Question

In Exercises \(11-18,\) identify and sketch the quadric surface. Use a computer algebra system to confirm your sketch. $$ \frac{x^{2}}{16}+\frac{y^{2}}{25}+\frac{z^{2}}{25}=1 $$

Short answer

The quadric surface represented by the given equation is an ellipsoid with semi-axes lengths 4, 5 and 5 along the x, y, and z axis respectively. Verify this using a computer algebra system.

Step-by-step solution

Identify the quadric surface

Examine the given equation. It is in the standard form of an ellipsoid, since all the squares of the variables are added together and equal to 1. And all the denominators (16, 25, 25) are positive constants. Therefore, the quadric surface is an ellipsoid.

Find the semi-axes

The lengths of the semi-axes of the ellipsoid are given by the square roots of the denominators. Let's compute those: \(a=\sqrt{16}=4\), \(b=\sqrt{25}=5\), and \(c=\sqrt{25}=5\). Thus, we have the points (±4, 0, 0), (0, ±5, 0), and (0, 0, ±5) that provide the lengths of the semi-axes on the x, y and z axis respectively.

Sketch the ellipsoid

We may sketch the ellipsoid now. The points determined in Step 2 will form the basis of our sketch: Draw an x, y, and z axis. Then mark the points (±4, 0, 0) on the x-axis, (0, ±5, 0) on the y-axis and (0, 0, ±5) on the z-axis. Now, imagine a vertically elongated sphere that 'touches' the x-axis at ±4, the y-axis at ±5 and the z-axis at ±5. That is our ellipsoid.

Confirm with a computer algebra system

You may use software like Desmos, GeoGebra or Wolfram Alpha to draw the given equation and see a 3D model of an ellipsoid. This will confirm the accuracy of your sketch.

Key concepts

Ellipsoid

An ellipsoid is a 3-dimensional analogue of an ellipse and is a type of quadric surface. It has a shape similar to a stretched or squashed sphere. Each cross-section of an ellipsoid parallel to a coordinate plane is an ellipse or a circle. An ellipsoid is defined by three perpendicular axes that intersect at the center of symmetry: the major axis, the minor axis, and the intermediate axis.

These axes give rise to the lengths known as semi-axes, which determine how stretched the sphere is in the x, y, and z directions. Ellipsoids are often visualized in fields like astronomy, geodesy, and physics to model celestial bodies, the Earth, or equipotential surfaces in fields.

Equation of an Ellipsoid

The general equation of an ellipsoid centered at the origin in a 3-dimensional Cartesian coordinate system is given by:
\[\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\]
where 'a', 'b', and 'c' are the lengths of the semi-axes along the x, y, and z axes, respectively. The equation represents the set of all points (x, y, z) that satisfy the ellipsoid shape. When 'a', 'b', and 'c' are all equal, the ellipsoid becomes a sphere. The values under the squared terms indicate how far the surface stretches in each direction from the center, influencing the ellipsoid's proportions.

Sketching Quadric Surfaces

Sketching quadric surfaces such as ellipsoids involves understanding how the equation of the surface dictates its shape in 3D space. Once the equation is recognized as a standard form, the key steps involve identifying the semi-axes lengths and using these to sketch the surface’s profile in 2D. The semi-axes dictate the extent to which the surface stretches along the coordinate axes.

When manually sketching an ellipsoid, it's helpful to first draw the coordinate axes and then mark the lengths of the semi-axes. Next, you draw ellipses in the three principal planes (xy, xz, and yz planes) using the semi-axis lengths. Finally, you complete the 3D shape by smoothly connecting these ellipses to form the ellipsoid. This helps to visualize the curvature along all axes and provides a good approximation of the true shape.

Computer Algebra Systems

Computer algebra systems (CAS) are software tools that mathematicians, scientists, engineers, and students use to solve mathematical expressions symbolically. They are capable of manipulating mathematical equations and expressions in ways that allow for algebraic rearrangements, providing exact answers rather than numerical approximations.

CAS can plot complex geometric objects, including quadric surfaces like ellipsoids, hyperboloids, and paraboloids. By inputting the equation of an ellipsoid into a CAS, you can get a precise 3D representation that can be manipulated and viewed from different angles. This visual aid is invaluable for confirming manual sketches and developing intuition about these surfaces. Well-known computer algebra systems include Wolfram Mathematica, Maple, and MATLAB, as well as free online tools like Desmos and GeoGebra that are easily accessible to students.