Double Curvature Surfaces

Doubly curved surfaces (for example, spherical, toroidal, etc.) are unrolled approximately into flat sheets.

• Unrolling a doubly curved surface introduces some distortion in angles and/or lengths

• Doubly curved surfaces that have small curvature in one of the directions will unroll

  • A sphere would distort too much to flatten (picture a flattened map of the world)
  • A big enough torus, and many free-form designs can be flattened without too much distortion

• Angles, perimeter, and area are preserved as closely as possible (within 15%)

  • The accuracy increases as the curvature decreases
  • The simpler the shape, the more accurate the perimeter and area

This example is a parabola swept along a slightly curved trajectory. Notice that the Area and Perimeter are both within 0.3% of the original.

Original Surface Properties
Unrolled Surface Properties Area within 0.3% Perimeter within 0.2%

This example is a parabola swept along a highly curved trajectory. Notice that the Area and Perimeter are also both within 0.3% of the original. This is because the shape is a simple parabola.

Original Surface Properties
Unrolled Surface Properties Area within 0.3% Perimeter within 0.2%

This example Is a more organic shape. Notice that the Area is within 0.4% and the Perimeter is within 6.5%.

Original Surface Properties
Unrolled Surface Properties Area within 0.3% Perimeter within 0.2%

This example is a more complex perimeter. Notice that the Area is within 0.8% and the Perimeter is within 6.0%.

Original Surface Properties
Unrolled Surface Properties Area within 0.3% Perimeter within 0.2%