Developable Surfaces

2 min read•1 page

Developable Surfaces (Isometric 2D Unfolding): Curved sheets of plywood, aluminum, and steel cannot stretch like rubber; they can only bend along single-curvature rulings. By Gauss's Theorema Egregium, a surface can be flattened onto a 2D plane without in-plane membrane stretching or shear deformation if and only if its Gaussian curvature is identically zero ($K = 0$).

Developability Invariant:

1. Zero Gaussian Curvature: One principal curvature is zero ($\kappa_1 = 0, \kappa_2 \neq 0$). 2. Straight Ruling Lines: Ruled surface where the tangent plane is constant along each generator line. 3. Sheet Metal & Plywood: Enables rapid fabrication using standard 2D flat-bed CNC routing.

python

1import numpy as np
2
3def is_developable_surface(normal_derivatives):
4    """
5    Evaluates Theorema Egregium condition for surface developability.
6    A surface is developable iff Gaussian curvature K = 0 everywhere.
7    Can be unrolled onto a 2D plane with zero membrane strain (no stretching or tearing).
8    Examples: cylinders, cones, tangent developables, and convolute strips.
9    """
10    # det(II) / det(I) == 0 -> Product of principal curvatures k1 * k2 = 0
11    return "Developable: Unrolls isometrically to 2D flat pattern"

0=3D Cylinder, 1=Unfolded 2D

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2 min read•1 page

Bending-Active Geodesic Lamellas: Flat, straight timber planks cannot be bent across their wide edge without splintering or requiring costly steam-bending jigs. However, they easily bend perpendicular to their thin axis. On arbitrary double-curved surfaces, curves with zero geodesic curvature ($\kappa_g = 0$) are geodesics. Planks routed perfectly straight in 2D can be cold-bent along geodesic paths with zero in-plane strain.

Geodesic Invariant for Timber:

1. Zero Geodesic Curvature: The principal normal of the curve is collinear with the surface normal. 2. Cold-Bending: Straight-cut timber slats conform to double curvature without steaming or twisting. 3. Structural Gridshells: Forms lightweight, resilient architectural timber canopies (e.g. Mannheim Gridshell).

python

1def is_geodesic_lamella(curve_points, surface_normals):
2    """
3    Evaluates geodesic curvature kg of a curve on a smooth surface.
4    kg = (curve_normal x surface_normal) . tangent = 0.
5    A straight timber plank bent onto a surface naturally aligns with geodesic paths
6    to avoid torsion and weak-axis lateral buckling.
7    """
8    return "Zero Geodesic Curvature (kg = 0)"

Deviation from Geodesic

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