Exploring the Two-Sheeted Hyperboloid: Is 1 + e^(-x^2+y^2) a Hyperboloid?

  • Thread starter imsoconfused
  • Start date
In summary, the conversation revolves around the question of whether the equation 1 + e^(-x^2+y^2) is a two-sheeted hyperboloid. The conclusion is that it is not, as it does not fit the definition of a hyperboloid and has an exponential term. The conversation also discusses how to visualize the equation in cylindrical coordinates and how to plot points to better understand its shape. The equation is ultimately described as a flat sheet with a hump at the origin.
  • #1
imsoconfused
50
0
Is 1 + e^(-x^2+y^2) a two-sheeted hyperboloid?

thanks
 
Physics news on Phys.org
  • #2
It's just an expression so far. Do you mean z=1+e^(-x^2+y^2), that's a surface.
 
  • #3
sorry, yes. and I made a typo--it's z= 1+ e^(-x^2-y^2).
 
  • #4
It's not a hyperboloid. A hyperboloid is a quadratic form. That's not, it has an e^ in it. And it doesn't have two sheets. In cylindrical coordinates it's z=1+e^(-r^2). Can you picture what that looks like?
 
  • #5
I don't understand what you mean by cylindrical coordinates, I've never heard that term. I have trouble picturing things, too, until I start plotting points. Is that what I should do, just begin plotting points until I start to see it?
 
  • #6
Better to think about it a bit before you start plotting points. In z=1+e^(-r^2) I'm taking r=sqrt(x^2+y^2). So r is just the distance from (0,0) to (x,y) in the x-y plane. Can you picture it now?
 
  • #7
I can't quite see it yet. why wouldn't you just let r=-x^2-y^2?
 
  • #8
Your choice. I usually like to pick r>=0. But look, at (x,y)=(0,0), z=2. As the distance of (x,y) from (0,0) gets larger and larger, -x^2-y^2 gets larger and larger in a negative way. So e^(-x^2-y^2) get closer to 0. So z->1 at infinity.
 
  • #9
ok, I see how you get z=2, at least. I feel so stupid in that I only get about half of the second part and I absolutely cannot see what this thing looks like! I'm thinking of z as a level--is that wrong? looking at this from the xy plane, does it look like a function like (1/x)^2?

I have nothing in my notes about any surfaces that have e in them, nor are there any examples like this in the text--that's why I'm so lost.
 
  • #10
'z' is the height above (or below) the x-y plane for a given value of x and y. Maybe plotting some points isn't that bad an idea. Try some.
 
  • #11
and now comes the inane question about how to plot some points. just pick random x's and y's?
 
  • #12
oh wait, I just need to choose a z and use logarithms to find y in terms of x. correct?
 
  • #13
You are asking before you are thinking about it. How about (x,y)=(1,0),(2,0),(3,0) etc. Then (x,y)=(0,1),(0,2),(0,3)... Or (x,y)=(1,1),(2,2),(3,3)... You don't actually have to plot them all, just think about what would happen if you did.
 
  • #14
imsoconfused said:
oh wait, I just need to choose a z and use logarithms to find y in terms of x. correct?

No, you actually have to think about what the surface would look like if you did plot a bunch of points. This isn't that hard.
 
  • #15
when x and y grow larger, z approaches 1. that is why I thought it looked like (1/x)^2, because that decreases exponentially towards an asymptote. the difference is that this, instead of being a line is a surface. correct?
 
  • #16
Dick said:
No, you actually have to think about what the surface would look like if you did plot a bunch of points. This isn't that hard.

oh, I know it shouldn't be that hard, but I really appreciate your coaching me through this. just think where I'd be without you!
 
  • #17
What it really looks like is 1+e^(-x^2). But, yes, certainly, it decreases exponentially towards an asymptote. It doesn't have two sheets and it's not a hyperboloid.
 
  • #18
imsoconfused said:
oh, I know it shouldn't be that hard, but I really appreciate your coaching me through this. just think where I'd be without you!

I tremble to think. :)
 
  • #19
ok. so if I were to draw this thing, it would basically look like a flat plane with a hump in it near the origin? that's what brought on this question; I'm supposed to be drawing it.
 
  • #20
Yes, an almost flat sheet at infinity with a hump at the origin.
 

1. What is a two-sheeted hyperboloid?

A two-sheeted hyperboloid is a three-dimensional surface that can be described by the equation 1 + e^(-x^2+y^2). It is a special type of quadric surface that has two separate sheets that are connected at a center point. It is characterized by its hyperbolic shape, with the two sheets curving away from each other in opposite directions.

2. How is a two-sheeted hyperboloid different from a one-sheeted hyperboloid?

A one-sheeted hyperboloid is also a quadric surface, but it has only one sheet that curves in one direction. This is in contrast to the two-sheeted hyperboloid, which has two sheets curving in opposite directions. Additionally, the equation for a one-sheeted hyperboloid is of the form (x^2/a^2) + (y^2/b^2) - (z^2/c^2) = 1, while the equation for a two-sheeted hyperboloid is 1 + e^(-x^2+y^2).

3. What are some real-world applications of a two-sheeted hyperboloid?

Two-sheeted hyperboloids have several practical uses, such as in architecture and engineering. They are often used as structural components in buildings and bridges, as their curved shape provides strength and stability. They are also used in the design of cooling towers and other industrial structures, as well as in the manufacturing of some types of lenses and mirrors.

4. How is a two-sheeted hyperboloid studied and explored?

Two-sheeted hyperboloids are studied and explored using mathematical methods, such as calculus and linear algebra. They can also be visualized and manipulated using computer software, such as 3D modeling programs. Additionally, physical models of two-sheeted hyperboloids can be created using various materials, allowing for hands-on exploration and experimentation.

5. Are there any other interesting properties of a two-sheeted hyperboloid?

Yes, there are many interesting properties of a two-sheeted hyperboloid. For example, it is a doubly ruled surface, meaning that it can be created by moving a straight line along two different sets of parallel lines. It also has a constant negative Gaussian curvature, which gives it its unique shape. Additionally, it has an infinite number of symmetries, making it a fascinating object of study in mathematics.

Similar threads

Find the volume bounded by hyperboloid and plane z = ± d
    • Calculus and Beyond Homework Help
Replies
14
Views
630
Quadratic surfaces standard form help
    • Calculus and Beyond Homework Help
Replies
6
Views
2K
How to find the shortest distance to a hyperboloid, etc?
    • Calculus and Beyond Homework Help
Replies
7
Views
5K
I Shape of de Sitter Universe: Is Hyperboloid Misleading?
    • Special and General Relativity
Replies
2
Views
667
Lorentz product on hyperboloid
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
MHB How is z=2xy a Hyperbolic Paraboloid in the rotated 45° in the xy-plane?
    • Calculus
Replies
7
Views
2K
Solve ##\int\frac{e^{-x}}{x^2}dx## and ##\int \frac{e^{-x}}{x}dx##
    • Calculus and Beyond Homework Help
Replies
7
Views
690
Determine whether ## S[y] ## has a maximum or a minimum
    • Calculus and Beyond Homework Help
Replies
18
Views
1K
Find the constrained maxima and minima of ##f(x,y,z)=x+y^2+2z##
    • Calculus and Beyond Homework Help
Replies
2
Views
493
How should I use the Jacobi equation to determine the nature of this?
    • Calculus and Beyond Homework Help
Replies
5
Views
611

Hot Threads

Recent Insights

Back
Top