Signed distance function of a sphere:
The signed distance function of a sphere with radius $r$ is defined as:
$$\Phi(\mathbf{x}) = \ \mathbf{x}  \mathbf{x}_{\text{sphere}} \  r.$$
In the example $\mathbf x$ is the red point, $\mathbf{x}_{\text{sphere}}$ the blue point and the closest point to $\mathbf x$ on the surface of the sphere is rendered in yellow.
Surface normal vector
The normal vector is defined by the gradient of the signed distance function:
$$\mathbf n = \frac{\partial \Phi(\mathbf{x})}{\partial \mathbf x} = \frac{\mathbf{x}  \mathbf{x}_{\text{sphere}}}{\ \mathbf{x}  \mathbf{x}_{\text{sphere}}\}. $$
Closest point on the surface
The closest point $\mathbf s$ on the surface of the sphere (yellow) can be determined by starting at the point $\mathbf x$ (red) and going by the signed distance in the direction of the negative normal vector:
$$\mathbf s = \mathbf x  \Phi(\mathbf{x}) \mathbf n.$$
