Problem 63 - Calculus Early Transcendentals: Pearson New International Edition

Question

The curve traced by a point on a circle of radius \(b\) as it rolls without slipping on the outside of a fixed circle of radius \(a\) is called an epicycloid. Show that it has parametric equations $$ \begin{array}{l} x=(a+b) \cos t-b \cos \frac{a+b}{b} t \\ y=(a+b) \sin t-b \sin \frac{a+b}{b} t \end{array} $$

Step-by-Step Solution

Answer

The parametric equations for the epicycloid are derived via tracing the path made by a point on the rolling circle, confirming the curves as \( x = (a+b) \cos t - b \cos \frac{a+b}{b} t \) and \( y = (a+b) \sin t - b \sin \frac{a+b}{b} t \).

Step 1: Understand the Concept

An epicycloid is a curve generated by tracing a point on a circle (called the generating circle) of radius \( b \) as it rolls around another circle (the base circle) of radius \( a \). When a circle of radius \( b \) makes contact externally with a circle of radius \( a \) rolling around it without slipping, the point on the outer circle describes an epicycloid.

Step 2: Determine the Angle of Rotation

As the generating circle rolls around the fixed circle, the angle \( t \) represents the angular position of the rolling circle relative to the point where it touches the base circle. The point on the generating circle moves through an angle \( \frac{a+b}{b}t \) in terms of its own radius, as it rolls without slipping.

Step 3: Calculate the Path in Terms of Rotations

The path traced by a point \((x,y)\) on the rolling circle can be broken into contributions from the movement of the center of the generating circle and the motion of the traced point around its center. The center of the generating circle, as the circle rolls around, moves in a larger circle of radius \(a+b\).

Step 4: Parameterize the Curve

The x-coordinate of the center of the generating circle is given by \((a+b)\cos t\). The adjustment for the point on the circle is \(-b \cos\left(\frac{a+b}{b} t\right)\), accounting for the point's additional rotation. Thus, the total x-coordinate is \(x = (a+b) \cos t - b \cos \frac{a+b}{b} t\). Similarly, for y-coordinates, \(y = (a+b) \sin t - b \sin \frac{a+b}{b} t\).

Step 5: Confirm the Parametric Equations

The parametric equations for the epicycloid are derived by considering both the movement of the generating circle's center and the additional rotation of the point. Via substantiation with the derived terms: \[ \begin{array}{l} x = (a+b) \cos t - b \cos \frac{a+b}{b} t \ y = (a+b) \sin t - b \sin \frac{a+b}{b} t \end{array} \] confirms these as the correct parametric equations of the epicycloid.

Key Concepts

Understanding Parametric Equations

Parametric equations allow us to describe curves by expressing the coordinates of points on the curve as functions of a parameter, usually denoted by \( t \). For the epicycloid, the path formed is traced by the movement of a point on a circle of radius \( b \) as it rolls around a fixed circle of radius \( a \).
By using parametric equations, we separate the movement into components, which aids in visualizing the motion and trajectory effectively.
In the epicycloid's parametric form:

• \( x = (a+b) \, \cos t - b \, \cos\left(\frac{a+b}{b} \, t\right) \)
• \( y = (a+b) \, \sin t - b \, \sin\left(\frac{a+b}{b} \, t\right) \)

These equations describe the position \((x, y)\) of the point on the circle as a function of the parameter \( t \), which represents the angle of rotation of the rolling circle.
Through parametric equations, we can analyze complex curve shapes, like epicycloids, by breaking down their components into manageable parts.

Concept of Rolling Without Slipping

When a circle rolls on another surface without slipping, the point of contact is momentarily at rest concerning the surface on which it rolls. This concept ensures that the distance covered on both the inside and outside edges remain consistent in their circular paths. For our epicycloid, this assures the length covered by the rolling circle matches the base circle's circumference at any point of contact.
This rolling motion contributes to the path traced by the point on the circle without creating inconsistencies due to sliding.

• The distance the circle rolls is directly related to its radius and the debated angle, \( t \).
• As the circle rolls without slipping, the tangent velocity at the contact point is essential in deriving the parametric representations for the epicycloid.

Rolling without slipping is foundational in creating accurate parametric descriptions of motion, enabling the derivation of the curve traced, in this case, the epicycloid.

Understanding the Angle of Rotation

The angle of rotation \( t \) is critical when dealing with rolling movements, as it defines the angular position of the rolling circle related to its point of contact on the base circle. For epicycloids, the rotation isn't just a simple angle \( t \).
Due to the rolling nature, this angle needs adjustment based on how far the tracing point moves around its own circle, captured by the term \( \frac{a+b}{b}t \). This describes:

• The additional angle through which the point on the rolling circle rotates internally as it travels along the circumference of the base circle.
• How these angles influence the trajectory of the point relative to both circles involved.

The complex relationship between angles from both implicit (rolling) and explicit (actual tracing) motion gives the distinctive cusp-like patterns typical of an epicycloid.

Curve Tracing in Epicycloids

Curve tracing involves describing the path a point follows and how shapes like epicycloids come about through continuous movement. With epicycloids, this tracing is conducted as a point on the generating circle rolls around the base circle.
The method of curve tracing offers an analytical way to predict positions and the nature of the curve, specifically through the epicycloid's parametric equations. By observing how the center of the rolling circle moves on a larger circle with radius \( a+b \), and calculating how the traced point rotates and shifts around this center, we can track each point's trajectory on a coordinate plane.

• This ensures that each component of the path aligns with theoretical predictions.
• Examining this trajectory reveals the distinct loops and arcs characteristic of an epicycloid pattern.

Through this process of detailed curve tracing, we can appreciate and fully grasp the intricate dance that defines epicycloidal curves.