Problem 36 - Thomas Calculus

Question

Hypocycloid When a circle rolls on the inside of a fixed circle, any point \(P\) on the circumference of the rolling circle describes a hypocycloid. Let the fixed circle be \(x^{2}+y^{2}=a^{2},\) let the radius of the rolling circle be \(b,\) and let the initial position of the tracing point \(P\) be \(A(a, 0) .\) Find parametric equations for the hypocycloid, using as the parameter the angle \(\theta\) from the positive \(x\) -axis to the line joining the circles' centers. In particular, if \(b=a / 4\) as in the accompanying figure, show that the hypocycloid is the astroid $$x=a \cos ^{3} \theta, \quad y=a \sin ^{3} \theta$$

Step-by-Step Solution

Answer

The parametric equations for the astroid are \(x = a \cos^3 \theta\) and \(y = a \sin^3 \theta\).

Step 1: Understanding the Geometry

To find the equations for the hypocycloid, understand the geometry of the system. A fixed circle with equation \(x^2 + y^2 = a^2\) has a center at the origin and radius \(a\). A rolling circle of radius \(b\) rolls inside this fixed circle. The distance from the center of the fixed circle to the center of the rolling circle is \(a - b\). The tracing point \(P\) will start at \((a, 0)\) on the fixed circle.

Step 2: Rolling Circle Center Position

Let the center of the rolling circle be \((h, k)\). As the rolling circle rolls inside the fixed circle, its center follows a circular path with radius \(a - b\) around the origin. The angle \(\theta\) is the parameter that measures the angle from the positive \(x\)-axis to the line joining the centers of the circles. Thus, \(h = (a - b) \cos \theta\) and \(k = (a - b) \sin \theta\).

Step 3: Point P on the Rolling Circle

The point \(P\) traces a path on the rolling circle with radius \(b\). The position of \(P\) relative to the center \((h, k)\) of the rolling circle describes a rotation through an angle \(-\left(\frac{a-b}{b}\right)\theta\) due to an inner rolling rotation. So, the parametric position in terms of \(b\) on the rolling circle is \(P_x = b \cos \left(-\frac{a-b}{b} \theta\right)\) and \(P_y = b \sin \left(-\frac{a-b}{b} \theta\right)\).

Step 4: Combine into Parametric Form

Combine the position of \(P\) relative to the center \((h, k)\) to get the full parametric expressions: \[x = h + b \cos \left(-\frac{a-b}{b} \theta\right) = (a-b) \cos \theta + b \cos \left(\frac{a-b}{b} \theta\right)\] and \[y = k + b \sin \left(-\frac{a-b}{b} \theta\right) = (a-b) \sin \theta + b \sin \left(\frac{a-b}{b} \theta\right)\].

Step 5: Simplify for Specific Case b = a/4

Specifically, if \(b = \frac{a}{4}\), simplify the expressions: \[x = \frac{3a}{4} \cos \theta + \frac{a}{4} \cos(3\theta)\] and \[y = \frac{3a}{4} \sin \theta + \frac{a}{4} \sin(3\theta)\]. After simplifying using trigonometric identities for \(\cos(3\theta)\) and \(\sin(3\theta)\): \(\cos(3\theta) = 4\cos^3\theta - 3\cos\theta\) and \(\sin(3\theta) = 3\sin\theta - 4\sin^3\theta\), these reduce to \(x = a \cos^3 \theta\) and \(y = a \sin^3 \theta\). These are the parametric equations of an astroid.

Key Concepts

Understanding Parametric Equations

Parametric equations are a way of expressing a set of related quantities called parameters. In the context of a hypocycloid, they provide a way to express the coordinates of any point on the curve as functions of a single parameter, which is typically an angle. Such equations are quite useful for describing complex motion or paths like those traced by rolling objects.

A general parametric equation has the form

• \(x = f(t)\)
• \(y = g(t)\)

where \(t\) is the parameter. In our case, \(\theta\) is the angle from the positive x-axis to the line joining the centers of the circles. This angle changes as the rolling circle moves, which means the parameter \(\theta\) continuously varies. For a hypocycloid, these parameter formulas take into account not only the motion of the rolling circle but also its rotation.

This approach gives an elegant and efficient way to describe complicated, looping curves like an astroid, which can be challenging to express in standard Cartesian coordinates.

Parametric Equations for Hypocycloids

When deriving the parametric equations for a hypocycloid, we make use of the rolling circle's position relative to the fixed circle. We start with the relationship:\[(h, k) = ((a-b)\cos\theta, (a-b)\sin\theta)\]
This expression defines the center of the rolling circle as it rolls inside the fixed circle. Each position of this center tells us a bit about the rolling circle's journey along the hypocycloid path.

The point \(P\) on the rolling circle will make a different rotation in the opposite direction. This accounts for both the linear movement along its circular path and the circle's own internal rotation. The parametric coordinates of the tracing point \(P\) on the circle are then\[x = (a-b) \cos \theta + b \cos \left(-\frac{a-b}{b} \theta\right)\] and \[y = (a-b) \sin \theta + b \sin \left(-\frac{a-b}{b} \theta\right)\]
These equations clearly illustrate the influence of both the moving center and the counter-rotation. Each distinct value of \(\theta\) corresponds to a different position along the hypocycloid curve.

Geometry of Circles in a Hypocycloid

Understanding the geometry involved in a hypocycloid setup is pivotal. The configuration consists of two circles: a fixed circle and a rolling circle, which traces the path.

• The fixed circle has its center at the origin of a coordinate system and a large radius \(a\).
• The rolling circle has a smaller radius \(b\) and rolls within the fixed circle without slipping.

For the hypocycloid, the rolling circle stays entirely inside the fixed circle, continually touching its interior edge. This rolling action around the origin creates a unique, looping path for any point on the circumference of the rolling circle.

The motion is dictated by the radius difference \((a - b)\), which determines the path's arching distance from the center to the focal point. The beauty of the hypocycloid is its ability to form complex paths, like the astroid, which can be modeled by manipulating this geometric setup.

Using Trigonometric Identities

Trigonometric identities are core tools which simplify the expressions in parametric equations. They help convert complex trigonometric expressions into more manageable forms.

In the case of the hypocycloid, simplifying our parametric formulas requires understanding compound angle identities.

• The identity \[\cos(3\theta) = 4\cos^3\theta - 3\cos\theta\] allows us to shift from a triple angle formula back to expressions involving simpler power-based terms.
• Similarly, \[\sin(3\theta) = 3\sin\theta - 4\sin^3\theta\] works the same way for the sine function.

These identities are not only crucial for finding clean parametric equations like \(x = a\cos^3\theta\) and \(y = a\sin^3\theta\) for the astroid but also for revealing hidden regularities in seemingly complex paths. Trigonometry thus plays a significant role in unpacking and understanding the inherent symmetries and elegance in geometric configurations like the hypocycloid.