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Text of the page (random words):
ture κ of a surface at a point is the product of the principal curvatures κ 1 and κ 2 at the given point k κ 1 κ 2 displaystyle k kappa _ 1 kappa _ 2 the gaussian radius of curvature is the reciprocal of κ for example a sphere of radius r has gaussian curvature 1 r 2 everywhere and a flat plane and a cylinder have gaussian curvature zero everywhere the gaussian curvature can also be negative as in the case of a hyperboloid or the inside of a torus gaussian curvature is an intrinsic measure of curvature depending only on distances that are measured within or along the surface not on the way it is isometrically embedded in euclidean space this is the content of the theorema egregium gaussian curvature is named after carl friedrich gauss who published the theorema egregium in 1827 contents 1 informal definition 2 relation to geometries 3 relation to principal curvatures 4 alternative definitions 5 total curvature 6 important theorems 6 1 theorema egregium 6 2 gauss bonnet theorem 7 surfaces of constant curvature 8 alternative formulas 9 see also 10 references 11 books 12 external links informal definition edit saddle surface with normal planes in directions of principal curvatures at any point on a surface we can find a normal vector that is at right angles to the surface planes containing the normal vector are called normal planes the intersection of a normal plane and the surface will form a curve called a normal section and the curvature of this curve is the normal curvature for most points on most smooth surfaces different normal sections will have different curvatures the maximum and minimum values of these are called the principal curvatures call these κ 1 κ 2 the gaussian curvature is the product of the two principal curvatures κ κ 1 κ 2 the sign of the gaussian curvature can be used to characterise the surface if both principal curvatures are of the same sign κ 1 κ 2 0 then the gaussian curvature is positive and the surface is said to have an elliptic point at such points the surface will be dome like locally lying on one side of its tangent plane all sectional curvatures will have the same sign if the principal curvatures have different signs κ 1 κ 2 0 then the gaussian curvature is negative and the surface is said to have a hyperbolic or saddle point at such points the surface will be saddle shaped because one principal curvature is negative one is positive and the normal curvature varies continuously if you rotate a plane orthogonal to the surface around the normal to the surface in two directions the normal curvatures will be zero giving the asymptotic curves for that point if one of the principal curvatures is zero κ 1 κ 2 0 the gaussian curvature is zero and the surface is said to have a parabolic point most surfaces will contain regions of positive gaussian curvature elliptical points and regions of negative gaussian curvature separated by a curve of points with zero gaussian curvature called a parabolic line relation to geometries edit when a surface has a constant zero gaussian curvature then it is a developable surface and the geometry of the surface is euclidean geometry when a surface has a constant positive gaussian curvature then the geometry of the surface is spherical geometry spheres and patches of spheres have this geometry but there exist other examples as well such as the football when a surface has a constant negative gaussian curvature then it is a pseudospherical surface and the geometry of the surface is hyperbolic geometry relation to principal curvatures edit the two principal curvatures at a given point of a surface are the eigenvalues of the shape operator at the point they measure how the surface bends by different amounts in different directions at that point we represent the surface by the implicit function theorem as the graph of a function f of two variables in such a way that the point p is a critical point that is the gradient of f vanishes this can always be attained by a suitable rigid motion then the gaussian curvature of the surface at p is the determinant of the hessian matrix of f being the product of the eigenvalues of the hessian recall that the hessian is the 2 2 matrix of second derivatives this definition allows one immediately to grasp the distinction between a cup cap versus a saddle point alternative definitions edit it is also given by k 2 1 1 2 e 1 e 2 det g displaystyle k frac bigl langle nabla _ 2 nabla _ 1 nabla _ 1 nabla _ 2 mathbf e _ 1 mathbf e _ 2 bigr rangle det g where i e i is the covariant derivative and g is the metric tensor at a point p on a regular surface in r 3 the gaussian curvature is also given by k p det s p displaystyle k mathbf p det s mathbf p where s is the shape operator a useful formula for the gaussian curvature is liouville s equation in terms of the laplacian in isothermal coordinates total curvature edit the sum of the angles of a triangle on a surface of negative