Question

. Let a circle of radius \(b\) roll, without slipping, inside a fixed circle of radius \(a, a>b .\) A point \(P\) on the rolling circle traces out a curve called a hypocycloid. Find parametric equations of the hypocycloid. Hint: Place the origin \(O\) of Cartesian coordinates at the center of the fixed, larger circle, and let the point \(A(a, 0)\) be one position of the tracing point \(P\). Denote by \(B\) the moving point of tangency of the two circles, and let \(t\), the radian measure of the angle \(A O B\), be the parameter (see Figure 11 ).

Short answer

The parametric equations of the hypocycloid are \(x(t) = (a-b)\cos(t) + b\cos\left(\frac{a-b}{b}t\right)\) and \(y(t) = (a-b)\sin(t) - b\sin\left(\frac{a-b}{b}t\right)\).

Step-by-step solution

Understand the Problem Setup

Let's picture the scenario: A smaller circle of radius \(b\) rolls inside a larger circle of radius \(a\) without slipping. The key points involved are the origin \(O\) at the center of the larger circle, \(A(a,0)\) is a position on the rolling circle, and \(B\) is the tangency point between the two circles. We want to find the path traced by a point \(P\) on the rolling circle, called a hypocycloid.

Analyze Angles and Distances

The arcs swept by both circles maintain a constant proportional relationship due to the non-slipping condition. Specifically, if the larger circle rolls a distance equal to \(t\times a\), the smaller circle rolls a distance of \(t\times b\). This implies that as the smaller circle rolls by angle \(t\) inside the larger circle, its own angle of rotation due to rolling is \(\theta = \frac{a-b}{b}t\).

Determine the Center of the Rolling Circle

The center of the rolling circle moves around a path centered at \(O\) of the fixed circle. When the angle \(AOB\) is \(t\), the coordinates of the center \(C\) of the rolling circle are given by the polar forms: \((a-b)\text{cos}(t),\ (a-b)\text{sin}(t)\). This traces the path of \(C\) around the origin.

Incorporate the Point P on the Rolling Circle

The point \(P\) on the rolling circle will rotate with respect to its center as the circle rolls. Therefore, its position relative to \(C\) will rotate by \(\theta = \frac{a-b}{b}t\). The coordinates of point \(P\) relative to \(C\) are \(b\text{cos}(\theta)\) and \(b\text{sin}(\theta)\).

Combine the Components to Find Parametric Equations

Now, combine the position of \(C\) and the relative position of \(P\) to find the global coordinates of \(P\). The x-coordinate of \(P\) is given by \(x(t) = (a-b)\text{cos}(t) + b\text{cos}\left(\frac{a-b}{b}t\right)\). Similarly, the y-coordinate of \(P\) is \(y(t) = (a-b)\text{sin}(t) - b\text{sin}\left(\frac{a-b}{b}t\right)\).

Key concepts

Non-Slipping Condition

Imagine two circles, one rolling within the other like gears fitting perfectly. The non-slipping condition is crucial here. It means the smaller circle, with radius \(b\), rolls inside the larger circle of radius \(a\) without sliding. This ensures a pure rolling motion.
This condition can be visualized by imagining a point on the edge of the rolling circle contacting the inner surface of the fixed circle continuously and without losing contact. Mathematically, this implies the arc lengths covered by both circles are equal. This results in the relationship: \(t \times a = \theta \times b\).
Here \(t\) is the parameter (angle in radians) describing the position on the larger circle, while \(\theta\) denotes the internal angle of rotation of the smaller circle. Consequently, \(\theta = \frac{a-b}{b}t\) describes how much the rolling circle rotates as it rolls inside the fixed circle.

Rolling Circle Dynamics

Picture the dynamics as the smaller circle, centered on a point \(C\), tracing a path inside the larger circle. The center \(C\) follows a circular path with radius \((a-b)\), derived from the subtraction of the circle radii.
The position of \(C\) at any point can be defined using polar coordinates, since this system naturally accommodates circular motion. Thus, the coordinates of \(C\) are: \((a-b)\cos(t), (a-b)\sin(t)\).
This equation vividly illustrates how the center of the smaller circle orbits the central point of the larger circle continuously, always maintaining a radius \((a-b)\) from the center.

Coordinate Geometry

As the circle rolls, the point \(P\) on its edge traces a path described in the coordinate plane, forming a hypocycloid. To fully appreciate this, let's dive into coordinate geometry which allows precise tracking of \(P\)'s movement.
While the center of the rolling circle moves along a defined path, \(P\) concurrently rotates around its center. This rotation is captured in the coordinates relative to \(C\) as \(b\cos(\theta)\) and \(b\sin(\theta)\).
The equations of the hypocycloid combine the positions of \(C\) and the relative offset of \(P\):

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

These parametric equations give a clear map of where \(P\) will be at any angle \(t\), blending the motions of both circles harmoniously.

Polar Coordinates

Converting circular motion into polar coordinates simplifies visualizing complex rotations. Polar coordinates revolve around the radius and angular position from a central point, making them perfect for circular paths.
In our scenario, the center \(C\) of the rolling circle follows a polar coordinate system around the origin \(O\). This is describes as: \[(r, \theta) = (a-b, t)\]
Here, \(r\) is the radius dictated by the inner circle's smaller radius revolving around the outer circle's center. \(\theta\), measured in radians, is the angle that describes \(C\)'s current position.
Polar coordinates are instrumental in translating rotational dynamics into easily understandable terms, enhancing comprehension of the path traced by \(P\) and bringing the rolling motion to life from a geometric perspective.