Problem 30 - APEX Calculus

Question

Sketch the quadric surface. \(16 x^{2}-16 y^{2}-16 z^{2}=1\)

Step-by-Step Solution

Answer

It's a hyperboloid of two sheets centered at the origin.

Step 1: Identify the type of quadric surface

The equation given is \( 16x^2 - 16y^2 - 16z^2 = 1 \). This equation is similar to the standard form of a hyperboloid, specifically a hyperboloid with one sheet or two sheets. To match the standard form, compare this with \( Ax^2 + By^2 + Cz^2 = D \) format.

Step 2: Simplify the equation form

To simplify the equation, divide the entire equation by 16 to match a standard form: \( x^2 - y^2 - z^2 = \frac{1}{16} \). This identifies the type clearly.

Step 3: Recognize the hyperboloid shape

Upon simplifying, it's clear the equation takes the form \( x^2 - y^2 - z^2 = \frac{1}{16} \). This format of \( x^2 - y^2 - z^2 = -1/k^2 \) is consistent with the typical form of a hyperboloid of two sheets.

Step 4: Sketch the quadric surface

To sketch this hyperboloid of two sheets, note that the surface is symmetrical about the x-axis. Cross-sections parallel to the yz-plane are hyperbolas with the transverse axis along the x direction. The sheets are separated and extend on either side along the x-axis.

Key Concepts

Hyperboloid of Two Sheets

A hyperboloid of two sheets is a type of quadric surface. Imagine two separate curved surfaces that extend infinitely and seemingly never connect. These surfaces look like two bowls or dishes facing away from each other.
A hyperboloid of two sheets is described mathematically by a specific family of equations. Such equations will have some variables with positive signs and others with negative signs.

• In the standard form of this equation, the two sheets indicate that two separate symmetrical components exist.
• Typically, one variable will have a different sign, indicating the axis of symmetry.

In our case, the hyperboloid of two sheets involves the equation: \[ 16x^2 - 16y^2 - 16z^2 = 1 \] This becomes more apparent upon simplification.

Equation Simplification

Simplifying an equation helps to easily identify the type of surface it represents. In the given exercise, you start with the equation: \[ 16x^2 - 16y^2 - 16z^2 = 1 \] To simplify, divide every term by 16 to ease its comparison with standard forms: \[ x^2 - y^2 - z^2 = \frac{1}{16} \]

• This simplification makes the form more recognizable and easier to analyze.
• In this simpler form, it becomes apparent that this surface is indeed a hyperboloid of two sheets.

Recognizing how simplification works is crucial because it transforms the problem into something easier to interpret and solve.

Cross-Sections

Cross-sections are slices of the 3D surface made by cutting along a particular plane. They help visualize the shape and structure of quadric surfaces. For a hyperboloid of two sheets:

• When a plane is parallel to the yz-plane, the cross-sections appear as hyperbolas.
• This observation assists us in understanding how the surface behaves along each axis.

The cross-section’s shape and orientation give insights into the geometry of the hyperboloid. In this particular hyperboloid, the yz cross-section is hyperbolic with the transverse axis in the direction of the x-axis.

Symmetry in Surfaces

Symmetry is a critical feature in surfaces, telling much about their geometric properties. In the case of a hyperboloid of two sheets:

• It is symmetric about the axis corresponding to the variable with only one positive coefficient, in this case, the x-axis.
• Each point on one 'sheet' has a mirrored point on the other sheet, but reflected across the axis of symmetry.

Symmetry provides cues about how to visualize and sketch the surface. In our scenario, symmetry helps in sketching, as knowing the symmetry axis gives you a starting place for the graph.