Module sheet 

A flat sheet laid onto a face: the chart from the plane to the face’s surface that carries a sketch onto it.

The sheet is laid by geodesic polar coordinates, the exponential map at its origin: the sheet point at distance r and angle t from the origin lands where the geodesic leaving the origin’s point at angle t from the sheet’s +x is after length r. Lengths along every straight line through the origin are kept; across those lines a length is stretched by J(r) / r, J the Jacobi field along the geodesic (J'' = -K J, J(0) = 0, J'(0) = 1, K the Gaussian curvature), and so is an area, since the radial length is kept. Where the surface is flat (a plane, a cylinder, a cone) the map is the surface’s development and stretches nothing.

A plane, a cylinder, a cone and a sphere are laid in closed form. Every other surface has its geodesics integrated in its own chart, the geodesic equation and the Jacobi equation together, by classical Runge-Kutta steps whose error is estimated by step doubling and held, per unit of length, to the share of the accuracy asked that the whole length leaves it. A chart position is reached continuously along the sheet’s straight line from the origin, so it runs on past a periodic seam unwrapped; a carried curve’s chart curve is continued along the curve itself instead, so one going round a pole of the chart winds a whole period, as a parallel does.

Structs

SheetDistortion

How far a laid sheet stretches, over the sheet points it has carried.

SheetPlacement

How a flat sheet is laid on a face.

SurfaceSheet

A flat sheet laid on a face’s surface, as placed.

Enums

SheetMethod

How the flat sheet is carried away from its origin.