Problem 58 - Calculus Volume 3

Question

Sketch the curve known as an epitrochoid, which gives the path of a point on a circle of radius \(b\) as it rolls on the outside of a circle of radius \(a\). The equations are $$ \begin{array}{l} x=(a+b) \cos t-c \cdot \cos \left[\frac{(a+b) t}{b}\right] \\ y=(a+b) \sin t-c \cdot \sin \left[\frac{(a+b) t}{b}\right] \end{array} $$ Let \(a=1, b=2, c=1\).

Step-by-Step Solution

Answer

Sketch by substituting values into the equations and plotting points for \(t = 0\) to \(4\pi\).

Step 1: Substitute Given Values into Equations

To sketch the epitrochoid, first substitute the given values: \(a=1\), \(b=2\), and \(c=1\), into the parametric equations:\[x = (1+2) \cos t - 1 \cdot \cos \left[\frac{(1+2)t}{2}\right]y = (1+2) \sin t - 1 \cdot \sin \left[\frac{(1+2)t}{2}\right]\] This simplifies to:\[x = 3 \cos t - \cos \left( \frac{3t}{2} \right)y = 3 \sin t - \sin \left( \frac{3t}{2} \right)\]

Step 2: Choose a Range for Parameter t

The parameter \(t\) usually ranges from \(0\) to \(2\pi\) to complete one full rotation. However, since the inner circle makes multiple rotations around the larger circle, we need a larger range. Let's use \(t = 0\) to \(t = 4\pi\) to ensure we capture the full behavior of the curve.

Step 3: Create a Table of Values

Using software, a calculator, or by hand, compute coordinates \((x,y)\) for several values of \(t\) from \(0\) to \(4\pi\). Here are some sample points:- For \(t = 0\), \( (x, y) = (2, 0)\)- For \(t = \frac{\pi}{2}\), \( (x, y) \approx (-2.93, 3)\)- For \(t = \pi\), \( (x, y) = (-4, 0)\)- For \(t = \frac{3\pi}{2}\), \( (x, y) \approx (-2.93, -3)\)- For \(t = 2\pi\), \( (x, y) = (2, 0)\)Continue computing until \(t = 4\pi\).

Step 4: Sketch the Curve

Plot the calculated points \((x, y)\) on a graph. Connect these points smoothly to illustrate how the epitrochoid looks. Observe the pattern and symmetry formed as it represents the path of the point. It should display a pattern of loops and possibly repeats every few cycles, given that \(b > a\).

Key Concepts

Parametric Equations

Parametric equations are an essential tool in mathematics that allow us to represent curves in a plane. These equations express the coordinates of the points on a curve as functions of a parameter, typically denoted by "t". For the epitrochoid, the parametric equations used are:\[x = (a+b) \cos t - c \cdot \cos \left[\frac{(a+b) t}{b}\right]\]\[y = (a+b) \sin t - c \cdot \sin \left[\frac{(a+b) t}{b}\right]\]This means each point \((x, y)\) on the epitrochoid is a specific outcome based on the parameter "t". By substituting different values of "t", we can find a variety of points across the curve.

• "\(t\)" acts as the parameter that progresses the drawing of the curve.
• Parametric equations are particularly useful for curves that are not functions, as they can express more complex paths.

Curve Sketching

Curve sketching is an important skill in mathematics, especially when dealing with curves defined by parametric equations. It involves determining the general shape of the curve by plotting several calculated points from the equations. The exercise provides specific parametric equations for an epitrochoid, a type of roulette curve.
To sketch such a curve:

• Select a range for the parameter "\(t\)", often extended beyond \(0\) to \(2\pi\) given the rolling nature of the epitrochoid.
• Compute numerous values of \((x, y)\) and plot these points on a graph.
• Connect these points smoothly to visualize the epitrochoid's path.

The curve can be intricate with loops and repeating patterns. An epitrochoid's beauty is captured in the connection of these calculated points.

Radius

Radius plays a vital role in defining the shape and size of an epitrochoid. In this context, there are two radii to consider:

• "\(a\)" - the radius of the fixed circle.
• "\(b\)" - the radius of the rolling circle.

The path of the point is also influenced by "\(c\)", impacting the distance from the center of the rolling circle to the tracing point. The larger the values of these radii, the larger and more spread out the curve will appear.

• The ratio of \(b\) to \(a\) affects the number of loops the curve will have.
• An understanding of how each radius influences the overall shape is crucial for accurate sketching.

Path of a Point

The journey of a specific point on the rolling circle defines the epitrochoid's path. As the circle with radius \(b\) rolls along the outside of the circle of radius \(a\), a point (located a distance \(c\) from the center of the rolling circle) traces out a path known as the epitrochoid.Think of this point as a passenger on a ferris wheel that is rolling along a circular track. The path taken by this passenger provides the unique pattern recognizable as an epitrochoid. As the rolling circle progresses, different loops and shapes are formed based on the interplay between \(a\), \(b\), and \(c\).

• This path is characterized by the functional outcomes of the parametric equations.
• Each part of the curve signifies a phase of the rolling motion.

Through this understanding, one grasps how each parameter helps define the overall path of this fascinating curve.