like. They all belong to a family of curves, which includes other types of trochoids. Hypotrochoids are graphed using the following parametric equationsĪll hypotrochoids are defined by these parametric equations. Note that the point P can be anywhere in relation to the interior circle.

We call this simply a hypocycloid Hypocycloid4.gif.Ī hypotrochoid is a roulette traced by a point P attached to a circle of radius b rolling around the inside of a fixed circle of radius a, where P is a distance h from the center of the interior circle. If P is on the rolling circle, then the distance from the center of the inner circle to P is equal to the radius of the inner circle. If P is outside the rolling circle, then the distance of the line from the center of the inner circle to P is greater than the radius of the inner circle. If P is inside the rolling circle, then the distance of the line from the center of the inner circle to P is less than the radius of the inner circle. The point P can be in three different general locations in relation to the rolling circle: inside, outside, or on the edge.

There are three subgroups within the Hypotrochoid family. If the blue circle rolled (like a wheel) along the edge of the larger circle and the red point acted as a pen, leaving a trace, then the red drawing created would be a hypotrochoid.

Graphmatica equation for hypotrochoids