Problem 9 - Essential Calculus Early Transcendentals

Question

(a) Find and identify the traces of the quadric surface \(x^{2}+y^{2}-z^{2}=1\) and explain why the graph looks like the graph of the hyperboloid of one sheet in Table \(1 .\) (b) If we change the equation in part (a) to \(x^{2}-y^{2}+z^{2}=1,\) how is the graph affected? (c) What if we change the equation in part (a) to \(x^{2}+y^{2}+2 y-z^{2}=0 ?\)

Step-by-Step Solution

Answer

(a) It forms a hyperboloid of one sheet. (b) It's a hyperboloid of two sheets. (c) The surface remains a hyperboloid of one sheet, shifted in y.

Step 1: Determine the xy-plane trace for \(x^{2}+y^{2}-z^{2}=1\)

When \(z = 0\), the equation becomes \(x^{2} + y^{2} = 1\). This describes a circle with radius 1 in the xy-plane.

Step 2: Determine the yz-plane trace for \(x^{2}+y^{2}-z^{2}=1\)

When \(x = 0\), the equation becomes \(y^{2} - z^{2} = 1\). This is a hyperbola in the yz-plane.

Step 3: Determine the xz-plane trace for \(x^{2}+y^{2}-z^{2}=1\)

When \(y = 0\), the equation becomes \(x^{2} - z^{2} = 1\). This is also a hyperbola in the xz-plane.

Step 4: Explain why it looks like a hyperboloid of one sheet

The traces in the xy-plane form circles whereas the traces in the yz-plane and xz-plane form hyperbolas, suggesting a continuous surface. These characteristics are indicative of a hyperboloid of one sheet.

Step 5: Analyze \(x^{2}-y^{2}+z^{2}=1\)

Now consider \(x^2 - y^2 + z^2 = 1\). For \(z = 0\), the equation becomes \(x^2 - y^2 = 1\), a hyperbola. For \(x = 0\), it becomes \(-y^2 + z^2 = 1\), another hyperbola. For \(y = 0\), it becomes \(x^2 + z^2 = 1\), a circle. This graph is a hyperboloid of two sheets, defined by opposite characteristics.

Step 6: Analyze \(x^{2}+y^{2}+2y-z^{2}=0\)

Complete the square for the y terms to rewrite the equation as \(x^2 + (y+1)^2 - z^2 = 1\). When \(z = 0\), the trace is \(x^2 + (y+1)^2 = 1\), a circle. For \(x = 0\) or \(y = 0\), the trace indicates a hyperboloid's central slice, suggesting the surface remains a hyperboloid of one sheet but shifted along the y-axis by 1 unit.

Key Concepts

Hyperboloid of One Sheet

A hyperboloid of one sheet is a fascinating type of quadric surface that's shaped kind of like an hourglass or a two-sided bowl. It's defined by the equation \[ x^{2} + y^{2} - z^{2} = 1 \] which shows its symmetrical properties with respect to the axes. This surface can appear in many physical settings, such as cooling towers of power plants.
The key aspect of this surface is that as you slice it with various planes, you get different shapes:

• Horizontal cuts, or traces in the xy-plane, reveal circles.
• Vertical cuts through the y or x axes, in the yz and xz planes, show hyperbolas.

As all sides of the hyperboloid curve outward away from the z-axis, the shape becomes more evident—a single, continuous surface curving around the z-axis.

Traces in Coordinate Planes

To fully understand the structure of a hyperboloid, examining its traces in the coordinate planes is crucial. These traces give you cross-sectional views of the surface.

• In the xy-plane: When you set \( z = 0 \), the equation \( x^{2} + y^{2} = 1 \) emerges, describing a circle.
• In the yz-plane: With \( x = 0 \), you get \( y^{2} - z^{2} = 1 \), a hyperbola oriented with its axis along the z-direction.
• In the xz-plane: Setting \( y = 0 \), leads to \( x^{2} - z^{2} = 1 \), once again a hyperbola.

These traces reinforce the shape's complexity but predictable nature. Recognizing the distinct trace patterns helps in visualizing how the hyperboloid stretches and compresses in three-dimensional space.

Completing the Square

Completing the square is a vital algebraic technique useful when manipulating equations to uncover more insights about surfaces like a hyperboloid.
Consider transforming the equation \( x^{2}+y^{2}+2y-z^{2}=0 \). By completing the square on the \( y \) terms, you can rewrite it as: \[ x^{2} + (y+1)^{2} - z^{2} = 1 \] This form reveals that the hyperboloid is centered around a point shifted from the origin in the y-direction by 1 unit.

• Completing the square clarifies the loci, or center, of rotational symmetry.
• This step simplifies the equation, making it easier to spot characteristics of the quadric surface.

Using this technique, it's easier to identify changes in shape or direction because it more clearly outlines the geometric properties of the surface.