A Surface

A 3D Surface

is mathematically defined using one of three primary forms, depending on its geometric complexity and how it needs to be analyzed:

• Explicit Form: \(z = f(x, y)\)

This is the most standard form in foundational multivariable calculus. It defines the height \(z\) explicitly as a function of two independent coordinates, \(x\) and \(y\). It is ideal for surfaces that do not fold back on themselves (i.e., they pass the vertical line test in 3D).

• Implicit Form: \(F(x, y, z) = 0\)

This equation groups all variables together without isolating one. It is necessary for defining closed geometry, such as a sphere (\(x^2 + y^2 + z^2 - r^2 = 0\)), where a single \(x,y\) coordinate pairs with multiple \(z\) values.

• Parametric Form: \(\vec{r}(u, v) = \langle x(u, v), y(u, v), z(u, v) \rangle\)

This form maps two independent parameters, typically \(u\) and \(v\), to generate a 2D sheet mapped into 3D space. Because of its flexibility in handling complex, overlapping geometry, it is the standard mathematical foundation used by 3D CAD software and surface modeling engines.