Parametric Surfaces

Define and explore surfaces through parametrization

Parametric Surfaces

A parametric surface is a 2D manifold embedded in 3D space, defined by a function S(u, v) = (x(u,v), y(u,v), z(u,v)). The parameters u and v serve as coordinates on the surface, like latitude and longitude on Earth.

At each point, the surface has a tangent plane spanned by the partial derivatives Su and Sv, and a normal vector N perpendicular to it. These form the foundation for measuring curvature.

Interactive: Surface Parametrization

The parameter grid shows how (u, v) coordinates map onto the surface. Cyan lines trace constant v (u-curves), magenta lines trace constant u (v-curves). Hover to see coordinates at any point.

Parameter Grid
u-lines (constant v)
v-lines (constant u)
Sphere

The classic 2-sphere

A unit sphere centered at the origin

Parametrization
S(u, v) → (x, y, z)
Hover over the surface to see coordinates

Interactive: Tangent Plane Explorer

Click anywhere on the surface to visualize the tangent plane at that point. The arrows show the partial derivative vectors Su (cyan) and Sv (pink), plus the normal vector N (green).

Click anywhere on the surface to see the tangent plane
Su — Tangent in u-direction
Sv — Tangent in v-direction
N — Surface normal
Tangent plane

Partial Derivatives

Su = ∂S/∂u
Sv = ∂S/∂v

These vectors are tangent to the surface: Su points in the direction of increasing u (along u-curves), Sv points in the direction of increasing v.

Surface Normal

N = (Su × Sv) / |Su × Sv|

The unit normal vector is perpendicular to both tangent vectors. It determines the “outward” direction at each point (for orientable surfaces).

The Tangent Plane

At each point p on a smooth surface, the tangent plane TpS is the best linear approximation to the surface near p. It contains all vectors tangent to curves passing through p.

Any tangent vector can be written as aSu + bSv for some coefficients a and b. This makes {Su, Sv} a natural basis for the tangent space — though not necessarily orthonormal!

Surface Gallery

Sphere
(sin φ cos θ, sin φ sin θ, cos φ)
K > 0
Torus
((R + r cos v) cos u, ...)
K varies
Saddle
(u, v, u² - v²)
K < 0
Cylinder
(cos u, sin u, v)
K = 0

Key Takeaways

  • Parametric surfaces map a 2D parameter domain (u, v) into 3D space
  • Partial derivatives Su and Sv span the tangent plane
  • The normal vector N = Su × Sv is perpendicular to the surface
  • Parameter curves form a coordinate grid on the surface
  • The tangent plane is the best linear approximation to the surface at each point

Next: The First Fundamental Form — measuring lengths, angles, and areas on surfaces.