Question

An epitrochoid is the path of a point on a circle of radius \(b\) as it rolls on the outside of a circle of radius \(a\). It is described by the equations $$\begin{array}{l}x=(a+b) \cos t-c \cos \left(\frac{(a+b) t}{b}\right) \\\y=(a+b) \sin t-c \sin \left(\frac{(a+b) t}{b}\right)\end{array}$$ Use a graphing utility to explore the dependence of the curve on the parameters \(a, b,\) and \(c.\)

Short answer

Based on the exploration of the epitrochoid curve using a graphing utility and varying the parameters \(a, b,\) and \(c\), describe how these parameters affect the shape and features of the curve. Answer: The parameters \(a, b,\) and \(c\) affect the shape and features of the epitrochoid curve in different ways. Parameter \(a\) influences the overall size of the epitrochoid, making it expand outward as the value of \(a\) increases. Parameter \(b\) alters the number of "loops" in the curve, with more loops appearing as the value of \(b\) increases. Parameter \(c\) changes the size and elongation of these loops, making them larger and more elongated as the value of \(c\) increases.

Step-by-step solution

Understand the parametric equations

We are given two parametric equations for the epitrochoid curve. These equations give the x and y coordinates of a point on the curve based on parameter t. $$\begin{array}{l}x=(a+b) \cos t-c \cos \left(\frac{(a+b) t}{b}\right) \\\y=(a+b) \sin t-c \sin \left(\frac{(a+b) t}{b}\right)\end{array}$$ These equations depend on 3 parameters: \(a, b\), and \(c\).

Use a graphing utility

To explore the dependence of the curve on the parameters, we will use a graphing utility that supports parametric equations, such as Desmos, GeoGebra, or a graphing calculator. First, we need to set the values for the parameters \(a, b\), and \(c\). Let's start with some arbitrary values, for example \(a = 5, b = 3\), and \(c = 2\). Now, plot the parametric equations with these initial values on the graphing utility, and observe the shape of the curve. Remember that the parameter t should vary between \(0\) and \(2\pi\) to cover the entire curve.

Vary parameter \(a\)

Now we will see how the curve changes when we vary the parameter \(a\). Keep the values for parameters \(b\) and \(c\) fixed (\(b = 3\) and \(c = 2\)), and change the value of \(a\) to different values, such as 3, 4, 6, 8, and 10. Observe the changes in the curve as you increase the value of \(a\). We can notice that increasing \(a\) makes the epitrochoid expand outward.

Vary parameter \(b\)

Now we will see how the curve changes when we vary the parameter \(b\). Keep the values for parameters \(a\) and \(c\) fixed (for example, \(a = 5\) and \(c = 2\)), and change the value of \(b\) to different values, such as 1, 2, 3, 4, and 5. Observe the changes in the curve as you increase the value of \(b\). We can notice that increasing \(b\) changes the number of "loops" of the epitrochoid.

Vary parameter \(c\)

Now we will see how the curve changes when we vary the parameter \(c\). Keep the values for parameters \(a\) and \(b\) fixed (for example, \(a = 5\) and \(b = 3\)), and change the value of \(c\) to different values, such as 1, 2, 3, 4, and 5. Observe the changes in the curve as you increase the value of \(c\). We can notice that increasing \(c\) makes the "loops" of the epitrochoid larger and more elongated.

Conclusion

In this exercise, we have used a graphing utility to explore the dependence of an epitrochoid curve on the parameters \(a, b,\) and \(c\). We have observed that: 1. Parameter \(a\) affects the overall size of the epitrochoid. 2. Parameter \(b\) alters the number of "loops" in the curve. 3. Parameter \(c\) changes the size and elongation of these loops.

Key concepts

Parametric Equations

Imagine drawing a curve as if you were using a mathematical pen connected to a rolling disk. Parametric equations make this possible by expressing the coordinates of a point in terms of a parameter, often denoted as \(t\). Instead of directly relating \(x\) and \(y\), parametric equations provide a pair of functions that define each axis separately using \(t\).For the epitrochoid, our parametric equations look like this:\[x = (a+b) \cos t - c \cos \left(\frac{(a+b) t}{b}\right), \quad y = (a+b) \sin t - c \sin \left(\frac{(a+b) t}{b}\right)\]These equations allow us to plot the curve in the \(xy\)-plane, with each value of \(t\) producing unique \(x\) and \(y\) coordinates. This way, as \(t\) varies from \(0\) to \(2\pi\), it traces out the entire curve. When exploring the possible shapes, you'll notice the influence of three parameters: \(a\), \(b\), and \(c\). Each one alters the curve’s properties in distinct ways, like adjusting the overall size, number of loops, and their elongation. Parametric equations cater to dynamic, intricate curves that aren't easily captured by simpler \(y = f(x)\) equations.

Graphing Utility

A graphing utility is essential for visualizing complex mathematical curves like the epitrochoid. Tools such as Desmos or GeoGebra are great for this purpose as they let you input parametric equations directly to produce stunning graph visualizations. Here’s a simple approach to use a graphing utility:

• Input the parametric equations into the utility.
• Set the range for \(t\). Commonly, \(t\) ranges from \(0\) to \(2\pi\).
• Experiment with different values for \(a\), \(b\), and \(c\) to see how the curve adapts.

Using a graphing utility helps turn abstract equations into tangible shapes, and it provides immediate feedback on how parameter changes influence the curve. They are powerful educational tools that make learning about complex curves interactive and engaging, letting you visualize mathematical relationships quickly.

Mathematical Curves

Mathematical curves, such as the epitrochoid, offer fascinating insights into geometry and how figures can be represented analytically. These curves are not just academic exercises but have real-world applications. For instance, similar patterns can be found in engineering and nature.The epitrochoid itself is the path traced by a point on the boundary of a disk rolling outside another disk. Its mathematical elegance comes from how changing its parameters \(a\), \(b\), and \(c\) can generate a family of curves, each with unique characteristics.

• Parameter \(a\): This can be thought of as the radius of the smaller disk plus the larger disk. Increasing \(a\) stretches the curve, making it larger overall.
• Parameter \(b\): The base frequency component. Changing \(b\) alters how many loops the epitrochoid contains.
• Parameter \(c\): Represents the distance from the center of the smaller circle to the tracing point, which elongates the loops when altered.

Understanding these mathematical curves helps demystify the interplay between geometric shapes and their algebraic representations. These curves also form the groundwork for studying more complex systems in physics and engineering, making their exploration both rewarding and relevant.