Problem 18 - Thomas Calculus

Question

Show that the ellipse \(x=a \cos t, y=b \sin t, a > b > 0,\) has its largest curvature on its major axis and its smallest curvature on its minor axis. (As in Exercise \(17,\) the same is true for any ellipse.)

Step-by-Step Solution

Answer

Largest curvature on major axis, smallest on minor.

Step 1: Parametric Equations of Ellipse

The ellipse is given by the parametric equations: \( x = a \cos t \) and \( y = b \sin t \), where \( a > b > 0 \). These equations describe the elliptical path in terms of the parameter \( t \).

Step 2: Derivative of Parametric Equations

Find the derivatives of \( x \) and \( y \) with respect to \( t \): \[ \frac{dx}{dt} = -a \sin t \] \[ \frac{dy}{dt} = b \cos t \]

Step 3: Second Derivatives

Find the second derivatives of \( x \) and \( y \):\[ \frac{d^2x}{dt^2} = -a \cos t \]\[ \frac{d^2y}{dt^2} = -b \sin t \]

Step 4: Curvature Formula

The curvature \( \kappa \) of a parametric curve is given by:\[ \kappa = \frac{|\frac{dx}{dt} \cdot \frac{d^2y}{dt^2} - \frac{dy}{dt} \cdot \frac{d^2x}{dt^2}|}{\left(\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2\right)^{3/2}} \]

Step 5: Substitute Derivatives Into Curvature Formula

Substitute the first and second derivatives into the curvature formula:\[\kappa = \frac{| (-a \sin t)(-b \sin t) - (b \cos t)(-a \cos t) |}{\left(( -a \sin t)^2 + (b \cos t)^2\right)^{3/2}}\] Simplify the expression:\[\kappa = \frac{| ab \sin^2 t + ab \cos^2 t |}{(a^2 \sin^2 t + b^2 \cos^2 t)^{3/2}}\] Using \( \sin^2 t + \cos^2 t = 1 \), simplifies further to:\[\kappa = \frac{ab}{(a^2 \sin^2 t + b^2 \cos^2 t)^{3/2}}\]

Step 6: Analyze Curvature Behavior

The expression for \( \kappa \) is dependent on \( t \). Analyze where it reaches its maximum and minimum values.- For maximum curvature, evaluate at endpoints of major axis where \( t = 0 \) or \( t = \pi \) (\( x=a \) or \( x=-a \)): \[ \kappa_{max} = \frac{ab}{b^3} = \frac{a}{b^2} \]- For minimum curvature, evaluate at endpoints of minor axis where \( t = \frac{\pi}{2} \) or \( t = \frac{3\pi}{2} \) (\( y=b \) or \( y=-b \)): \[ \kappa_{min} = \frac{ab}{a^3} = \frac{b}{a^2} \]

Step 7: Conclusion

Since \(a > b > 0\), it follows that \( \frac{a}{b^2} > \frac{b}{a^2} \). Therefore, the largest curvature occurs along the major axis, and the smallest curvature occurs along the minor axis.

Key Concepts

Parametric Equations

Parametric equations are a way to describe curves in a plane using parameters. In the case of an ellipse, we use two equations: \( x = a \cos t \) and \( y = b \sin t \). Here, \( t \) is the parameter, and it varies from \( 0 \) to \( 2\pi \). This way of expressing an ellipse helps us plot it easily by just changing the values of \( t \).
Each pair of \( x \) and \( y \) coordinates corresponds to a specific point on the ellipse.
The parameters \( a \) and \( b \) are constants that define the shape and size of the ellipse. The term "parametric" simply means we're using one or more parameters to describe our curve.

Derivatives of Parametric Curves

When dealing with parametric equations, taking derivatives is essential to understanding the behavior of the curve. To find these derivatives, we differentiate each parametric equation with respect to the parameter \( t \).
For an ellipse expressed as \( x = a \cos t \) and \( y = b \sin t \), the derivatives are:

• \( \frac{dx}{dt} = -a \sin t \)
• \( \frac{dy}{dt} = b \cos t \)

Taking these derivatives helps in computing the curvature and analyzing how the slope of the tangent vector changes along the path of the ellipse.

Major and Minor Axes

The axes of an ellipse are crucial in determining its geometry. The major axis is the longest diameter that runs through the center, and the minor axis is the shortest. The lengths of these axes are defined by the constants \( a \) and \( b \), where \( a \) is greater than \( b \). This means the major axis stretches along the direction where \( x \) values range from \(-a\) to \(a\), and the minor axis stretches from \(-b\) to \(b\) along \( y \) values.
Understanding the orientation of these axes is vital for analyzing the curvature behavior, as the ellipse's curvature will differ along these axes.

Curvature Formula

The curvature of a parametric curve tells us how sharply it is bending at any given point. For parametric equations like the ones describing an ellipse, we use the formula:\[\kappa = \frac{|\frac{dx}{dt} \cdot \frac{d^2y}{dt^2} - \frac{dy}{dt} \cdot \frac{d^2x}{dt^2}|}{\left(\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2\right)^{3/2}}\]This formula takes into account both the first and second derivatives of the parametric equations.
By plugging in the derivatives, you can compute how the curvature changes as you move around the ellipse.

Analyzing Curvature Behavior

Understanding curvature behavior helps identify where the curvature is strongest or weakest on a curve. When analyzing an ellipse, we find that the curvature is largest at the endpoints of the major axis and smallest at the endpoints of the minor axis. This is due to the nature of elliptical shapes.
For maximum and minimum curvature, evaluate the curvature formula at specific values of \( t \):

• Maximum curvature: \( t = 0 \) or \( t = \pi \) results in \( \kappa_{max} = \frac{a}{b^2} \)
• Minimum curvature: \( t = \frac{\pi}{2} \) or \( t = \frac{3\pi}{2} \) results in \( \kappa_{min} = \frac{b}{a^2} \)

Given that \( a > b > 0 \), it is clear \( \frac{a}{b^2} > \frac{b}{a^2} \). This analysis proves that the largest curvature appears on the major axis and the smallest on the minor axis.