Formal Definition of All Cycloids in Relation to Their Generating Circles

Some context, discussing Evangelista Torricelli’s Opera Gometrica (1644), is omitted.

how might someone formally define all cycloids in relation to their generating circles?

Formal Definition of All Cycloids in Relation to Their Generating Circles

A cycloid is a curve generated by a fixed point on, inside, or outside a circle as it rolls without slipping along a straight line. A generalized cycloid can be formally defined using parametric equations based on the position of the tracing point relative to the generating circle.

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1. Generalized Cycloid Definition

Let:

• $R$ be the radius of the generating circle.
• $v$ be the position of the tracing point relative to the center of the circle.
  • If $v=R$, we obtain the ordinary (standard) cycloid.
  • If $0<v<R$, we obtain a shortened (contracted) cycloid.
  • If $v>R$, we obtain an elongated (protracted) cycloid.
  • If $v<0$, the tracing point is inside the circle.
• The circle rolls along a straight line, moving with uniform velocity.
• $\theta$ is the angle of rotation of the circle as it rolls along the horizontal axis.

The parametric equations for a generalized cycloid (including standard, shortened, and elongated cases) are:

$$x(\theta )=R\theta -v\sin \theta$$

$$y(\theta )=R-v\cos \theta$$

where:

• $x(\theta )$ describes the horizontal displacement.
• $y(\theta )$ describes the vertical displacement.
• $\theta$ increases as the circle rolls.

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2. Special Cases

(i) Standard Cycloid (Ordinary Cycloid)

• Definition: A standard cycloid is the trajectory of a point on the circumference of a rolling circle.
• Condition: $v=R$.
• Equations:

$$x(\theta )=R(\theta -\sin \theta )$$

$$y(\theta )=R(1-\cos \theta )$$

• Properties:
  • The base length (one full period) is $2\pi R$.
  • The height of one arch is $2R$.
  • The area under one arch is $3\pi R^2$.
  • The arc length of one full arch is $8R$.

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(ii) Shortened Cycloid (Contracted Cycloid)

• Definition: The trajectory of a point inside the rolling circle.
• Condition: $0<v<R$.
• Equations:

$$x(\theta )=R\theta -v\sin \theta$$

$$y(\theta )=R-v\cos \theta$$

• Properties:
  • The curve remains concave downward.
  • The height is less than $2R$.
  • The length of one complete arch is less than $8R$.
  • Appears as a flattened wave.

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(iii) Elongated Cycloid (Protracted Cycloid)

• Definition: The trajectory of a point outside the rolling circle.
• Condition: $v>R$.
• Equations:

$$x(\theta )=R\theta -v\sin \theta$$

$$y(\theta )=R-v\cos \theta$$

• Properties:
  • The curve develops loops.
  • The height exceeds $2R$.
  • The arcs become elongated.
  • Resembles a series of loops touching the baseline.

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(iv) Curtate Cycloid (Point Inside the Circle)

• Definition: If $v<0$, the tracing point is inside the circle.
• Condition: $v<0$.
• Equations:

$$x(\theta )=R\theta +|v|\sin \theta$$

$$y(\theta )=R+|v|\cos \theta$$

• Properties:
  • Unlike other cases, this curve has a wavy sinusoidal appearance.
  • The tracing point never touches the ground.
  • Used in mechanical gears and pendulums.

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3. Unified Representation for All Cycloids

We can express all types of cycloids in a single parametric form using a scaling parameter $k$:

$$x(\theta )=R\theta -kR\sin \theta$$

$$y(\theta )=R-kR\cos \theta$$

where:

• $k=1$ for a standard cycloid.
• $0<k<1$ for a shortened cycloid.
• $k>1$ for an elongated cycloid.
• $k<0$ for a curtate cycloid.

This form allows for a general mathematical classification of cycloids based on the relative distance of the tracing point from the center of the rolling circle.

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4. Applications of Cycloids

• Standard Cycloid:
  • Used in bicycle wheels, pendulums, and brachistochrone problems.
  • Huygens’ clock used a cycloidal pendulum to ensure isochronous oscillations.
• Shortened Cycloid:
  • Found in gear tooth design and optimal track curves.
• Elongated Cycloid:
  • Seen in roller coaster loops and roller bearing motion.
• Curtate Cycloid:
  • Describes planetary motion in certain conditions.

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5. Conclusion

A cycloid’s shape depends on the location of the tracing point relative to its generating circle.
The parametric equations provide a unified mathematical framework for all variations, making cycloids central to mechanical physics, engineering, and mathematics.