How to find the shortest distance to a hyperboloid, etc?

• RJLiberator
In summary, the conversation discussed finding the shortest distance between the hyperboloid of two sheets and a point P(0,1,0). The distance function was used with the point P and the partial derivatives were taken to find the critical point at (0, 1/2, sqrt(5)/2). This is the only critical point and the coordinates of all points on the surface for which this distance is attained. However, there is also an opposite point at (0, 1/2, -sqrt(5)/2) due to the symmetry of the hyperboloid.
RJLiberator
Gold Member

Homework Statement

Consider the hyperboloid of two sheets: z^2=x^2+y^2+1
and a point P(0, 1, 0). Find the shortest distance between the hyperboloid and the point P. Also, find coordinates of all points on the surface for which this distance is attained.

The Attempt at a Solution

So, I first used the distance function with the point P.
(x-0)^2+(y-1)^2+(z-0)^2
and this simplified (with the use of the equation z^2=x^2+y^2+1 to:
2x^2+2y^2-2y+3=f(x,y)
I then took the partial derivatives with respects to x and y
fx = 4x
fy = 4y-2

These equal 0 when x = 0 and y = 1/2. This is the ONLY critical point.
I then found the z value with original function so (0, 1/2, sqrt(5)/2) is the point.

I then found the distance to (0,1,0) through subtraction. (0, 1/2, -sqrt(5)/2) = Shortest distance from hyperboloid to point P.

Now I'm pretty sure that this is correct, but how do I go about finding the coordinates of all points on the surface for which the distance is attained? I'm struggling to start here. Any tips on this would be great.

RJLiberator said:
This is the ONLY critical point.
I think you are done.

RJLiberator
Interesting. I had this hunch as well. It didn't make sense that there would be more coordinates if that was the only critical point.
In a sense, they were simply testing conceptual understanding with that add-on then.

Let me just clarify my understanding: Since there is only one critical point on the hyperboloid that minimizes the distance in such a way that it becomes the shortest possible distance from the hyperboloid and the point P, then that is all possible points on the surface for which the distance is attained.

Excellent.

Thank you.

RJLiberator said:
These equal 0 when x = 0 and y = 1/2. This is the ONLY critical point.
I then found the z value with original function so (0, 1/2, sqrt(5)/2) is the point.
Is that the only z value?

RJLiberator
Oi, perhaps +/- sqrt(5)/2 Eh? Yes, that exists because z^2. But then, because that exists that is a second point similar distance to point P.

RJLiberator said:
Oi, perhaps +/- sqrt(5)/2 Eh? Yes, that exists because z^2. But then, because that exists that is a second point similar distance to point P.
Yes. (Picturing it helps. You see the symmetry about the XY plane.)

RJLiberator
Yes, I was curious about this as I made a poster for hyperboloids with two sheets and noticed that there should be an opposite point that is symmetrical. Thank you.

Oops, good point, I missed the other sheet as well.

1. What is a hyperboloid?

A hyperboloid is a three-dimensional geometric shape that resembles two infinite cones connected at their bases. It can be thought of as a three-dimensional analog of the hyperbola in a two-dimensional plane.

2. How do you find the shortest distance to a hyperboloid?

The shortest distance to a hyperboloid is known as the perpendicular distance. It can be calculated by finding the distance between a point and the closest point on the surface of the hyperboloid. This can be done by using the formula for the distance between a point and a plane.

3. What is the equation for a hyperboloid?

The equation for a hyperboloid can be written as x^2/a^2 + y^2/b^2 - z^2/c^2 = 1, where a, b, and c are constants that determine the size and shape of the hyperboloid. This is known as the standard form of a hyperboloid.

4. Are there different types of hyperboloids?

Yes, there are two types of hyperboloids - elliptic and hyperbolic. Elliptic hyperboloids have a positive coefficient for the z^2 term in the standard form equation, while hyperbolic hyperboloids have a negative coefficient for the z^2 term.

5. What are some real-world applications of hyperboloids?

Hyperboloids have various applications in engineering and architecture. They are commonly used in the design of cooling towers, as the shape allows for efficient air circulation. Hyperboloids can also be found in the construction of bridges and other structures that require strong, lightweight materials.

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