1. Alright, I have a conjecture that;

The shortest distance from some point to the surface of a sphere is the distance from the point to the center of the sphere minus the radius.

I have a feeling it's already been proven true, but I was just curious what other peoples thoughts on this are.

2.

3. if you look at your statement again you will see that problem is basically.

"show that the shortest distance between 2 points is a straight line"

4. Originally Posted by Arcane_Mathamatition
Alright, I have a conjecture that;

The shortest distance from some point to the surface of a sphere is the distance from the point to the center of the sphere minus the radius.
This is only true for points outside or on the surface of the sphere.

5. no, it's true for ALL points in . I'm not quite sure how to prove it, but I'd love a counter example if you could provide one.

6. Originally Posted by Arcane_Mathamatition
no, it's true for ALL points in . I'm not quite sure how to prove it, but I'd love a counter example if you could provide one.
Sphere centered at (0,0,0), radius 1.

Point (0,0,.5).

Distance to surface: .5

Distance to center: .5

Distance to center minus rarius: -.5

-.5 is not equal .5

How was that for a counterexample?

7. shouldn't distance always be positive? Wouldn't you need an absolute value there? Or, conversely, since the point is WITHIN the sphere, I could say it's distance is negative, as you would assume the distance to be towards the center of the sphere as opposed to away from it. so you would need to move a distance of .5 negative to the center.

8. Originally Posted by Arcane_Mathamatition
shouldn't distance always be positive? Wouldn't you need an absolute value there?
This is precisely what was wrong about your conjecture. A distance should always be non-negative (positive or zero). For points inside the sphere, the difference in your conjecture is not. Put in an absolute value and your conjecture will be right.

9. but isn't a distance DEFINED as an absolute value? so by claiming I want a distance, don't I already imply absolute value?

10. Originally Posted by Arcane_Mathamatition
but isn't a distance DEFINED as an absolute value? so by claiming I want a distance, don't I already imply absolute value?
You gave a recipe which was in fact a formula except it was written with words rather than symbols: "this" minus "that", period. This left no room for taking into account what you may have implied elsewhere. Mathematics is an exact science.

Besides, there are other ways of turning a possibly negative value into a guaranteed nonnegative one: taking its square or any other even power, or exp(x) (i.e. inverse natural logarithm), and lots more. Unless you specify which of those you want, your hints at "wanting a distance" (and therefore implicitly a nonnegative value) don't make a completely defined algorithm.

11. Originally Posted by Arcane_Mathamatition
Alright, I have a conjecture that;

The shortest distance from some point to the surface of a sphere is the distance from the point to the center of the sphere minus the radius.

I have a feeling it's already been proven true, but I was just curious what other peoples thoughts on this are.
Your conjecture is equivalent to the shortest distance to the sphere is along a line that is normal to the sphere. This is true.

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