In this paper we focus on the estimation of curvature magnitudes and principal directions, and discuss surface normals only to the extent that they a. Editors gaussian curvature is a curvature intrinsic to a twodimensional surface, something youd never expect a. By the tangent space of m at p, denoted by t pm, we mean the set of all vectors v in r3 such that for each vector v there exists a smooth curve. Gaussian curvature is an intrinsic surface property which refers to an isometric invariant of a surface 4. May 24, 2007 such a surface is often called a monge patch in the theory of surfaces. For a minimal surface, the mean curvature is zero at every. Surface curvature for depth images can be calculated using monge patch formulas. The main types of curvature that emerged from this were mean curvature and gaussian curvature. Ruled surfaces with vanishing second gaussian curvature. Derive the formula for gaussian curvature of the surface. For a monge patch, the gaussian curvature and mean curvature are 8 9 see also. 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.
In this lecture we explore the geometric meaning of k. Differential geometrygauss curvature of graphs of a. In order to calculate the k1 and k2, you need to use the first file mean curvature. This is a so called monge representation of the surface patch. There are three classes of such surfaces, the least obvious but most interesting being the class of tangent developables. 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.
This form is often referred to as monge form, and the surface is called a monge patch. The continuum limits of discrete definitions of the membrane curvatures are studied. Surface shape and curvature scales deep learning course. A facial feature tracker for humancomputer interaction. The gauss map in local coordinates develop effective methods for computing curvature of surfaces. Jun 15, 2004 the input should be matrix containing points in x,y,z. Index termsnonrigid motion, correspondence estimation, differential geometry, gaussian curvature. Questions, no matter how basic, will be answered to the best ability of the online. Gaussian curvature article about gaussian curvature by the. With the ricci scalar we may derive the gaussian curvature6. The euclidean plane and the cylinder both have constant gaussian curvature 0.
The gaussian curvature is calculated from the sum of vertex incident angles, weighted by same voronoi areas as for the mean curvature. At a local min, the sign of the two principal curvatures cannot be opposite. A surface on which the gaussian curvature is everywhere positive is called synclastic, while a surface on which is everywhere negative is called anticlastic. Gaussian curvature is named after carl friedrich gauss, who published the theorema egregium in 1827. Further, the gaussian curvature and gaussian torsion of a regular patch. Reverse engineering of pipe layouts and 3d point set. Both gaussian and mean curvatures have the attractive characteristics of translational and rotational invariance. Of these, the cone and cylinder are the only flat surfaces of revolution. Exploration of intrinsic curvature developed after the study of the extrinsic. The normal curvature is therefore the ratio between the second and the. If is a regular patch, then the gaussian curvature is given by. 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. Differential geometry discussion compute the gaussian and mean curvatures of a monge patch, being open set, r. The calculation is based on the first and second fundamental form.
For a monge patch, the gaussian curvature and mean curvature are 8 9 see also monge s form, patch. Iterative point matching for registration of freeform. For a surface free of points of vanishing gaussian curvature in euclidean space the second gaussian curvature is defined formally. Mean curvature h and gaussian curvature k are defined as sum and. Surfaces with constant gaussian curvature include the cone, cylinder, kuen surface, plane, pseudosphere, and sphere. Surface shape and curvature scales jan j koenderink and andrea j van doorn the classical surface curvature measures, such as the gaussian and the mean curvature at a point of a surface, are not very indicative of local shape. Now the mean h curvature and gaussian k curvature for a monge patch can be. Gaussian curvature is intrinsic to the surface, and does not depend on the embedding in 3d space. Surfaces of revolution and constant curvature surfaces of revolution form the most easily recognized class of surfaces. Differentialgeometric constraints derived in the paper allow one to estimate parameters of the local affine motion model given the values of gaussian curvature before and after motion. 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. 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.
Ur3 of the form xu,vu,v,hu,v, 1 where u is an open set in r2 and h. Shape and curvature in this chapter, we study the relationship between the geometry of a regular. The two principal curvatures taken as a pair are more informative, but. It is the algebraic area of the image of the region on the unit sphere under the gauss map. We know that ellipsoids and hyperboloids are surfaces of revolution provided that two of their axes are equal. It is shown that the simplified gaussian and mean curvatures and usual ones are. Iterative point matching for registration of freeform curves and surfaces 121 cessed during the registration.
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. Compactness results for immersions of prescribed gaussian curvature i analytic aspects dedicated to hennie smith 2212191408122009 with affection author links open overlay panel graham smith. Straight pipes, for example, are equivalent to cylinders which can be effectively identi. Like all simple that is, onepatch surfaces, the helicoid and saddle surfaces are orientable, since computations as above provide a unit normal on the whole surface.
Arslan department of mathematics, uludag university. It is, therefore, convenient to have analytic equations for the gaussian and mean curvatures expressed in terms of the derivatives of the height function. The gaussian curvature k and mean curvature h can be approximated via the following. In particular the gaussian curvature is an invariant of the metric, gausss celebrated theorema egregium. Thus the gaussian curvature of m depends only on distance to the z axis, rising from k. Surface curvature analysis is a powerful method for identifying pipes and pipe features in range images.
