Problem 80 - Thomas Calculus

Question

Find parametric equations and a parameter interval for the motion of a particle that starts at \((a, 0)\) and traces the ellipse \(\left(x^{2} / a^{2}\right)+\left(y^{2} / b^{2}\right)=1\). $$ \begin{array}{ll}{\text { a. once clockwise. }} & {\text { b. once counterclockwise. }} \\ {\text { c. twice clockwise. }} & {\text { d. twice counterclockwise. }}\end{array} $$ (As in Exercise \(79,\) there are many correct answers.)

Step-by-Step Solution

Answer

For clockwise motion: use \(x = a \cos(\theta), y = -b \sin(\theta)\). Counterclockwise: use \(x = a \cos(\theta), y = b \sin(\theta)\). Adjust \(\theta\) range for one or two rotations.

Step 1: Understand the Ellipse Equation

The equation of an ellipse centered at the origin is given by \( \left(\frac{x^2}{a^2}\right) + \left(\frac{y^2}{b^2}\right) = 1 \). This equation can be represented parametrically using angles.

Step 2: Basic Parametric Equations

To parameterize the ellipse, use the parametric equations \( x = a \cos(\theta) \) and \( y = b \sin(\theta) \). This will describe the ellipse when \( \theta \) ranges from \( 0 \) to \( 2\pi \) for a full counterclockwise traversal.

Step 3: Adjust for Clockwise Orientation

For a clockwise traversal, reverse the direction by using \( x = a \cos(-\theta) \) and \( y = b \sin(-\theta) \). Simplifying this, it becomes \( x = a \cos(\theta) \) and \( y = -b \sin(\theta) \).

Step 4: Once Clockwise Motion (a)

Set \( \theta \) from \( 0 \) to \( 2\pi \) in \( x = a \cos(\theta) \) and \( y = -b \sin(\theta) \) for the particle to travel once clockwise.

Step 5: Once Counterclockwise Motion (b)

Set \( \theta \) from \( 0 \) to \( 2\pi \) for \( x = a \cos(\theta) \) and \( y = b \sin(\theta) \) for the particle to travel once counterclockwise.

Step 6: Twice Clockwise Motion (c)

Extend \( \theta \) from \( 0 \) to \( 4\pi \) in the clockwise parametric equations: \( x = a \cos(\theta) \), \( y = -b \sin(\theta) \).

Step 7: Twice Counterclockwise Motion (d)

Extend \( \theta \) from \( 0 \) to \( 4\pi \) in the counterclockwise equations: \( x = a \cos(\theta) \), \( y = b \sin(\theta) \).

Key Concepts

Ellipse Equation

The ellipse equation is a mathematical representation of an ellipse, describing its shape and orientation. When centered at the origin, the equation is expressed as \[ \left(\frac{x^2}{a^2}\right) + \left(\frac{y^2}{b^2}\right) = 1 \] where

• \(a\) is the semi-major axis length,
• \(b\) is the semi-minor axis length.

This equation indicates that the total stretch along the x-axis is \(2a\) and along the y-axis is \(2b\). The ellipse differs from a circle, which has equal radii, as the axes lengths of an ellipse can be different. Understanding the basics of the ellipse equation is essential to work with parametric equations involving ellipses.

Clockwise Traversal

Clockwise traversal refers to moving around an ellipse in the direction that mimics the motion of a clock's hands. Parametrically, this requires adjusting the angle factor in the equations. Typically, the parametric equations \( x = a \cos(\theta) \) and \( y = b \sin(\theta) \) are adjusted for clockwise motion by reversing the direction of traversal. To achieve this, we modify the y-component by flipping its sign:

• Use \( x = a \cos(\theta) \),
• Set \( y = -b \sin(\theta) \).

This change reflects a mirror image of the counterclockwise plot, causing the particle to start from the positive x-axis and move through the lower half of the ellipse before completing the loop in the upper segment. Controlling the parameter \(\theta\) from 0 to \(2\pi\) ensures the particle completes one full clockwise traversal.

Counterclockwise Traversal

Counterclockwise traversal is the standard direction for moving around an ellipse, moving opposite to a clock's hand movement. This is the natural way to parameterize an ellipse using parametric equations. The standard parametric equations are\( x = a \cos(\theta) \) and \( y = b \sin(\theta) \), where:

• \(\theta\) typically ranges from 0 to \(2\pi\) for a full traversal.
• The motion starts at \((a, 0)\) and goes through the upper half of the ellipse first.

Counterclockwise traversal guarantees the particle completes the ellipse in a full, smooth path that encompasses the top and bottom halves of the ellipse in one traverse.

Parametrization

Parametrization is a technique that represents geometric figures like ellipses using parameters, commonly angles like \(\theta\). It provides a systematic way to describe every point on the ellipse through equations expressed in terms of these parameters. For an ellipse, the standard parametric form is:

• \( x = a \cos(\theta) \)
• \( y = b \sin(\theta) \)

where \(\theta\) is the parameter. This method simplifies the description of complex motions such as ellipses by condensing them into manageable equations. Parametrization enables visualizing the motion of a particle as it traverses the ellipse. The choice of \(\theta\)'s range can tailor the traversal for desired directions and repetitions; for instance, extending to \(4\pi\) achieves two complete traversals, either clockwise or counterclockwise. This flexibility makes parametrization a powerful tool in geometry and physics.