Problem 24 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the general equation of a central hyperboloid of one sheet that is symmetric with respect to the following: (a) origin (b) \(y\)-axis (c) \(x y\)-plane

Step-by-Step Solution

Answer

The general equation is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1 \).

Step 1: Identify the General Equation for a Hyperboloid of One Sheet

The general equation for a hyperboloid of one sheet oriented along the coordinate axes is: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1 \). It represents a central surface with symmetry properties depending on the variable axes.

Step 2: Applying Symmetry with Respect to the Origin

To be symmetric with respect to the origin, the equation must remain unchanged if all signs of the variables are swapped. For a hyperboloid of one sheet, the standard form already satisfies this symmetry.

Step 3: Ensure Symmetry with Respect to the y-axis

The hyperboloid equation should remain unchanged when \(x\) is replaced with \(-x\) and \(z\) with \(-z\), ensuring symmetry about the \(y\)-axis. The existing form already satisfies these conditions because no cross-terms (e.g., \(xy\), \(yz\), or \(zx\)) are present in a properly oriented equation along the coordinate axes.

Step 4: Check Symmetry with Respect to the xy-plane

Replacing \(z\) with \(-z\) in the equation should yield the original equation for symmetry across the \(xy\)-plane. Since the current form of the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1 \) satisfies this condition, it is symmetric with respect to the \(xy\)-plane.

Key Concepts

Central Symmetry

Central symmetry is a fascinating concept in mathematics, especially when studying geometric shapes like hyperboloids. It refers to a situation where a shape looks the same when seen from two opposite directions of a central point. For a hyperboloid of one sheet, central symmetry means if you were to flip every coordinate sign, the equation \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1 \]remains unchanged.
This is because the structure is perfectly symmetrical around an imaginary origin point.

Understanding central symmetry helps in identifying why the hyperboloid's properties seem evenly distributed in all directions from that central point. It's like looking into a mirror. What you see left and right should mirror each other perfectly.

Coordinate Axes

In three-dimensional geometry, the coordinate axes are crucial. They form the backbone of any spatial calculation or description by providing directions: the x, y, and z-axis.
Since the hyperboloid of one sheet equation\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1\]utilizes these axes, one can think of them as the guiding lines that define the shape in space.

Here's a simple way to understand their role:

• x-axis: Represents horizontal direction.
• y-axis: Represents the vertical direction.
• z-axis: Depth or the back-and-forth direction.

By aligning the equation with these axes, we ensure the symmetry and balance of the hyperboloid in space.

Symmetry Properties

Symmetry properties provide essential insights into the geometric nature of objects like hyperboloids. For instance, when an equation like\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1\]remains unchanged despite transforming or rotating it, it shows inherent symmetry.

Different types of symmetry include:

• Origin Symmetry: The presence of even powers of x, y, and z ensures the form is symmetric around the center.
• Axis Symmetry: Ensuring transformations like replacing x with -x or z with -z retain the unchanged equation; this reflects symmetry around the y-axis.
• Plane Symmetry: Replacing z with -z in the equation validates symmetry along the xy-plane.

These symmetries are crucial in explaining why a hyperboloid of one sheet appears uniform from different perspectives.

Three-Dimensional Geometry

Three-dimensional geometry is like the art of visualizing and interpreting the spatial world using mathematics. It's the study of shapes that have depth, breadth, and height. The hyperboloid of one sheet is a classic example of the beauty in three-dimensional geometry.

The equation\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1\]describes a shape that stretches across all three dimensions, resembling twisted circular tubes connecting across the axes.

• Height: Denoted by the vertical component.
• Width: Represented by the horizontal reach.
• Depth: Suggested by how far back it seems to go.

This multidimensionality not only looks elegant but embodies critical symmetry about which students must learn, influencing fields ranging from architecture to physics.