If at least one of the principal curvatures is zero at every point, then the gaussian curvature will be 0 and the surface is a developable surface. Gaussian curvature is an intrinsic surface property which refers to an isometric invariant of a surface 4. Let a metric with negative gaussian curvature be given, and let. Then the gaussian curvature kand gaussian torsion kn of mbecome k cfuufvv. For a monge patch, the gaussian curvature and mean curvature are 8 9 see also. Abstractwe present a novel technique for utilizing the gaussian curvature information in 3d nonrigid motion estimation in the absence of known correspondence. The gauss map in local coordinates develop effective methods for computing curvature of surfaces. It is the algebraic area of the image of the region on the unit sphere under the gauss map. This form is often referred to as monge form, and the surface is called a monge patch. However, since we have trouble even visualizing and understanding. A depth surface is a range image observed from a single view which can be re presented by a digital graph monge patch surface.
Surface curvature analysis is a powerful method for identifying pipes and pipe features in range images. For a regular height eld, curvature can be calculated directly by using monge patch gray, 1997. I know the gaussian curvature of a monge patch can be. Gaussian curvature in differential geometry, the gaussian curvature or gauss curvature. Both gaussian and mean curvatures have the attractive characteristics of translational and rotational invariance. General description of surface with zero gaussian curvature.
If the gauss curvature is 0, the circumference is 2. A regularization technique is proposed for unstructured polyhedral membranes. For a surface free of points of vanishing gaussian curvature in euclidean space the second gaussian curvature is defined formally. Mean and gaussian curvature can be computed from these derivatives. Mean and gaussian curvature for a gaussian hill seem wrong calculating the gaussian.
Several conventions are helpful in avoiding confu sions. Principal curves 467 as an important application of theorem 15. This is perhaps the most elegant of the dozens of formulas that have been found for gaussian curvature. Hence we obtain the geodesic equations for a monge patch. If the gaussian curvature changes its sign, the gauss map may fold the patch many times over the region. Generalized aminov surfaces given by a monge patch in the.
The equations of weingarten express the entries in the matrix for dn p in terms of the coefficients of the first and second fundamental forms. It is a straightforward matter to compute the gaussian and mean curvature of a ruled surface. This quantity is the gaussian curvature and is denoted as k. The gaussian curvature of a regular surface in r3 at a point p is formally defined as kpdetsp, 1 where s is the shape operator and det denotes the determinant. For a monge patch, the gaussian curvature and mean curvature are. So are you are seeking an expression for path curvature itex\kappaitex and torsion itex\tauitex for a space curve which belongs to a monge patch. It is, therefore, convenient to have analytic equations for the gaussian and mean curvatures expressed in terms of the derivatives of the height function. A depth surface of e3 is a range image observed from a single view can be represented by a digital graph monge patch surface. Straight pipes, for example, are equivalent to cylinders which can be effectively identi. It is therefore not necessary to describe the curvature properties of a. Now i am assuming that this problem is referring to a monge patch i. The gaussian curvature is calculated from the sum of vertex incident angles, weighted by same voronoi areas as for the mean curvature.
Gaussian curvature is regarded as an intrinsic property of space that is independent of the coordinate system that is used to describe that space. The gaussian curvature of a regular surface in at a point p is formally defined as 1 where is the shape operator and det denotes the determinant. A complete surface of gaussian curvature zero in euclidean three space is a cylinder where a cylinder means the surface generated by the lines parallel to a given axis passing through a fixed curve in the subspace perpendicular to the axis. Ruled surfaces with vanishing second gaussian curvature. A flat surface is a regular surface and special class of minimal surface on which gaussian curvature vanishes everywhere. Ruled surfaces for which a linear combination of the second gaussian curvature and the. Principal, gaussian and mean curvature of triangulated mesh. Gaussian curvature, sometimes also called total curvature kreyszig 1991, p.
What is the simplest way to compute principal curvature for a mesh triangle. Surface shape and curvature scales deep learning course. If is a regular patch, then the gaussian curvature is given by 2. It is first pointed out that a minimal surface has vanishing second gaussian curvature but that a surface with vanishing second gaussian curvature need not be minimal. If the gaussian curvature does not change its sign over the patch,the closer the total gaussian curvatureis to zero,the. Grimm department of computer science, washington university, st. With the ricci scalar we may derive the gaussian curvature6. Surface curvature for depth images can be calculated using monge patch formulas. Generalized aminov surfaces given by a monge patch 55 kn and the vector of mean curvature h on the behavior of surfaces is an actual problem. Let us compute the gaussian and mean curvatures of the hyperbolic. Both gaussian and mean curva tures have the attractive characteristics of translational and rotational invariance.
