# How do I find the nearest point on a sphere?

Say I have a sphere of radius $6$ centered at $(3, 4, 5)$.

What's the nearest point on the surface of the sphere to point $(1, 2, 3)$, which is within the sphere?

I feel that this is a minimization problem involving calculus since we can minimize the distance. Is there a way to do this problem without calculus? I want to program it, and using calculus to solve a minimization problem may not be efficient.

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## 2 Answers

Find the unit vector from (3,4,5) to (1,2,3), multiply it by 6, that is the point you are looking for.

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Find a line equation passing through the points $(3,4,5)$ and $(1,2,3)$. This line intersect the sphere at two points.(we already know the equation of sphere $(x-3)^2+(y-4)^2+(z-5)^2=6^2$) Chose the point which is closest to $(1,2,3)$ using distance formula.

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Perhaps simpler: To get from $(3,4,5)$ to $(1,2,3)$, you can add $(-2,-2,-2)$, which takes you a distance $2\sqrt3$ from the center. To find the point you want, go 6 units from the center instead, in the same direction, by adding $\frac{6}{2\sqrt3}(-2,-2,-2)$ to $(3,4,5)$. This saves you time that can be spent later arguing with your teacher, if she objects to your having avoided calculus. –  Steve Kass Mar 8 '13 at 2:35