Parametric Surface Plotter | Mathematical Visualization Lab

Surface Presets

Custom Parametric Equations

x(u, v) =

y(u, v) =

z(u, v) =

Parameter Ranges

u min

u max

v min

v max

Resolution (u) 80

Resolution (v) 80

Visualization Options

Color Mode

Show Wireframe Parameter Grid Lines Normal Vectors Coordinate Axes Auto Rotate

Lighting & Material

Ambient Light 0.3

Diffuse Light 0.7

Specular 0.5

Shininess 32

Initializing WebGL...

Torus

A torus is a surface of revolution generated by revolving a circle in 3D space about an axis.

Current Equations

r(u, v) =

x = (2 + cos(v)) · cos(u)

y = (2 + cos(v)) · sin(u)

z = sin(v)

Color Scale

0 2π

Surface Statistics

12,800

Vertices

25,088

Triangles

FPS

1.0×

Zoom Level

Understanding Parametric Surfaces

Parametric Representation

A parametric surface is defined by a vector-valued function r(u,v) that maps a 2D parameter domain to 3D space. Each point on the surface corresponds to a unique (u,v) pair.

r(u,v) = [x(u,v), y(u,v), z(u,v)]

Surface Normal

The normal vector at each point is computed using the cross product of partial derivatives, enabling proper lighting calculations.

N = ∂r/∂u × ∂r/∂v

Gaussian Curvature

Measures how the surface curves in different directions. Positive for spheres, negative for saddles, zero for planes.

K = κ₁ · κ₂