Problem 53 - Calculus Early Transcendentals

Question

Use a graphing utility to graph the following curves. Be sure to choose an interval for the parameter that generates all features of interest. Evolute of an ellipse \(x=\frac{a^{2}-b^{2}}{a} \cos ^{3} t, y=\frac{a^{2}-b^{2}}{b} \sin ^{3} t\) \(a=4\) and \(b=3.\)

Step-by-Step Solution

Answer

In summary, to graph the evolute of an ellipse given by the parametric equations \(x=\frac{a^{2}-b^{2}}{a} \cos ^{3} t, y=\frac{a^{2}-b^{2}}{b} \sin ^{3} t\) with \(a=4\) and \(b=3\), we first identify any important features and that the parameter \(t\) is the only parameter. We then choose an interval for the parameter \(t\) to be \(0 \leq t \leq 2\pi\), since it is inside trigonometric functions sin and cos. Finally, we use a graphing utility to graph the parametric equations within the chosen interval for \(t\), resulting in a curve that represents the evolute of the ellipse.

Step 1: Identify important features

Before graphing, let's identify any important features that we need to consider while choosing an interval for the parameter \(t\). The parametric equations are: \(x=\frac{a^{2}-b^{2}}{a} \cos ^{3} t, y=\frac{a^{2}-b^{2}}{b} \sin ^{3} t\) with \(a=4\) and \(b=3\). In this case, note that the parameter \(t\) is indeed the only parameter.

Step 2: Choose an interval for the parameter t

Since the parameter \(t\) is inside trigonometric functions sin and cos, it is clear that we need to consider the full period of those functions (from 0 to \(2\pi\)) to capture all of the relevant features in the graph. Hence, we choose the interval for the parameter \(t\) to be \(0 \leq t \leq 2\pi\).

Step 3: Use a graphing utility and graph the parametric equations

Now, we will use a graphing utility to graph the parametric equations for the given ellipse's evolute within the chosen interval for \(t\). Plug in the given values of \(a\) and \(b\), and graph the parametric equations: \(x=\frac{4^{2}-3^{2}}{4} \cos ^{3} t, y=\frac{4^{2}-3^{2}}{3} \sin ^{3} t\) within the interval \(0\leq t \leq 2\pi\). After graphing the parametric equations, you will obtain a distinct shaped curve that represents the evolute of the given ellipse.

Key Concepts

Graphing Ellipse

Graphing an ellipse involves understanding its fundamental shape and characteristics. An ellipse can be thought of as a stretched circle, characterized by two main axes: the major axis (longest diameter) and the minor axis (shortest diameter). The basic standard form of an ellipse centered at the origin is given by the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a\) represents the semi-major axis length and \(b\) represents the semi-minor axis length.
When graphing an ellipse, you need to determine:

• The center, which in the standard form is \((0,0)\).
• The lengths of the axes, where \(a\) is the distance from the center to the edge along the x-axis, and \(b\) along the y-axis.
• The orientation, which depends on whether \(a > b\) (horizontal) or \(b > a\) (vertical).

Using a graphing utility, ensure all these parameters are correctly input to visualize the ellipse fully. In the context of parametric equations, ranges for the parameter \(t\) typically extend from \(0\) to \(2\pi\) to complete the full shape of an ellipse.
Exploring the graph iteratively helps in understanding the changes in real-time, which can be adjusted according to the practical needs to match the theoretical predictions about the ellipse. This results not only in creating precise graphs for studies or illustrations but also aids in a deeper understanding of the curve's dynamics.

Trigonometric Functions

Trigonometric functions like \(\sin\) and \(\cos\) are foundational in creating and analyzing parametric curves like ellipses. They describe periodic behaviors, which are essential in defining the x and y coordinates on a plane over specific intervals. When using these functions:

• The sine function \(\sin(t)\) models vertical oscillations. As \(t\) varies, \(\sin(t)\) moves from -1 to 1, and completes a full cycle every \(2\pi\).
• The cosine function \(\cos(t)\) models horizontal oscillations. Similar to sine, \(\cos(t)\) ranges from -1 to 1 within one complete cycle spanning interval \(0\) to \(2\pi\).

In the given exercise, using \(\cos^3(t)\) and \(\sin^3(t)\) showed how trigonometric functions alter the standard ellipse to describe an evolute's curves. Cubing these values emphasizes the peaks and valleys in the oscillation, which deforms the ellipse into an evolute.
It is vital to consider these peaks when graphing for proper evolution. One full cycle means the full shape is captured, highlighting the relationship between trigonometric functions and parametric forms. These trigonometric processes are crucial for conceptual appreciation of how complex curves are visualized and constructed.

Evolute of Ellipse

The evolute of an ellipse is a geometric locus that connects the centers of curvature of the original ellipse. When visualized, it exhibits a unique cusp-like pattern, reflecting the changing curvature along the ellipse's perimeter. Understanding its structure involves several steps:

• Recognize that it is formed by the curvature points where each portion of the ellipse has maximum twist.
• Parametrically, as given in this exercise, its expression becomes \(x=\frac{a^{2}-b^{2}}{a} \cos^{3} t\) and \(y=\frac{a^{2}-b^{2}}{b} \sin^{3} t\), where \(a\) and \(b\) refer to the semi-major and semi-minor axes, respectively.
• This expression helps in investigating the changes in curvature and how this affects the knotted shape of the evolute for different \(t\).

When \(a=4\) and \(b=3\), the sharp and swooping curves of the evolute are evidenced. Understanding how its unique form arises from a basic ellipse can augment comprehension of more general concepts like envelopes and central paths within geometry. Observing the evolute offers insights not only into advanced mathematical patterns but also their practical implications in architectural design and optics.
In conclusion, the evolute's twisted design is well-analyzed using parametric plots and understanding its construction yields deeper insights into geometric transformations beyond basic shapes.