# Weingarten surface

## Weingarten surface

[′wīn‚gärt·ən ‚sər·fəs]
(mathematics)
A surface such that either of the principal radii is uniquely determined by the other.
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Classically, a Weingarten surface or linear Weingarten surface (or briefly, a W-surface) is a surface on which there is a nontrivial functional relation [PHI] ([k.sub.1], [k.sub.2]) = 0 between its principal curvatures [k.sub.1] and [k.sub.2] or equivalently, there is a nontrivial functional relation [PHI] (K, H) = 0 between its Gaussian curvature K and mean curvature H.
Therefore, M is a Weingarten surface. We have the following theorem:
A tubular surface M about unit-speed curve in [E.sup.3] is a Weingarten surface.
We suppose that M is a linear Weingarten surface in [E.sup.3].
Therefore, [M.sub.1] is a Weingarten surface. We have the following theorem:
A tubular surface [M.sub.1] about unit-speed spacelike curve with timelike principal normal in Minkowski 3-space is a Weingarten surface.
Therefore, [M.sub.2] is a Weingarten surface. We have the following theorem:
A tubular surface [M.sub.2] about unit-speed spacelike curve with spacelike principal normal in Minkowski 3-space is a Weingarten surface.
Thus Bertrand curves may be regarded as 1-dimensional analogue of Weingarten surfaces. For application of Weingarten surfaces to CAGD, we refer to , .
Grant: Potential applications of Weingarten surfaces in CADG part I: Weingarten surfaces and surface shape investigation, Computer Aided Geometric Design, 13(1996), 569-582.
Thus Bertrand curves may be regraded as one-dimensional analogue of Weingarten surfaces .
Grant, Potential application of Weingarten surfaces in CADG, Part I: Weingarten surfaces and surface shape investigation, Computer Aided Geometric Design, 13(1996), 569-582.
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