formula for a sphere?

Question

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?

Answer 1

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.

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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