Understanding the Path of a Point on a Rolling Circle

When a circle moves along a straight line without slipping, the path traced by a specific point located on the edge (circumference) of the circle creates a unique curve. This phenomenon is a fundamental concept in geometry and kinematics.

Let's analyze the options provided:

• Tractrix: A tractrix is a curve defined by the path of an object pulled by a rope of fixed length, where the pulling end moves along a straight line. This is different from a rolling circle.
• Cycloid: A cycloid is precisely defined as the curve traced by a point on the circumference of a circle that rolls along a straight line without slipping. As the circle rolls, the point on its edge follows this characteristic arched path.
• Epicycloid: An epicycloid is the curve traced by a point on the circumference of a small circle that rolls around the outside of a larger circle. This involves rolling on another circle, not a straight line.
• Hypocycloid: A hypocycloid is the curve traced by a point on the circumference of a small circle that rolls around the inside of a larger circle. Like an epicycloid, this involves rolling inside another circle, not on a straight line.

Based on the definitions, the path taken by any point on the circumference of a circle when it rolls along a straight line without slipping is known as a Cycloid.