Distance between a point and a surface (sphere)

In summary, the sphere has a point (1/3, 2/3, 2/3) as its nearest point to (1, 2, 2) and a point (-1/3, -2/3, -2/3) as its farthest point from (1, 2, 2).
  • #1
JustAChemist
5
0

Homework Statement



What function represents the distance between the point P(1, 2, 2) and any point on the sphere
x^2 + y^2 + z^2 = 1?


Homework Equations



My solutions manual says that the answer is f = (x - 1)^2 + (y - 2)^2 + (z - 2)^2

The Attempt at a Solution



I tried applying pythagoras' theorem and came up with f = sqrt[ (1 - x)^2 + (2 - y)^2 + (2 - z)^2 ] :/

It's actually part of an optimisation problem (what are the shortest and longest distances from P and the sphere?), but I'm pretty sure I've got it sorted once I can get the function... can anyone help?... please? ): (without vectors if possible :/)

thankyou (:
 
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  • #2
Who cares if you do the work differently than the solution manual? There are lots of ways to do problems, and it's usually best to do things in a way you understand rather than trying to mimic exactly something you don't understand.

And sometimes solutions manuals are flat-out wrong.

(But after the fact, it is often worth studying something you don't understand to learn why it works and how it compares to what you do understand)


Anyways, your function looks like it really is the distance formula. I imagine the method of the solution manual and the method you intend to use are similar, with the only difference is that they simplified the problem early on, so that their later work will be easier.
 
Last edited:
  • #3
Thankyou (:

> The solutions manual, however, used an algebraic trick that is commonly useful.

I apologise for asking, but would you mind sharing the trick with me? (:
 
  • #4
Well, you do recognize the similarities between the two functions you wrote? I bet part of the work in solutions manual is to find the optima of that function?
 
  • #5
yeah it was. I got those correctly using the function I 'borrowed' from the solution manual, but I didn't use mine because I assumed it was wrong lol :P

thanks a lot for the help (:

embarassingly, I had been stuck with this for a couple hours lol... :/
 
  • #6
The simplest way to do this is to write out the equation of the line through the given point and the center of the sphere. The points at which that line crosses the sphere will be the points on the sphere nearest to and farthest from the given point.

The sphere given by [itex]x^2 +y^2+ z^2+ 1[/itex] has center at (0, 0, 0) and the line through (0, 0, 0) and (1, 2, 2) is given by x= t, y= 2t, z= 2t.

The points where that line crosses the sphere must satisfy [itex](t)^2+ (2t)^2+ (2t)^2= 1[/itex] or [itex] 9t^2= 1[/itex] so that [itex]t= 1/3[/itex] or [itex]t= -1/3[/itex].

Since (1, 2, 2) has t= 1, the point with t= 1/3, (1/3, 2/3, 2/3), is the point on the sphere nearest to (1, 2, 2) and the point with t= -1/3, (-1/3, -2/3, -2/3) is the point on the sphere farthest from (1, 2, 2).
 

1. What is the formula for finding the distance between a point and a surface (sphere)?

The formula for finding the distance between a point and a surface (sphere) is d = |r - r0| - r, where d is the distance, r is the radius of the sphere, and r0 is the distance from the center of the sphere to the point.

2. How is the distance between a point and a surface (sphere) calculated in three-dimensional space?

In three-dimensional space, the distance between a point and a surface (sphere) is calculated using the Pythagorean theorem, where the distance is the hypotenuse of a right triangle with the radius of the sphere and the distance from the center of the sphere to the point as the other two sides.

3. Can the distance between a point and a surface (sphere) be negative?

No, the distance between a point and a surface (sphere) cannot be negative. It is always a positive value, representing the shortest distance between the point and the surface of the sphere.

4. How does the distance between a point and a surface (sphere) change as the point moves closer or farther away from the sphere?

The distance between a point and a surface (sphere) decreases as the point moves closer to the sphere and increases as the point moves farther away. When the point is on the surface of the sphere, the distance is equal to the radius of the sphere.

5. Can the distance between a point and a surface (sphere) be greater than the radius of the sphere?

Yes, the distance between a point and a surface (sphere) can be greater than the radius of the sphere. This occurs when the point is outside of the sphere, and the distance is measured from the center of the sphere to the point.

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