Surfaces

• Feb 1st 2007, 11:39 PM
Ideasman
Surfaces
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?
• Feb 2nd 2007, 12:20 AM
earboth
Quote:

Originally Posted by Ideasman
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: $x = 4y^2$

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

EB
• Feb 2nd 2007, 05:34 AM
Soroban
Hello, Ideasman!

I use the standard orientation of the three coordinate axes.
Code:

                  z                   |                   |                   |                   |                   * - - - - - - y                 /               /             /           x
Quote:

Consider the following surface: . $x \:= \:4y^2 + 4z^2$

What's this surface called?

It is a paraboloid.

Quote:

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

Quote:

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

Let $y = 0$ and we have: . $x = 4z^2$

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

Quote:

What's the trace of this surface in the planes $x=k$ look like?

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

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