Problem 28 - APEX Calculus

Question

Sketch the quadric surface. \(z^{2}=x^{2}+\frac{y^{2}}{4}\)

Step-by-Step Solution

Answer

The quadric surface is a hyperboloid of two sheets, symmetric around the z-axis, forming ellipses in z-parallel planes.

Step 1: Recognize the Form

The given equation \(z^2 = x^2 + \frac{y^2}{4}\) is a standard form of a hyperboloid of two sheets. In three dimensions, it's important to first recognize the quadratic form it represents by comparing with known quadric forms.

Step 2: Identify the Axis of Symmetry

Check which variable is distinguished in the equation. Here, \(z^2\) is on one side of the equation alone, indicating that the symmetry for this hyperboloid is along the z-axis.

Step 3: Determine Cross Sections

To better understand the shape, find the cross sections by setting one variable at a constant value. For instance, setting \(z=c\) will give us the ellipse equation \(c^2 = x^2 + \frac{y^2}{4}\), showing ellipses in planes parallel to the xy-plane.

Step 4: Analyze Special Cases

Set \(x=0\) and \(y=0\) to understand additional details. Setting \(x=0\), results in \(z^2 = \frac{y^2}{4}\) (a pair of lines that separate as \(|z|\) increases). Setting \(y=0\), we get \(z^2 = x^2\) (another set of lines in a different orientation). These indicate the shape's opening.

Step 5: Interpret the Shape

Combine all insights: Each constant \(z\) cross-section is an ellipse. No real values for \(z\) produce a circle, suggesting two disconnected 'sheets'; hence the name 'hyperboloid of two sheets'. Visualization involves plotting several such ellipses, showing openings for large \(|z|\).

Key Concepts

Hyperboloid of Two Sheets

Imagine a hyperboloid of two sheets as two separate surfaces facing away from each other, mirroring each other's curves. This geometric structure arises from the equation \( z^2 = x^2 + \frac{y^2}{4} \). These surfaces do not touch or overlap.
They are well separated, creating two 'sheets' hence the name. Unlike its cousin, the hyperboloid of one sheet, which forms a single continuous surface, this form creates two distinct parts.
This distinction is important as it defines many properties of the surface like intersection with planes and symmetry.

Axis of Symmetry

In three-dimensional geometry, symmetry simplifies understanding. To find the axis of symmetry in the equation \( z^2 = x^2 + \frac{y^2}{4} \), notice how the term \( z^2 \) is distinct.
The symmetry axis is where this term stands alone on one side of the equation. Here, it is clearly the z-axis. The z-axis acts like the spine of a book, around which the other parts are evenly balanced.
In practical terms, this symmetry means every point is mirrored along this axis, which is pivotal for visualizing the shape and understanding its properties.

Cross Sections Analysis

Cross sections help us visualize and understand complex 3D surfaces. For the hyperboloid of two sheets, cross sections are interpreted by fixing one variable.
Let's fix \( z = c \); substituting into our equation results in \( c^2 = x^2 + \frac{y^2}{4} \), which is an ellipse. This shows that all cross sections parallel to the xy-plane form elliptical shapes.
By changing the value of \( z \), you slide these ellipses up or down along the z-axis, revealing the shape's curvature and breadth as it spans across the two sheets.
Additionally, setting \( x = 0 \) or \( y = 0 \) provides further insights, yielding hyperbolic structures that illustrate how sheets separate or come closer.

Ellipse in 3D

Ellipses in 3D give us a better perspective of how shapes are structured in space. With our equation \( z^2 = x^2 + \frac{y^2}{4} \), each slice \( z = c \) results in an elliptical path.
These slices are critical because they show how the surface's width changes, constraining our understanding of the volume it encompasses. The ellipse reveals scale and proportions just as a circle does, but in a stretched form.
Visualizing ellipses helps in drafting the two branching sheets that define the hyperboloid's spatial structure. Their alignment determines the shape's spread and overall geometry.