Surface normals

The unit vector perpendicular to a surface at a point. For an implicit surface $F = 0$ it is $\nabla F/\norm{\nabla F}$; for a parametric one $r(u,v)$, the normalized cross product $r_u \times r_v$. Every lighting computation starts from it.

Light reflects depending on the angle between the incoming ray and the surface. That angle is measured against the normal — and the normal is a derivative: the gradient points across level sets, so it is perpendicular to the surface $F = 0$.

Sphere $F = x^2 + y^2 + z^2 - R^2$: $\nabla F = 2(x, y, z)$, so the normal at a point is the point itself divided by $R$ — radial, as expected. A bump map perturbs $h$ with a texture and recomputes $n$ from its partial derivatives: flat geometry that looks rough.

Wrong normals are the most visible rendering bug; normal maps are how games show detail without millions of triangles.