Sphere Signed Distance Function

1. Background Knowledge

A signed distance function (SDF) is a scalar function that, given a point in space $p \in \mathbb{R}^3$, returns the (signed) shortest distance to a surface. For a solid object, the sign convention is usually: negative inside the object, zero on the surface, and positive outside. The surface itself is the zero level set of the SDF: all points $p$ such that $SDF(p) = 0$. This makes SDFs powerful implicit shape representations in graphics, 3D reconstruction, and robotics.

For a sphere, the geometry is particularly simple: a sphere is the set of points at a fixed distance $r$ (the radius) from a center $c$. The Euclidean distance $\|p - c\|$ already tells you how far a point $p$ is from the center. To get distance to the surface, you subtract the radius: if $\|p - c\| > r$, you are outside; if it’s equal to $r$, you are on the surface; if it’s less than $r$, you are inside. Making this signed naturally leads to the formula given in the problem.

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2. Algorithm/Approach

The general pattern for SDF queries is:

• Compute the unsigned distance from the query point $p$ to the shape’s defining primitive (for a sphere: distance to center).
• Adjust this distance so that it becomes zero on the surface, and its sign matches the “inside vs outside” convention.
• Return this scalar value as the SDF value.

For a sphere, this boils down to:

• Compute $d = \|p - c\|$.
• Return $d - r$ as the SDF value.

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3. Step-by-Step Strategy

Assume 3D vectors with components $(x, y, z)$:

1. Input:

• Query point $p = (p_x, p_y, p_z)$
• Sphere center $c = (c_x, c_y, c_z)$
• Sphere radius $r$