Trochoid

Trochoid

A trochoid is a type of curve generated by a point on a circle as it rolls along a straight line. The term "trochoid" comes from the Greek word "trochos," meaning "wheel." Trochoids are a broader class of curves that include cycloids, epicycloids, and hypocycloids.

Types of Trochoids

Trochoids can be classified into three main types based on the position of the point relative to the circle:

• Cycloid: A cycloid is a special type of trochoid where the point is on the circumference of the circle. It is the path traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slipping.

• Curtate Trochoid: In a curtate trochoid, the point is inside the circle. This results in a curve that loops inward, creating a series of arches.

• Prolate Trochoid: A prolate trochoid occurs when the point is outside the circle. This type of trochoid has loops that extend outward from the path of the circle.

Mathematical Representation

The parametric equations for a trochoid can be expressed as:

\[ x(\theta) = a\theta - b\sin(\theta)\ y(\theta) = a - b\cos(\theta)\ \]

where:

• \(a\) is the radius of the rolling circle,
• \(b\) is the distance from the center of the circle to the tracing point,
• \(\theta\) is the parameter, representing the angle of rotation of the circle.

Applications

Trochoids have applications in various fields such as mechanical engineering, physics, and computer graphics. They are used in the design of gear teeth, cam profiles, and in the study of motion and wave patterns.

Related Pages

• Cycloid
• Epicycloid
• Hypocycloid
• Spirograph

Gallery

A cycloid, a special case of a trochoid.

A prolate trochoid with the point outside the circle.

A curtate trochoid with the point inside the circle.

References

• Gray, A. (1997). Modern Differential Geometry of Curves and Surfaces with Mathematica. CRC Press.
• Lawrence, J. D. (1972). A Catalog of Special Plane Curves. Dover Publications.

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  Cycloid animation

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  Trochoid with h=1.25

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  Trochoid with h=0.8