Problem 58 - Calculus Single Variable

Question

Suppose that \(h\) and \(k\) are constants and that \(a\) and \(b\) are positive constants. Parameterize the ellipse with center \((h, k)\) and semi-axes \(a\) and \(b\) parallel to the \(x\) -axis and \(y\) -axis.

Step-by-Step Solution

Answer

The parameterization is \( x = h + a \cos(t), \; y = k + b \sin(t) \) with \( t \in [0, 2\pi] \).

Step 1: Understand the Structure of an Ellipse

An ellipse centered at (h, k) with semi-major axis length a (along the x-axis) and semi-minor axis length b (along the y-axis) has the equation:\[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\]Our goal is to parameterize this ellipse.

Step 2: Use Sine and Cosine for Parameterization

The standard parameterization of an ellipse is based on the trigonometric identities involving circles. Start by letting \( x = h + a \cos(t) \) and \( y = k + b \sin(t) \). This uses the periodicity of sine and cosine to traverse the ellipse as t varies from 0 to 2\pi.

Step 3: Verify the Parameterization

Substitute \( x = h + a \cos(t) \) and \( y = k + b \sin(t) \) into the ellipse equation:\[\frac{((h + a \cos(t)) - h)^2}{a^2} + \frac{((k + b \sin(t)) - k)^2}{b^2} = 1\]This simplifies to:\[\frac{(a \cos(t))^2}{a^2} + \frac{(b \sin(t))^2}{b^2} = \cos^2(t) + \sin^2(t) = 1\]The identity holds, confirming our parameterization is correct.

Key Concepts

Ellipse Equation

An ellipse is a geometric shape that resembles a stretched circle. It can be identified by its equation in the Cartesian coordinate system. The ellipse is centered at a point \(h, k\), and it stretches out along two axes – the major axis and the minor axis.

The major axis is the longest diameter of the ellipse, and it aligns along the x-axis if the equation is structured to prefer the x values. Its length is denoted by \(2a\), with \(a\) being the semi-major axis length. Similarly, the minor axis runs perpendicular to the major axis along the y-axis, with a length of \(2b\) where \(b\) is the semi-minor axis length.

The general equation of an ellipse with center \( (h, k) \) is given by:

• \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]

This equation captures the symmetry and proportions of the ellipse. It implies that for any point \(x, y\) lying on the ellipse, dividing the square of the distance from the center by the corresponding axis squared, then summing these values will result in 1. This property helps in determining the boundary points of the ellipse.

Trigonometric Identities

Trigonometric identities are essential mathematical expressions that relate angles to side lengths in right-angled triangles. They serve a significant role in the parameterization of curves like ellipses through sine and cosine functions.

A crucial identity involved in such derivations is the Pythagorean identity:

• \[ \cos^2(t) + \sin^2(t) = 1 \]

This identity maintains that the sum of the squares of sine and cosine of any angle \(t\) equals one. It provides the mathematical grounding needed to express curves using standard trigonometric functions. By employing this identity, one can assure that a parameterized form maintains its elliptical shape.

When parameterizing an ellipse, the identity ensures that the transformations applied (via sine and cosine) will conform to the elliptical path, as it ensures that \(\cos^2(t) + \sin^2(t)\) will always sum to one, encapsulating the nature of circular/elliptical rotations.

Sine and Cosine Functions

The sine and cosine functions are fundamental trigonometric functions frequently used to describe periodic or oscillating behavior, as seen in waves or circles. They are indispensable in parameterizing an ellipse.

The functions are defined as follows:

• The cosine function, \( \cos(t) \), represents the x-coordinate of a point on the unit circle as it traverses around.
• The sine function, \( \sin(t) \), represents the y-coordinate of that same point.

When applied to an ellipse, these functions help map out points along its edge. By using the following parameterization:

• \( x = h + a \cos(t) \)
• \( y = k + b \sin(t) \)

The pair \(x, y\) at any angle \(t\) can be defined, effectively traversing the ellipse’s boundary.

These parameterized equations make use of the ellipse’s axes' lengths: the semi-major axis length \(a\) and the semi-minor axis length \(b\). As \(t\) varies from 0 to \(2\pi\), the full outline of the ellipse is realized, thanks to the cyclical nature of trigonometric functions.