curvature is less than that of a plane triangle the surface integral of the gaussian curvature over some region of a surface is called the total curvature the total curvature of a geodesic triangle equals the deviation of the sum of its angles from π the sum of the angles of a triangle on a surface of positive curvature will exceed π while the sum of the angles of a triangle on a surface of negative curvature will be less than π on a surface of zero curvature such as the euclidean plane the angles will sum to precisely π radians i 1 3 θ i π t k d a displaystyle sum _ i 1 3 theta _ i pi iint _ t k da a more general result is the gauss bonnet theorem important theorems edit theorema egregium edit main article theorema egregium gauss s theorema egregium latin remarkable theorem states that gaussian curvature of a surface can be determined from the measurements of length on the surface itself in fact it can be found given the full knowledge of the first fundamental form and expressed via the first fundamental form and its partial derivatives of first and second order equivalently the determinant of the second fundamental form of a surface in r 3 can be so expressed the remarkable and surprising feature of this theorem is that although the definition of the gaussian curvature of a surface s in r 3 certainly depends on the way in which the surface is located in space the end result the gaussian curvature itself is determined by the intrinsic metric of the surface without any further reference to the ambient space it is an intrinsic invariant in particular the gaussian curvature is invariant under isometric deformations of the surface in contemporary differential geometry a surface viewed abstractly is a two dimensional differentiable manifold to connect this point of view with the classical theory of surfaces such an abstract surface is embedded into r 3 and endowed with the riemannian metric given by the first fundamental form suppose that the image of the embedding is a surface s in r 3 a local isometry is a diffeomorphism f u v between open regions of r 3 whose restriction to s u is an isometry onto its image theorema egregium is then stated as follows the gaussian curvature of an embedded smooth surface in r 3 is invariant under the local isometries for example the gaussian curvature of a cylindrical tube is zero the same as for the unrolled tube which is flat 1 page needed on the other hand since a sphere of radius r has constant positive curvature r 2 and a flat plane has constant curvature 0 these two surfaces are not isometric not even locally thus any planar representation of even a small part of a sphere must distort the distances therefore no cartographic projection is perfect gauss bonnet theorem edit main article gauss bonnet theorem the gauss bonnet theorem links the total curvature of a surface to its euler characteristic and provides an important link between local geometric properties and global topological properties surfaces of constant curvature edit two surfaces which both have constant positive gaussian curvature but with either an open boundary or singular points minding s theorem 1839 states that all surfaces with the same constant curvature k are locally isometric a consequence of minding s theorem is that any surface whose curvature is identically zero can be constructed by bending some plane region such surfaces are called developable surfaces minding also raised the question of whether a closed surface with constant positive curvature is necessarily rigid liebmann s theorem 1900 answered minding s question the only regular of class c 2 closed surfaces in r 3 with constant positive gaussian curvature are spheres 2 if a sphere is deformed it does not remain a sphere proving that a sphere is rigid a standard proof uses hilbert s lemma that non umbilical points of extreme principal curvature have non positive gaussian curvature 3 hilbert s theorem 1901 states that there exists no complete analytic class c ω regular surface in r 3 of constant negative gaussian curvature in fact the conclusion also holds for surfaces of class c 2 immersed in r 3 but breaks down for c 1 surfaces the pseudosphere has constant negative gaussian curvature except at its singular cusp 4 there are other surfaces which have constant positive gaussian curvature manfredo do carmo considers surfaces of revolution ϕ v cos u ϕ v sin u ψ v displaystyle phi v cos u phi v sin u psi v where ϕ v c cos v displaystyle phi v c cos v and ψ v 0 v 1 c 2 sin 2 v d v textstyle psi v int _ 0 v sqrt 1 c 2 sin 2 v dv an incomplete elliptic integral of the second kind these surfaces all have constant gaussian curvature of 1 but for c 1 displaystyle c neq 1 either have a boundary or a singular point do carmo also gives three different examples of surface with constant negative gaussian curvature one of which is pseudosphere 5 there are many other possible bounded surfaces with constant gaussian curvature whilst the sphere is rigid and can not be bent using an isometry