Hypotrochoid
Encyclopedia
A hypotrochoid is a roulette
traced by a point attached to a circle
of radius
r rolling around the inside of a fixed circle of radius R, where the point is a distance
d from the center of the interior circle.
The parametric equation
s for a hypotrochoid are:
Where
is the angle formed by the horizontal and the center of the rolling circle (note that these are not polar equations because 
is not the polar angle).
Special cases include the hypocycloid
with d = r and the ellipse
with R = 2r.
The classic Spirograph
toy traces out hypotrochoid and epitrochoid
curves.
Roulette (curve)
In the differential geometry of curves, a roulette is a kind of curve, generalizing cycloids, epicycloids, hypocycloids, trochoids, and involutes....
traced by a point attached to a circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
of radius
Radius
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...
r rolling around the inside of a fixed circle of radius R, where the point is a distance
Distance
Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical distance...
d from the center of the interior circle.
The parametric equation
Parametric equation
In mathematics, parametric equation is a method of defining a relation using parameters. A simple kinematic example is when one uses a time parameter to determine the position, velocity, and other information about a body in motion....
s for a hypotrochoid are:
Where
is the angle formed by the horizontal and the center of the rolling circle (note that these are not polar equations because is not the polar angle).Special cases include the hypocycloid
Hypocycloid
In geometry, a hypocycloid is a special plane curve generated by the trace of a fixed point on a small circle that rolls within a larger circle...
with d = r and the ellipse
Ellipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
with R = 2r.
Spirograph
Spirograph is a geometric drawing toy that produces mathematical curves of the variety technically known as hypotrochoids and epitrochoids. The term has also been used to describe a variety of software applications that display similar curves, and applied to the class of curves that can be produced...
toy traces out hypotrochoid and epitrochoid
Epitrochoid
An epitrochoid is a roulette traced by a point attached to a circle of radius r rolling around the outside of a fixed circle of radius R, where the point is a distance d from the center of the exterior circle....
curves.
External links
- Flash Animation of Hypocycloid
- Hypotrochoid from Visual Dictionary of Special Plane Curves, Xah Lee
- Interactive hypotrochoide animation