Developable surfaces along frontal curves on embedded surfaces
TL;DR: In this article, two types of developable surfaces along a frontal curve on an embedded surface in the Euclidean 3-space are considered. But the frontal curve may have singular points.
Abstract: We consider two types of developable surfaces along a frontal curve on an embedded surface in the Euclidean 3-space. One is called the osculating developable surface, and the other is called the normal developable surface. The frontal curve may have singular points. We give new invariants of the frontal curve which characterize singularities of the developable surfaces. Moreover, a frontal curve is a contour generator with respect to an orthogonal projection or a central projection if and only if one of these invariants is constantly equal to zero.
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TL;DR: In this paper, by using moving frame along frontal of Legendre curve, the authors define frontal partner curves on unit sphere S2. And they give the relationships between curvatures of the Legendre curves and frontal partners curves.
Abstract: In this study, by using moving frame along frontal of Legendre curve, we define frontal partner curves on unit sphere S2. We give the relationships between curvatures of Legendre curves and frontal partner curves are strengthen by an example.
1 citations
TL;DR: In this paper , a free-form surface flattening algorithm that minimizes geometric deformation by combining the advantages of the geometric flattening method and the mechanical energy flattening was proposed.
Abstract: Based on the analysis of the advantages and disadvantages of existing surface flattening algorithms, this paper proposes a free-form surface flattening algorithm that minimizes geometric deformation by combining the advantages of the geometric flattening method and the mechanical energy flattening method to address the problems of many iterations, large changes in convergence, and weak visualization of deformation. The point cloud surface is meshed using the triangular slice search method, and the 3D surface mesh is wrapped around the surface using geometric mapping relationships. The initial correction and deformation analysis of the ring flatten graphic are carried out according to the average flattening error, and a global geometric deformation energy model is established to obtain the energy-minimizing unfolding conditions and optimize the initial flattening graph by iterative optimization. The verification of the algorithm shows that the algorithm is general, has good robustness, high flattening accuracy, and visualizes the flattening deformation. The method is suitable for some design occasions of machining, such as the flattening calculation of automobile outer cladding parts and sheet metal parts. It is especially suitable for the flattening calculation of ring-like parts, and is a good guide for the design calculation and stamping process of sheet-metal-like parts.
1 citations
TL;DR: In this article , the authors define three different rotation-minimizing frames by rotating the moving frame around its coordinate axis vectors and derive the Darboux vectors associated with these frames.
Abstract: In this paper, we define three different rotation-minimizing frames by rotating the moving frame around its coordinate axis vectors. Darboux vectors associated with these frames are obtained as special cases of the Darboux vector associated with the moving frame. Using these Darboux vectors and the moving frame we define six different developable surfaces. For each of these surfaces we give two invariants of curves on these surfaces to characterize their singularities. Moreover, we show that the base curves of these surfaces are contour generators with respect to an orthogonal projection or a central projection if and only if one of the invariants given for each surface is constantly equal to zero. Examples are provided to illustrate our theorems and results.
25 Oct 2022
TL;DR: In this article , it was shown that the equation of the principal surfaces is a multiple of the equations of the developable surfaces, and that the multiplicative factor is associated to the singular set of ξ .
Abstract: . This paper is a first step in order to extend Kummer’s theory for line congruences to the case { x, ξ } , where x : U → R 3 is a smooth map and ξ : U → R 3 is a proper frontal. We show that if { x, ξ } is a normal congruence, the equation of the principal surfaces is a multiple of the equation of the developable surfaces, furthermore, the multiplicative factor is associated to the singular set of ξ .
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01 Jan 2000TL;DR: The epipolar parametrization is reviewed and shows how the degenerate cases can be used to recover surface geometry and unknown viewer motion from apparent contours of curved surfaces.
Abstract: For smooth curved surfaces the dominant image feature is the apparent contour, or outline. This is the projection of the contour generator, the locus of points on the surface which separate visible and occluded parts. The contour generator is dependent of the local surface geometry and the viewpoint. Each viewpoint will generate a different contour generator. This paper addresses the problem of recovering the three–dimensional shape and motion of curves and surfaces from image sequences of apparent contours.
For known viewer motion the visible surfaces can then be reconstructed by exploiting a spatio–temporal parametrization of the apparent contours and contour generators under viewer motion. A natural parametrization exploits the contour generators and the epipolar geometry between successive viewpoints. The epipolar parametrization leads to simplified expressions for the recovery of depth and surface curvatures from image velocities and accelerations and known viewer motion.
The parametrization is, however, degenerate when the apparent contour is singular since the ray is tangent to the contour generator and at frontier points when the epipolar plane is a tangent plane to the surface. At these isolated points the epipolar parametrization can no longer be used to recover the local surface geometry. This paper reviews the epipolar parametrization and shows how the degenerate cases can be used to recover surface geometry and unknown viewer motion from apparent contours of curved surfaces. Practical implementations are outlined.
263 citations
TL;DR: In this article, it was shown that any complete flat front admits only cuspidal edges and swallowtails, provided that the singularity of the front is not rotationally symmetric.
Abstract: It is well-known that the unit cotangent bundle of any Riemannian manifold has a canonical contact structure. A surface in a Riemannian 3-manifold is called a front if it is the projection of a Legendrian immersion into the unit cotangent bundle. We give easily computable criteria for a singular point on a front to be a cuspidal edge or a swallowtail. Using this, we prove that generically flat fronts in hyperbolic 3-space admit only cuspidal edges and swallowtails. We also show that any complete flat front (provided it is not rotationally symmetric) has associated parallel surfaces whose singularities consist of only cuspidal edges and swallowtails.
213 citations
TL;DR: In this paper, the singularities of spacelike maximal surfaces in Lorentz-Minkowski 3-space generically consist of cuspidal edges, swallowtails and cross caps.
Abstract: We show that the singularities of spacelike maximal surfaces in Lorentz–Minkowski 3-space generically consist of cuspidal edges, swallowtails and cuspidal cross caps. The same result holds for spacelike mean curvature one surfaces in de Sitter 3-space. To prove these, we shall give a simple criterion for a given singular point on a surface to be a cuspidal cross cap.
154 citations
TL;DR: In this paper, the curvature of a framed curve is defined, similarly to the curvatures of a regular curve and of a Legendre curve in the unit tangent bundle.
Abstract: Abstract A framed curve in the Euclidean space is a curve with a moving frame. It is a generalization not only of regular curves with linear independent condition, but also of Legendre curves in the unit tangent bundle. We define smooth functions for a framed curve, called the curvature of the framed curve, similarly to the curvature of a regular curve and of a Legendre curve. Framed curves may have singularities. The curvature of the framed curve is quite useful to analyse the framed curves and their singularities. In fact, we give the existence and the uniqueness for the framed curves by using their curvature. As applications, we consider a contact between framed curves, and give a relationship between projections of framed space curves and Legendre curves.
52 citations
TL;DR: In this article, the uniqueness and singularities of osculating developable surfaces along a curve on the surface were investigated and two new invariants of curves on a surface which characterize these singularities were discovered.
Abstract: We consider a developable surface tangent to a surface along a curve on the surface. We call it an osculating developable surface along a curve on the surface. We investigate the uniqueness and the singularities of such developable surfaces. We discover two new invariants of curves on a surface which characterize these singularities. As a by-product, we show that a curve is a contour generator with respect to an orthogonal projection or a central projection if and only if one of these invariants vanishes. We also classify the singularities. This is a joint work with Saki Otani.
33 citations