A curtate cycloid is a type of curve generated by a point on the interior of a circle as it rolls along a straight line. It differs from a regular cycloid in that the tracing point is not on the circumference but inside the circle.
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The parametric equations for a curtate cycloid are $x = r \theta - d\sin(\theta)$ and $y = r - d\cos(\theta)$, where $r$ is the radius of the rolling circle, $d$ is the distance from the center to the tracing point, and $\theta$ is the angle of rotation.
A curtate cycloid will have cusps (sharp points) when $d < r$, and will not intersect itself.
The period of one complete cycle of a curtate cycloid is $2\pi$ radians.
Unlike a standard cycloid, which has arches with cusps at both ends, a curtate cycloid’s cusps do not touch the ground line due to the interior point tracing.
The arc length of one arch of a curtate cycloid can be found using integral calculus.