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When does a surface have negative Gaussian curvature?
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differential geometry
An annular strip (the region between two concentric circles) can be cut and bent into a helical strake that follows approximately the contour of a cylinder. Techniques of differential geometry are employed to find the dimensions of the annular strip that will best match the required curvature of the strake.
In differential geometry, the Gaussian curvature of a surface at a given point is defined as the product of the two principal normal curvatures.12
A surface has negative Gaussian curvature at a point when its two principal normal curvatures curve in opposite directions. This geometric property indicates that the surface is locally shaped like a saddle, as opposed to a sphere (positive curvature) or a plane/cylinder (zero curvature). Surfaces with constant negative Gaussian curvature are locally isometric to a hyperbolic plane.2