The computation of the gaussian curvature of a surface is a requirement in many propagation problems in physics and engineering. Mean curvature h and gaussian curvature k are defined as sum and. Gaussian curvature is an intrinsic measure of curvature, depending only on distances that are measured on the surface, not on the way it is isometrically embedded in euclidean space. Modern differential geometry of curves and surfaces. For a monge patch, the gaussian curvature and mean curvature are 8 9 see also monges form, patch. Citeseerx document details isaac councill, lee giles, pradeep teregowda. In order to calculate the k1 and k2, you need to use the first file mean curvature. Such a surface is often called a monge patch in the theory of surfaces. Derive the formula for gaussian curvature of the surface. Global topological properties of images derived from local. The output is the gaussian curvature at each point. Pdf surfaces given with the monge patch in e4 researchgate.
Geodesic on surfaces of constant gaussian curvature. Let m be a smooth surface given with the monge patch 9. For gaussbolyailobachevsky space, the gaussian curvature is. Gaussian curvature in propagation problems in physics and. For a monge patch, the gaussian curvature and mean curvature are 8 9 see also monge s form, patch. Differentialgeometric constraints derived in the paper allow one to estimate parameters of the local affine motion model given the values of gaussian. Curved spaces with the necessity of curved geodesics within regions with signi. A depth surface is a range image observed from a single view can be represented by a digital graph monge patch surface. Arslan department of mathematics, uludag university bursa 16059, turkey email.
Citeseerx curvaturebased algorithms for nonrigid motion. The second fundamental form directional derivatives in ir3. Further, the mean curvature vector of a regular patch xu. Differential geometry discussion compute the gaussian and mean curvatures of a monge patch, being open set, r. A formula is developed for the calculation of the gaussian curvature by knowledge of two close geodesics on the surface, or alternatively from the projection i. Gaussian curvature is named after carl friedrich gauss, who published the theorema egregium in 1827. The normal curvature is therefore the ratio between the second and the. Note the use of the word algebraic since gaussian curvature can be either positive or negative, suppose the patch s. Consider a polygonal decomposition of r, that is a collection of polygonal patches that cover rin a. I suggest moving this thread to the differential geometry forum, where it obviously belongs. The calculation is based on the first and second fundamental form. A generalization of curvature known as normal section curvature. Negative curvature, surface of encyclopedia of mathematics.
Explicit formulas for principal curvatures, gaussian. The proposed approach makes use of monge patches of polyhedral surfaces. Ur3 of the form xu,vu,v,hu,v, 1 where u is an open set in r2 and h. Here we compute the gaussian and mean curvatures of a monge patch z. The paper deals with a discretetocontinuum approach to the curvatures of flexible membranes. A point p on a regular surface is classified based on the sign of as given in the following table gray 1993, p. The monge parameterization is the most straightforward one. From a regular height eld, derivatives can be estimated using neighboring points values, which are. I havent done too much differential geometry but ive needed to work with gaussianmean curvature for a simple 3d gaussian surface.
If there exists a surface in threespace, at a specific point, there is a plane tangent to that surface. A convenient way to understand the curvature comes from an ordinary differential equation, first. In the present study we consider the surfaces in euclidean 4space e4 given. There are three classes of such surfaces, the least obvious but most interesting being the class of tangent developables.
A discretetocontinuum approach to the curvatures of. In particular the gaussian curvature is an invariant of the metric, gausss celebrated theorema egregium. Mean and gaussian curvature for a gaussian hill seem. Gaussian curvature article about gaussian curvature by. The input should be matrix containing points in x,y,z. The gaussian curvature of the surface is then given by the second order deviation of the metric at the point from the euclidean metric.
This is a so called monge representation of the surface patch. Now, i am interested in calculating curvature values for each point from the data i have. For a minimal surface, the mean curvature is zero at every. What is the simplest way to compute principal curvature. Both gaussian and mean curva tures have the attractive.
That is, a depth or range value at a point u,v is given by a single valued function zfu,v. Modern differential geometry of curves and surfaces with mathematica, 2nd ed. Gaussian curvature is intrinsic to the surface, and does not depend on the embedding in 3d space. I have come across surface curvature matlab equivalent in python, but the implementation is said to work only when x, y, and z are 2d arrays. The mongeampere equation is a fully nonlinear degenerate elliptic equation arising in several problems in the areas of analysis and geometry, such as the prescribed gaussian curvature equation, affine geometry, and optimal transportation, says figalli. In the considering work we use the representation of surfaces in the explicit form. The continuum limits of discrete definitions of the membrane curvatures are studied. This parametrization x is called a monge parametrization or monge patch, and the corresponding surface a simple surface, so that a general surface in r3 can be. Suppose that you took a local patch around a point on the surface and squashed it. For the unit sphere, both principal curvatures are 1 and hence. Reverse engineering of pipe layouts and 3d point set.
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