Problem 51

Question

Sketch the quadric surface. $z^{2}-y^{2}=9$

Step-by-Step Solution

Verified

Answer

A hyperbolic cylinder with axis along the x-axis.

1Step 1: Identify the Type and Characteristics of the Surface

The given equation is $z^2 - y^2 = 9$. This is a hyperbolic cylinder because it is a quadratic equation in two variables $z$ and $y$, but no $x$ variable is present. The standard form of a hyperbola is $z^2/a^2 - y^2/b^2 = 1$, which matches this equation when written as $(z^2/9) - (y^2/9) = 1$. Therefore, $a^2 = 9$ so $a = 3$, and $b^2 = 9$ so $b = 3$.

2Step 2: Understand the Structure and Direction

In a hyperbolic cylinder, one variable is absent (in this case $x$), indicating that the hyperbola extends infinitely along the $x$-axis. The hyperbola opens along the coordinate axis of the present variables square root, which means our hyperbola will open along the $z$ and $y$ axes.

3Step 3: Plot the Surface on the xy, yz, and xz Planes

To sketch the surface, consider cross-sections parallel to each coordinate plane:1. **yz-plane**: Let $x = 0$. The equation becomes $z^2 - y^2 = 9$, which is the standard hyperbola in the $yz$-plane.2. **xz-plane**: Let $y = 0$. The equation becomes $z^2 = 9$, giving $z = \pm 3$, where the graph is horizontal lines at z = 3 and z = -3 extending along the x-axis.3. **xy-plane**: Let $z = 0$. The equation becomes $-y^2 = 9$, which has no real solutions for y, indicating no intersection with the $xy$-plane.

4Step 4: Describe the 3D Shape and Symmetry

The hyperbolic cylinder $z^2 - y^2 = 9$ has symmetry along the $z$-axis and is extended infinitely along the $x$-axis. The cross-sections in the $yz$-plane are hyperbolas, and in the $xz$-plane, they are two parallel lines $z = 3$ and $z = -3$. There is no real solution for $y$ in the $xy$-plane hence, it contributes no shape.

Key Concepts

Hyperbolic Cylinder3D GeometryCross-sections

Hyperbolic Cylinder

A hyperbolic cylinder is a type of quadric surface that can be identified by its distinct mathematical form. The equation is given as:

An expression involving squared variables, such as $ z^2 - y^2 = 9 $ in this case.
Notice there is no dependence on the $ x $ variable. This lack of the $ x $ variable means the surface extends indefinitely along the $ x $-axis.

To reshape the equation, compare it to the standard hyperbola formula:$ \frac{z^2}{9} - \frac{y^2}{9} = 1 $. This reveals a hyperbolic cylinder, as the equation forms a hyperbola in the $ zy $ plane.The graph opens along the $ z $ and $ y $ axes because the quadratic terms are $ z^2 $ and $ y^2 $. It mirrors and repeats itself endlessly along the missing variable's axis, which is $ x $ here.

3D Geometry

3D geometry helps us to visualize and analyze the structures that cannot be fully represented within the confines of a two-dimensional plane. In the case of a hyperbolic cylinder described by the equation $ z^2 - y^2 = 9 $, we explore the 3D structure and its positioning within the coordinate system.

The absence of the $ x $ variable allows this hyperbolic cylinder to extend infinitely back and forth along the $ x $-axis.
The solution is symmetric about the $ z $ axis, as reflected in the positive and negative values of $ z = \pm3 $, sanctioned by symmetry.

By evaluating at every point $ x $, we discover its consistent, repetitive behavior - a signature of its infinite extensionality within space. The visualization offers a better understanding of its continuous nature.

Cross-sections

To comprehend the geometric structure of the hyperbolic cylinder thoroughly, we often analyze its cross-sections. Cross-sections are the slices of a 3D object, captured by considering single planes.

The **yz-plane** cross-section confirms the hyperbolic nature, for $ x = 0 $ leads to $ z^2 - y^2 = 9 $, a classic hyperbola.
A cross-section in the **xz-plane** demonstrates horizontal slices at $ z = 3 $ and $ z = -3 $. These parallel lines extend along the $ x $-axis, further illustrating the infinite stretch inherent in the cylinder.
When examining the **xy-plane**, there's no viable intersection, as indicated by the $ -y^2 = 9 $ which lacks real solutions.

Breaking down these sections aids students in visualizing how certain aspects such as symmetry and trajectory govern the overall shape, providing more clarity and insight into 3D structures.

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