Fun with Math!: Elliptical Cones

A pair of elliptical cones is a type of quadric surfaces. If it is centered at the origin, its equationcould be one of the equations below:

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{z^2}{c^2}$ $\frac{x^2}{a^2}+\frac{z^2}{c^2}=\frac{y^2}{b^2}$ $\frac{y^2}{b^2}+\frac{z^2}{c^2}=\frac{x^2}{a^2}$
The opening of the double cones could be elliptical or circular, depends on the values of a, b and c. The equations above are referring to those centered at the origin. If it is centered at (h, k, l), the equations will look like the equations below.

$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=\frac{(z-l)^2}{c^2}$
$\frac{(x-h)^2}{a^2}+\frac{(z-l)^2}{c^2}=\frac{(y-k)^2}{b^2}$
$\frac{(y-k)^2}{b^2}+\frac{(z-l)^2}{c^2}=\frac{(x-h)^2}{a^2}$
The box below contains all the graphs mentioned above. Tick one category at a time to view the graph. You may slide the values of a, b, c, h, k and l to see the changes on the graph. Observe the following matters:

What is the axis of the graph?
What is the opening of the graph? Elliptical or circular? How do the values of a, b and c affect the opening.
What are the intercepts?

If you wish to have a better view and want to rotate the graph, just click here.

Betty, created with GeoGebra and Thomas Calculus, Pearson.

Fun with Math!

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Thursday, May 14, 2020

Elliptical Cones

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