Question

A surface \(\mathcal{S}\) in space is described that cannot be defined in terms of a function \(z=f(x, y)\). Give a parametrization of \(\mathcal{S}\). \(\mathcal{S}\) is the ellipsoid \(\frac{x^{2}}{9}+\frac{y^{2}}{4}+\frac{z^{2}}{16}=1\).

Short answer

The parametrization of the ellipsoid is \(x(\theta, \phi) = 3\cos(\theta)\sin(\phi),\ y(\theta, \phi) = 2\sin(\theta)\sin(\phi),\ z(\theta, \phi) = 4\cos(\phi)\).

Step-by-step solution

Understand the Ellipsoid Equation

The given ellipsoid is described by the equation \(\frac{x^2}{9} + \frac{y^2}{4} + \frac{z^2}{16} = 1\). This equation represents an ellipsoid centered at the origin. The semi-axes lengths are \(a = 3\), \(b = 2\), and \(c = 4\) for the \(x\), \(y\), and \(z\) dimensions, respectively.

Choose a Parametrization Method

For an ellipsoid, a common parametrization involves using spherical coordinates to describe any point on the surface in terms of two parameters, \(\theta\) and \(\phi\). These angles can be related to traditional spherical coordinates.

Define the Parametrization Equations

Using parameters \(\theta\) and \(\phi\), where \(\theta\) is the azimuthal angle and \(\phi\) is the polar angle, the parametrization equations for the ellipsoid are:\[x(\theta, \phi) = 3 \cos(\theta) \sin(\phi)\]\[y(\theta, \phi) = 2 \sin(\theta) \sin(\phi)\]\[z(\theta, \phi) = 4 \cos(\phi)\]where \(0 \leq \theta < 2\pi\) and \(0 \leq \phi \leq \pi\). This ensures traversing the entire surface of the ellipsoid.

Verify the Parametrization

To verify this parametrization, substitute these expressions for \(x\), \(y\), and \(z\) back into the ellipsoid equation:\[\frac{(3 \cos(\theta) \sin(\phi))^2}{9} + \frac{(2 \sin(\theta) \sin(\phi))^2}{4} + \frac{(4 \cos(\phi))^2}{16} = 1\]Simplifying each term shows that the original equation holds, confirming that the parametrization correctly describes the surface.

Key concepts

Ellipsoid

An ellipsoid is a three-dimensional shape that generalizes the form of an ellipse. Much like how an ellipse is a flattened circle, an ellipsoid can be considered as a stretched sphere. More formally, an ellipsoid is defined by an equation of the form:\[\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\]where \(a\), \(b\), and \(c\) are the semi-axis lengths in the \(x\), \(y\), and \(z\) directions, respectively. These values determine the extent of the ellipsoid along each axis. For instance, if all three are equal (i.e., \(a = b = c\)), we have a perfect sphere.In this specific problem, the ellipsoid is centered at the origin with semi-axes of length 3, 2, and 4. It’s important to note that these values influence the shape’s orientation and size. You can think of it as a balloon inflated variably in different directions.

Spherical Coordinates

Spherical coordinates offer a convenient way to describe positions on a sphere or spherical-like surface, such as an ellipsoid. Unlike the traditional Cartesian coordinates (\(x, y, z\)), spherical coordinates use angles and distance:- \(\rho\): The radial distance from the origin- \(\theta\): The azimuthal angle, which rotates around the vertical \(z\)-axis- \(\phi\): The polar angle, measured from the positive \(z\)-axisHowever, when parametrizing an ellipsoid, the distance \(\rho\) is fixed due to the surface's equation, and we focus primarily on the angles \(\theta\) and \(\phi\). By setting these angles, we can traverse any point on the surface:- \(x(\theta, \phi) = a \cos(\theta) \sin(\phi)\)- \(y(\theta, \phi) = b \sin(\theta) \sin(\phi)\)- \(z(\theta, \phi) = c \cos(\phi)\)Here, \(a\), \(b\), and \(c\) represent the semi-axes of the ellipsoid. This conversion lends itself well to complex surface geometry while simplifying calculations.

Surface Equation

The surface equation defines the shape and extent of a surface in mathematical terms. For ellipsoids, the equation connects the coordinate variables \(x, y, z\) through their respective semi-axes:\[\frac{x^2}{9} + \frac{y^2}{4} + \frac{z^2}{16} = 1\]This equation ensures that for any point \((x, y, z)\) on the surface, the relationship outlined is satisfied. The values in the denominators, 9, 4, and 16, are the squares of the semi-axis lengths (3, 2, and 4). This structure maintains the ellipsoid's dimensions.When using a parametrization such as through spherical coordinates, we substitute the parameter relations back into this equation to verify it holds true. If our parametrization is correct, the original surface equation should simplify back to a true statement, thus confirming we have accurately described the surface of the ellipsoid.