is there such a thing as a formula for a sphere? Is it $x^2+y^2+z^2=1$? if so, does the $1$ denotes a radius of $1$ for said sphere? what are the possible alterations for such a formula?
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It is not clear what you are asking. – Tim Raczkowski Mar 04 '15 at 18:23
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It might be better to check Wikipedia first. – Mnifldz Mar 04 '15 at 18:25
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Do you know how to find the distance between two points in 3D? (Specifically, between the origin and the point $(x,y,z)$.) Pythagoras tells us that, in 2D, we have $\text{distance}^2=x^2+y^2$. Use this to find and prove the formula for 3D. – Akiva Weinberger Mar 04 '15 at 19:03
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In general $(x-x_0)^2 + (y-y_0)^2 + (z-z_0)^2 = R^2$ represents a sphere centered in $(x_0, y_0, z_0)$ with radius $R$
It is important to understand what this actually means: It means that if you take the set of all the real numbers $x, y, z$ such that the above equation is satisfied, and you plot them, you'll end up with a sphere. Note that a priori there is no easy way to find which points actually satisfy the equation.
You may wonder why we represent a sphere in such a convoluted way, while for example a line is simply $y = mx$ and you plug whatever value of $x$ you choose and you know the value of $y$, without need to make too much calculations. The problem is that a sphere is not a function (much like a circle on the plane) so we need to give this "implicit" representation.
However you can give explicit representation for just part of the sphere. The relevant theorem here is Dini's theorem for implicit function
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