Pdf surfaces given with the monge patch in e4 researchgate. Gauss was the first to recognize the importance of the gaussian curvature. The gaussian curvature of a regular surface in at a point is formally defined as 1 where is the shape operator and det denotes the determinant. We present a novel technique for utilizing the gaussian curvature information in 3d nonrigid motion estimation in the absence of known correspondence. Both gaussian and mean curva tures have the attractive characteristics of translational and rotational invariance. A convenient way to understand the curvature comes from an ordinary differential equation, first. I know the gaussian curvature of a monge patch can be. 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. A discretetocontinuum approach to the curvatures of. Let a metric with negative gaussian curvature be given, and let. Explicit formulas for principal curvatures, gaussian. Mean and gaussian curvature for a gaussian hill seem wrong. The monge ampere 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.
If the gaussian curvature changes its sign, the gauss map may fold the patch many times over the region. The dirichlet problem for the equation of prescribed gauss. The proposed approach makes use of monge patches of polyhedral surfaces. Gausss view of curvature and the theorema egregium. Let m be a smooth surface given with the monge patch 9. Simplified gaussian and mean curvatures to range image. Then the gaussian curvature kand gaussian torsion kn of mbecome k cfuufvv. Ruled surfaces for which a linear combination of the second gaussian curvature and the mean. Just wondering if there is a general formula for the gaussian curvature at point x,y,fx,y in terms of x, y, and fx,y. Such a surface is often called a monge patch in the theory of surfaces. Note the use of the word algebraic since gaussian curvature can be either positive or negative, suppose the patch s.
Principal, gaussian and mean curvature of triangulated mesh. Gaussian curvature is regarded as an intrinsic property of space that is independent of the coordinate system that is used to describe that space. Geometric invariants for facial feature tracking with 3d. Modern differential geometry of curves and surfaces. It is therefore not necessary to describe the curvature properties of a. The information used in the primitivebased approach is much more concise than in the surfacebased approach, and is in gen eral preferable. The monge parameterization is the most straightforward one. 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. If the gaussian curvature does not change its sign over the patch,the closer the total gaussian curvatureis to zero,the. If there exists a surface in threespace, at a specific point, there is a plane tangent to that surface. A depth surface is a range image observed from a single view which can be re presented by a digital graph monge patch surface. Negative curvature, surface of encyclopedia of mathematics.
Here we compute the gaussian and mean curvatures of a monge patch z. From a regular height eld, derivatives can be estimated using neighboring points values, which are. Patches on the surface with zero curvature lines or areas correspond to a single point on the sphere. In this video we discuss gauss s view of curvature in terms of the derivative of the gauss rodrigues map the image of a unit normal n into the unit sphere, and expressed in terms of the. Gaussian curvature, sometimes also called total curvature kreyszig 1991, p.
Geodesic on surfaces of constant gaussian curvature. Math 497c oct 7, 20041 curves and surfaces fall 2004, psu lecture notes 9 2. Hence we obtain the geodesic equations for a monge patch. Apr 14, 2017 its parameterised using monge patch representation. Mean curvature was the relevant to applications of the time and was, as a result, the most studied. Notebook with reference to monge patches and other examples. In the considering work we use the representation of surfaces in the explicit form. Now i am assuming that this problem is referring to a monge patch i. If you take a flat piece of paper and bend it gently, it bends in only one direction at a time. Compactness results for immersions of prescribed gaussian. At any point on the paper, you can find at least one direction through which there is a straight line on the surface. It is a straightforward matter to compute the gaussian and mean curvature of a ruled surface. Curvature of surfaces in 3space goucher college blogs. Let us compute the gaussian and mean curvatures of the hyperbolic.
Global topological properties of images derived from local. Jan 15, 2014 principal, gaussian and mean curvature of triangulated mesh. Mean and gaussian curvature for a gaussian hill seem. A depth surface is a range image observed from a single view can be represented by a digital graph monge patch surface. Citeseerx curvaturebased algorithms for nonrigid motion. For a regular height eld, curvature can be calculated directly by using monge patch gray, 1997. For a monge patch, the gaussian curvature and mean curvature are. Generalized aminov surfaces given by a monge patch in the. I give another two examples here on surfaces of revolution and an application to the sphere where we see a different patch in contrast to the monge patch seen in part 2 of lecture 12. The paper deals with a discretetocontinuum approach to the curvatures of flexible membranes. Several conventions are helpful in avoiding confu sions. If input parametrization is given as gaussian curvature of xu,v cosu cosv, cosu sinv, sinu it simply outputs an assembly of three individual cartesian prismatic monge 3d u,v plots and their plotted k but does not refer to meridians and parallels of a single unit sphere surface. We also present a technique for curvilinear orthogonalization of quadratic monge patches that is essential in our derivation and useful in other applications. Reverse engineering of pipe layouts and 3d point set damage.
The second fundamental form directional derivatives in ir3. Both gaussian and mean curva tures have the attractive. The output is the gaussian curvature at each point. Loosely speaking, the curvature of a curve at the point p is partially due to the fact that the curve itself is curved, and partially because the surface is curved. As a result, the gaussian curvature at a local min is nonnegative. Gaussian curvature is regarded as an intrinsic property of space that is.
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