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Measures of surface curvature. (a) The principal curvatures are calculated from the intersections between the normal planes at a point and the surface. The intersections form curved lines in the surface, with a certain “normal curvature”. The maximum and minimum values of all possible normal curvatures are the principal curvatures, which are of opposite sign in this case given the fact that the surface curves “upward” in one direction and “downward” in the other. The color bar indicates the Gaussian curvature. (b) The mean and Gaussian curvatures could be calculated from the principal curvatures. Bending a planar surface could change the mean, or extrinsic, curvature (left figure, color bar indicates mean curvature), but not the Gaussian, or intrinsic, curvature (right figure, color bar indicates Gaussian curvature). (c) Transforming a planar surface (color bar indicates the Gaussian curvature). Top row: three types of developable surfaces, which could be flattened onto the plane through bending. From left to right: a cylindrical surface, a conical surface, and the tangent developable surface to a space curve (a helix in this case). Bottom row: three types of intrinsically curved surfaces. From left to right: a sphere withK > 0, a saddle withK < 0, and a vase surface with varying K. The sum of the internal angles of a triangle drawn on an intrinsically curved surface does not equal . (d) Interpreting the relationship between the metric and the Gaussian curvature (color bar indicates Gaussian curvature). Creating the bell-shaped surface from an initially flat plane requires distortion of the grid on the plane.
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Transforming flat sheets into three-dimensional structures has emerged as an exciting manufacturing
paradigm on a broad range of length scales. Among other advantages, this technique permits the use of
functionality-inducing planar processes on flat starting materials, which after shape-shifting, result in
a unique combination of macro-scale geomet...
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Citations
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Graphical abstract
... The average of the two principal curvatures determines the mean curvature (H), H = (k 1 + k 2 )/2. K is an intrinsic measure defined by the surface itself, whereas H is an extrinsic measure determined by the surface and its surrounding 209 . Therefore, it does not characterize the shape but gives the amount of curvature in one way or another. ...
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