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Chapter 11: Problem 21

Identify and sketch the quadric surface. \(v^{2} \quad q^{2}\) $$ 9 z^{2}-4 y^{2}-9 x^{2}=36 $$

Short Answer

Expert verified
The quadric surface is a hyperboloid of two sheets centered at the origin.

Step by step solution

01

Identify the Equation Type

First, recognize the standard form of a quadric surface. The given equation is \(9z^2 - 4y^2 - 9x^2 = 36\). This can be rewritten in the form \(Az^2 + By^2 + Cx^2 = D\), where \(A = 9\), \(B = -4\), and \(C = -9\). Look for matching patterns with standard quadric surfaces like ellipsoids, hyperboloids, etc.
02

Simplify the Equation

Divide every term in the equation \(9z^2 - 4y^2 - 9x^2 = 36\) by 36 to simplify:\[\frac{9z^2}{36} - \frac{4y^2}{36} - \frac{9x^2}{36} = 1 \], or:\[\frac{z^2}{4} - \frac{y^2}{9} - \frac{x^2}{4} = 1 \].
03

Compare with Standard Form

Compare this simplified equation with the standard equation of a hyperboloid of two sheets: \(\frac{z^2}{a^2} - \frac{y^2}{b^2} - \frac{x^2}{c^2} = 1\). Here, \(a^2 = 4\), \(b^2 = 9\), \(c^2 = 4\).
04

Sketch the Surface

The surface is a hyperboloid of two sheets. It is centered at the origin with symmetry on the z-axis. Each sheet has ellipses as cross-sections parallel to the yz and zx planes. The sheets are separated by regions where \( |z| < 2 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

hyperboloid of two sheets
A hyperboloid of two sheets is a type of quadric surface that looks like two separate curved surfaces, or "sheets," one above and one below a certain plane. In the equation given in the exercise, this surface appears as a distinctive form due to its characteristic properties. The equation \( \frac{z^2}{4} - \frac{y^2}{9} - \frac{x^2}{4} = 1 \) represents this type of surface. Here are some important features of a hyperboloid of two sheets:
  • Two Sheets: As the name suggests, the surface consists of two "sheets" that are symmetrical but do not touch each other.
  • Separation: The sheets are separated by some plane of symmetry, and in this case, the z-axis acts as the axis of symmetry.
  • Center and Orientation: The hyperboloid is centered at the origin, which means its symmetry is reflected across this origin point. Each sheet reflects an ellipse-like shape at different \( z \) values as you move along the z-axis.
Understanding the appearance and features of a hyperboloid of two sheets helps visualize its shape and recognize its equation's form.
equation of quadric surfaces
The equation of quadric surfaces presents a standardized way to describe various three-dimensional shapes using algebraic formulas. Quadrics include shapes such as ellipsoids, paraboloids, and hyperboloids, including the hyperboloid of two sheets discussed in the exercise. Typically, the general form of a quadric surface's equation is written as \( Ax^2 + By^2 + Cz^2 + D = 0 \) or some variant with mixed terms or linear terms. In our specific exercise:
  • The surface is described by the equation \( 9z^2 - 4y^2 - 9x^2 = 36 \).
  • It simplifies to \( \frac{z^2}{4} - \frac{y^2}{9} - \frac{x^2}{4} = 1 \) after dividing all terms by 36.
  • This matches the standard form of a hyperboloid of two sheets: \( \frac{z^2}{a^2} - \frac{y^2}{b^2} - \frac{x^2}{c^2} = 1 \), with specific values for \( a^2 \), \( b^2 \), and \( c^2 \).
Recognizing these forms helps not only in identifying the type of quadric surface but also in sketching or visualizing it.
symmetry in quadric surfaces
Symmetry is a fundamental characteristic in many natural and geometric forms, including quadric surfaces. By understanding the symmetry in a quadric surface, such as the hyperboloid of two sheets, we can more effectively describe its properties and appearance. Symmetry involves understanding the axes or planes about which a surface can be mirrored or rotated and look unchanged. In our case study:
  • The hyperboloid of two sheets has symmetry around the z-axis. This means that if you rotate the surface about the z-axis, it appears unchanged.
  • This symmetry makes the surface predictable in terms of its cross-sections, which can be ellipses along certain planes like yz and zx.
  • Furthermore, the surface is centered at the origin, implying symmetry across the origin point as well, adding another layer of simplicity to its geometric nature.
By examining a surface鈥檚 symmetries, we gain insights into its structure and can predict its behavior under transformations.

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