if a small region removed or even a cut along a small segment then the resulting surface can be bent such bending preserves gaussian curvature so any such bending of a sphere with a region removed will also have constant gaussian curvature 6 alternative formulas edit gaussian curvature of a surface in r 3 can be expressed as the ratio of the determinants of the second and first fundamental forms ii and i k det i i det i l n m 2 e g f 2 displaystyle k frac det mathrm i i det mathrm i frac ln m 2 eg f 2 the brioschi formula after francesco brioschi gives gaussian curvature solely in terms of the first fundamental form k 1 2 e v v f u v 1 2 g u u 1 2 e u f u 1 2 e v f v 1 2 g u e f 1 2 g v f g 0 1 2 e v 1 2 g u 1 2 e v e f 1 2 g u f g e g f 2 2 displaystyle k frac begin vmatrix frac 1 2 e_ vv f_ uv frac 1 2 g_ uu frac 1 2 e_ u f_ u frac 1 2 e_ v f_ v frac 1 2 g_ u e f frac 1 2 g_ v f g end vmatrix begin vmatrix 0 frac 1 2 e_ v frac 1 2 g_ u frac 1 2 e_ v e f frac 1 2 g_ u f g end vmatrix left eg f 2 right 2 for an orthogonal parametrization f 0 gaussian curvature is k 1 2 e g u g u e g v e v e g displaystyle k frac 1 2 sqrt eg left frac partial partial u frac g_ u sqrt eg frac partial partial v frac e_ v sqrt eg right for a surface described as graph of a function z f x y gaussian curvature is 7 k f x x f y y f x y 2 1 f x 2 f y 2 2 displaystyle k frac f_ xx cdot f_ yy f_ xy 2 left 1 f_ x 2 f_ y 2 right 2 for an implicitly defined surface f x y z 0 the gaussian curvature can be expressed in terms of the gradient f and hessian matrix h f 8 9 k h f f t f 0 f 4 f x x f x y f x z f x f x y f y y f y z f y f x z f y z f z z f z f x f y f z 0 f 4 displaystyle k frac begin vmatrix h f nabla f mathsf t nabla f 0 end vmatrix nabla f 4 frac begin vmatrix f_ xx f_ xy f_ xz f_ x f_ xy f_ yy f_ yz f_ y f_ xz f_ yz f_ zz f_ z f_ x f_ y f_ z 0 end vmatrix nabla f 4 for a surface with metric conformal to the euclidean one so f 0 and e g e σ the gauss curvature is given by δ being the usual laplace operator k 1 2 e σ δ σ displaystyle k frac 1 2e sigma delta sigma gaussian curvature is the limiting difference between the circumference of a geodesic circle and a circle in the plane 10 k lim r 0 3 2 π r c r π r 3 displaystyle k lim _ r to 0 3 frac 2 pi r c r pi r 3 gaussian curvature is the limiting difference between the area of a geodesic disk and a disk in the plane 10 k lim r 0 12 π r 2 a r π r 4 displaystyle k lim _ r to 0 12 frac pi r 2 a r pi r 4 gaussian curvature may be expressed with the christoffel symbols 11 k 1 e u γ 12 2 v γ 11 2 γ 12 1 γ 11 2 γ 11 1 γ 12 2 γ 12 2 γ 12 2 γ 11 2 γ 22 2 displaystyle k frac 1 e left frac partial partial u gamma _ 12 2 frac partial partial v gamma _ 11 2 gamma _ 12 1 gamma _ 11 2 gamma _ 11 1 gamma _ 12 2 gamma _ 12 2 gamma _ 12 2 gamma _ 11 2 gamma _ 22 2 right see also edit earth s gaussian radius of curvature sectional curvature mean curvature gauss map riemann curvature tensor principal curvature references edit porteous i r 1994 geometric differentiation cambridge university press isbn 0 521 39063 x kühnel wolfgang 2006 differential geometry curves surfaces manifolds american mathematical society isbn 0 8218 3988 8 gray alfred 1997 28 4 hilbert s lemma and liebmann s theorem modern differential geometry of curves and surfaces with mathematica 2nd ed crc press pp 652 654 isbn 9780849371646 hilbert theorem encyclopedia of mathematics ems press 2001 1994 carmo manfredo perdigão do 2016 first published 1976 differential geometry of curves and surfaces 2nd ed mineola ny dover publications p 171 isbn 978 0 486 80699 0 via zbmath hilbert david cohn vossen stephan 1952 geometry and the imagination 2nd ed chelsea p 228 isbn 0 8284 1087 9 general investigations of curved surfaces of 1827 and 1825 princeton the princeton university library 1902 goldman r 2005 curvature formulas for implicit curves and surfaces computer aided geometric design 22 7 632 658 citeseerx 10 1 1 413 3008 doi 10 1016 j cagd 2005 06 005 spivak m 1975 a comprehensive introduction to differential geometry vol 3 boston publish or perish a b bertrand diquet puiseux theorem struik dirk 1988 lectures on classical differential geometry courier dover publications isbn 0 486 65609 8 books edit grinfeld p 2014 introduction to tensor analysis and the calculus of moving surfaces springer isbn 978 1 4614 7866 9 rovelli carlo 2021 general relativity the essentials cambridge university press isbn 978 1 009 01369 7 external links edit gaussian curvature encyclopedia of mathematics ems press 2001 1994 v t e various notions of curvature defined in differential geometry differential geometry of curves curvature torsion of a curve frenet serret formulas radius of curvature applications affine curvature total curvature total absolute curvature differential geometry of surfaces principal curvatures gaussian curvature mean curvature darboux frame gauss codazzi equations first fundamental form second fundamental form third fundamental form riemannian geometry curvature...