A STEM Project: Slice Forms of Common Quadric Surfaces Part #2

Slice forms of a Hyperboloid of One Sheet, a Cone and a Hyperboloid of Two Sheets

Only one Cone and Hyperboloid of Two Sheets are shown above. To view the actual structure, 

use the app 3D Calculator- GeoGebra. 

View of the slice forms lying flat

I made slice forms of six different quadric surfaces: Hyperbolic Paraboloid aka Monkey Saddle, Ellipsoid, Hyperboloids of One Sheet, Cone, Hyperboloids of Two Sheets and Elliptic Paraboloid.  This post is the second of three posts about quadric surface slice forms.  It will include the slice forms of Hyperboloid of One Sheet, Cone and a Hyperboloid of Two Sheets. 

By using the app 3D Calculator- GeoGebra, all three of the quadric surfaces can be created by changing just one variable, k, by using this formula.

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=\mathrm{k}$$

When k > 0, a Hyperboloid of One Sheet is created.  Here is the graph at k = 1.

When k = 0, a Cone is created. Here is the graph at k = 0.

When k < 0, a Hyperboloid of Two Sheets is created.  Here is the graph at k = -1.

The slice forms that I created represents this transformation from the hyperboloid of one sheet to the cone and then to the hyperboloid of two sheets. It is interesting to note that as the figure transforms from the hyperboloid of one sheet, the base gets smaller. 

Here is the PDF.  I used a good quality 65 lb. cardstock.  The paper needs to be stiff and should not curl.  Otherwise the slices will curl. The slice forms are assembled (except the Hyperboloid of One Sheet) by sliding the two center slices together.  Next, all of the upward facing slices are slid onto the center slices.  Lastly, all of the downward facing slices are slid onto the center slices.

https://drive.google.com/file/d/1d9iOHShcFA7fc6_N4p04tJOVLKatQ74d/view?usp=sharing

Here is the .Studio file.

https://drive.google.com/file/d/1GV1mVp-qFk7eBftURZveOFqSGmjgFtcN/view?usp=sharing

Here is the SVG. The SVG file goes beyond the viewable area.  To see the entire file, zoom out.

https://drive.google.com/file/d/1RgSPHKX8Vaf-YqfVq-OTKAWZS1hkZQWO/view?usp=sharing

Hyperboloid of One Sheet

Slice form of a Hyperboloid of One Sheet

Side View

Equation of a Hyperboloid of One Sheet

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$$

I graphed the quadric surface with the Desmos online graphing calculator, https://www.desmos.com/calculator.  The image created represents the traces needed to form each slice. The slices were then divided by a constant interval of 0.5 inches to create the slits.

This graph represents the top half of the Hyperbola.  

To create the entire Hyperbola, I copied and flipped the image using the Silhouette software.

To assemble, the two center slices are slide into one another and then next set of four slices are slid together.  

Repeat for the remaining slices.

Cone

Slice form of a Cone

Side View

Equation of a Cone.

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=0$$

Graph of the Cone

(I only graphed one figure because the other is identical)

Hyperboloid of Two Sheets

Slice form of a Hyperboloid of Two Sheets 

Side View

Equation of a Hyperboloid of Two Sheets.

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=-1$$

Graph of the Hyperboloid of Two Sheets

(I only graphed one figure because the other is identical)