Problem 10 - 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 two sheets in Table \(1 .\) (b) If the equation in part (a) is changed to \(x^{2}-y^{2}-z^{2}=1,\) what happens to the graph? Sketch the new graph.

Step-by-Step Solution

Answer

The original graph is a hyperboloid of two sheets. Changing the equation forms a hyperboloid of one sheet.

Step 1: Identifying Traces: xy-plane Intersection

Substitute \( z = 0 \) in the equation \(-x^2 - y^2 + z^2 = 1\), which gives \(-x^2 - y^2 = 1\). No real solutions exist for this equation, hence no intersection with the xy-plane.

Step 2: Identifying Traces: xz-plane Intersection

Substitute \( y = 0 \) in the equation \(-x^2 - y^2 + z^2 = 1\), which simplifies to \-x^2 + z^2 = 1\. This equation describes two branches of hyperbolas with axes along the x and z directions.

Step 3: Identifying Traces: yz-plane Intersection

Substitute \( x = 0 \) in the equation \(-x^2 - y^2 + z^2 = 1\), and solve \(-y^2 + z^2 = 1\). This gives a hyperbola with its axes along the y and z directions.

Step 4: Identifying Quadric Surface Shape

The given equation, \(-x^2 - y^2 + z^2 = 1\), means the surface is centered among different planes with hyperbola traces, similar to a hyperboloid of two sheets. Each 'sheet' is symmetrical around the z-axis, as indicated by no real solution on the xy-plane.

Step 5: Modifying Equation and Analyzing New Shape

For \(x^2 - y^2 - z^2 = 1\), switch the signs, resulting in \(x^2 = y^2 + z^2 + 1\). The equation now represents a hyperboloid of one sheet, which is continuous and cylindrical in structure, unlike the two separate sheets in part (a).

Step 6: Sketching New Graph

In contrast to the two distinct sheets of the hyperboloid in part (a), the new equation results in one continuous sheet with vertices extending along the x-axis. This cylindrical shape reflects the hyperboloid of one sheet interpretted from the modified equation.

Key Concepts

Hyperboloid of Two Sheets

A hyperboloid of two sheets is a fascinating quadric surface that features two distinct components or "sheets." These sheets are symmetric around the central axis, typically the z-axis in standard mathematical models.
A key characteristic of the hyperboloid of two sheets is that it does not intersect with some coordinate planes, such as the xy-plane, due to specific algebraic properties.

• The general equation of a hyperboloid of two sheets is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = -1\).
• In this form, the minus sign with the z-term is crucial; it indicates that we are dealing with two sheets that open along parallel planes perpendicularly to the z-axis.
• The trace in a plane parallel to either the xz-plane or yz-plane results in hyperbolas, giving the surface a distinctive shape.

Moreover, the inability to find real solutions in the xy-plane confirms these separated nature of the sheets in space.

Hyperboloid of One Sheet

The hyperboloid of one sheet is another fascinating quadric surface. Unlike its two-sheet counterpart, this surface is continuous and extends infinitely.
This structure looks remarkably similar to a cooling tower of a nuclear power plant, with a smooth, curved surface wrapping around and tapering inward.

• The typical equation for a hyperboloid of one sheet is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1\).
• Here, the positive terms for both x and y axes allow the structure to connect these two directions smoothly, while the z terms form a circular or elliptical cross-section.
• Visualization of this shape along, let’s say, the xz-plane reveals an infinite, single-sheet hyperbola curving around itself in space.

The continuous nature makes it an elegant example of how algebraic expressions manifest into 3D forms, demonstrating diversity among quadric surfaces.

Traces of Surfaces

Traces of surfaces are crucial when identifying or understanding different quadric surfaces. Essentially, they are the curves we get when cutting through a 3D shape with a plane.
These traces help us visualize the 3D structure by reducing it to a more manageable 2D form.

• The trace in the xy-plane is found by setting \(z = 0\).
• The trace in the xz-plane requires setting \(y = 0\).
• The crucial role of traces is evident when manipulating equations, as they reveal essential features like symmetry and surface continuation.

Using traces, we can distinguish between different types of hyperboloids. For example, if we find a hyperbola in the xz and yz planes but no solution in the xy-plane, it points to a hyperboloid of two sheets. Meanwhile, having hyperbolas found in all three planes is indicative of a hyperboloid of one sheet. Traces give us a deeper understanding without needing to visualize the entire surface immediately.