Parametric Surfaces

Define and explore surfaces through parametrization

Parametric Surfaces

A surface is a sheet drawn in space by two dials, u and v, much like latitude and longitude. At every point it has a flat tangent plane and an arrow sticking straight out.

Surface Parametrization

Switch surfaces and hover to read (u, v) at any point. Cyan lines hold v fixed, magenta lines hold u fixed; notice the grid bunching up near the sphere's poles.

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
Go deeper: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.

Go deeper: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

Tangent Plane Explorer

Click anywhere on the surface to place its tangent plane, with Su (cyan), Sv (pink), and the normal N (green). Move the point and watch the plane tilt.

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
Go deeper: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.

Go deeper: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).

Go deeper: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!

Key Takeaways

  • A parametric surface maps a 2D parameter domain (u, v) into 3D space.
  • The partial derivatives Su and Sv span the tangent plane.
  • The normal N = Su × Sv is perpendicular to the surface.
  • The tangent plane is the best linear approximation to the surface at each point.

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