MATH 4230 · Differential Geometry

The Gauss Map & Weingarten Map

Hover over the surface to explore. Drag any view to rotate all four. Scroll over a view to zoom.

Self-contained · works offline

Custom parametrization Formulas and parameter ranges

x(u,v)y(u,v)z(u,v)

u minimumu maximumv minimumv maximum

Use u, v, pi, e, +, −, *, /, ^, parentheses, and sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, exp, log (natural log), ln, sqrt, abs, pow. Implicit products such as 2u are accepted. Use a smooth, regular parametrization on the chosen rectangle. No JavaScript is executed.

N⟶

W_p⟶

1Surface S ⊂ ℝ³

Hover to choose p

Preparing the geometry…

p + T_pSSelected point and its normal

2Gauss map N : S → S²

Unit sphere

N(p) + T_N(p)S²Gold patch = actual Gauss image

Tangent-space basis

Principal directions e₁, e₂ Coordinate basis σ_u, σ_v

Surface & sphereTangent planes

3Input tangent basis

Vectors based at 0

T_pS = N(p)^⊥ ⊂ ℝ³Selected basis vectors

4Images under W_p

Same plane · same scale

T_N(p)S² = N(p)^⊥ ⊂ ℝ³Wₚ = −TₚN

e₁ → Wₚe₁e₂ → Wₚe₂Unit normal N

W_p in coordinates e₁, e₂

2 × 2

B =

Columns give the images of e₁ and e₂ in this same basis.

Ambient extension in coordinates x, y, z

3 × 3

A =

Ambient extension: AN(p) = 0. Fixed standard basis.

Curvature at p

W = −TN

κ₁

—

κ₂

—

K = det B; H = ½ tr B. The ambient determinant det A is always zero.

Display options Vector scale, patch size, and keyboard-accessible point selection

Vector scale1.00×Patch size7.5%Coordinate gridTangent planesGauss patchUnit circle and its W-image

Select a point without hovering:u0.400v−0.250

The scale multiplies all tangent arrows by the same factor, never their numerical values. Unit normals stay unit length. The two lower views always have the same scale. Dragging holds the point fixed; moving off the surface retains the last selection. On touch screens, tap to select and drag to rotate. Arrow keys rotate a focused view; + and − zoom.

Initializing…