Surfaces

May 2006
369
2
Consider the following surface:

x = 4y^2 + 4z^2

What's this surface called?

What coordinate plane does z=0 define?

What's the trace of this surface in the xz plane look like?

What's the trace of this surface in the planes x=k look like?
 
Jan 2006
5,854
2,553
Germany
Consider the following surface:

x = 4y^2 + 4z^2

What's this surface called?
What coordinate plane does z=0 define?
What's the trace of this surface in the xz plane look like?
What's the trace of this surface in the planes x=k look like?
Hello,

I've attached a diagram of this surface.

a) It is a rotation paraboloid. The axis of rotation is the positive x-axis.

b) It's a parabola with the vertex in the origin: \(\displaystyle x = 4y^2\)

c) It's a circle: \(\displaystyle k = 4x^2+4y^2 \Longleftrightarrow x^2+y^2=\left( \frac{1}{2} \sqrt{k} \right)^2\)

EB
 

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May 2006
12,028
6,341
Lexington, MA (USA)
Hello, Ideasman!

I use the standard orientation of the three coordinate axes.
Code:
                  z
                  |
                  |
                  |
                  |
                  * - - - - - - y
                /
              /
            /
          x
Consider the following surface: .\(\displaystyle x \:= \:4y^2 + 4z^2\)

What's this surface called?
It is a paraboloid.


What coordinate plane does \(\displaystyle z=0\) define?
\(\displaystyle z = 0\) is the \(\displaystyle xy\)-plane (the "floor" of the graph).


What's the trace of this surface in the \(\displaystyle xz\)-plane look like?
The \(\displaystyle xz\)-plane is the "left wall" of the graph.

Let \(\displaystyle y = 0\) and we have: .\(\displaystyle x = 4z^2\)

This is a parabola on the "left wall", vertex at the origin,
. . opening in the positive \(\displaystyle x\)-direction.



What's the trace of this surface in the planes \(\displaystyle x=k\) look like?

Let \(\displaystyle x = k\). .We have: .\(\displaystyle 4y^2 + 4z^2\:=\:k\quad\Rightarrow\quad y^2 + z^2\:=\:\frac{k}{4}\)

These are circles: centered on the \(\displaystyle x\)-axis with radius \(\displaystyle \frac{\sqrt{k}}{2}\)

 
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