In geometry, a surface is generally a 2-dimensional manifold that may be embedded in a higher-dimensional space (typically 3D space). It can be thought of as a continuous, thin 'skin' or boundary of a solid object, or a generalization of a plane.

Unlike planes, surfaces can be curved in various ways.

Surfaces can be defined mathematically using different forms:

• Explicit Form: \(z = f(x, y)\). This describes a surface where each \((x, y)\) pair maps to a unique \(z\) value. E.g., a paraboloid \(z = x^2 + y^2\).
• Implicit Form: \(F(x, y, z) = 0\). This describes a surface as the set of points where a function of three variables equals zero. E.g., a sphere \(x^2 + y^2 + z^2 - r^2 = 0\).
• Parametric Form: Points on the surface are given by functions of two parameters, often \(u\) and \(v\). E.g., \(\vec{r}(u, v) = \langle x(u,v), y(u,v), z(u,v) \rangle\). This is highly flexible for complex surfaces.

Many specific types of surfaces are studied:

• Planes: Flat surfaces with zero curvature.
• Quadric Surfaces: Surfaces defined by second-degree equations in three variables (e.g., ellipsoids, paraboloids, hyperboloids, cones, cylinders).
• Ruled Surfaces: Surfaces formed by moving a straight line (e.g., cones, cylinders, hyperboloids of one sheet).
• Surfaces of Revolution: Formed by revolving a 2D curve around an axis (e.g., spheres, cones, cylinders, tori).
• Minimal Surfaces: Surfaces that locally minimize their area (like soap films).
• Non-orientable Surfaces: Surfaces that only have one side, like the Möbius strip or Klein bottle.

Key properties used to analyze surfaces include:

• Surface Area: The total area of the surface. Calculated using surface integrals.
• Curvature: A measure of how much a surface deviates from being flat (e.g., Gaussian curvature, mean curvature).
• Normal Vector: A vector perpendicular to the tangent plane at a point on the surface.
• Tangent Plane: A plane that 'just touches' the surface at a single point and best approximates the surface locally.

Surfaces are critical in:

• Computer-Aided Design (CAD) and Modeling: Creating complex 3D models for products, architecture, and animation.
• Engineering: Design of aerodynamic shapes, fluid flow systems, and structural components.
• Physics: Describing equipotential surfaces, wavefronts, and gravitational fields.
• Architecture: Designing curved roofs, domes, and modern building facades.
• Geology: Modeling terrain and geological formations.