2 edition of On the focal surfaces of the congruences of tangents to a given surface. found in the catalog.
On the focal surfaces of the congruences of tangents to a given surface.
|Other titles||Pamphlets. Mathematics. Quartos. v. 2, no. 9.|
|LC Classifications||QA608 .P38|
|The Physical Object|
|Pagination||1 p. L., 34 p., 1 L.|
|Number of Pages||34|
|LC Control Number||05033246|
Congruences associated with quad meshes are discrete versions of parametrized congruences associated with parametrized surfaces. In particular, the so-called torsal parametrizations are discussed from the integrable systems perspective by Bobenko and Suris .Cited by: 1. The tangent plane to a surface at a given point p is defined in an analogous way to the tangent line in the case of curves. It is the best approximation of the surface by a plane at p, and can be obtained as the limiting position of the planes passing through 3 distinct points on the surface close to p as these points converge to p.
oriented lines, or line congruences, which we view as surfaces in L. Thus we are lead to consider the geometry of immersed surfaces Σ ⊂ L. In the ﬁrst instance, since L can be identiﬁed with the tangent space to the 2-sphere, there is the natural bundle map π: L→ S2. If π|Σ: Σ → S2 is not an immersion, we say that Σ is ﬂat. You should prove that the tangent vector of the curve at $(1,1,1)$ is contained in the tangent plane of the surface, or is orthogonal to the normal vector to the surface at $(1,1,1)$ which is given by the gradient. You calculated the normal vector which is $(-1,-5,1)$.
The centres of those spheres having degenerate tangency with the surface (in Arnold's notation of type A2) form the focal set, and the set of points on the surface where there is a contact of type Author: Keisuke Teramoto. Focal surfaces are known in the eld of line congruences. Line congruences have been introduced in the eld of visualization by Hagen and Pottmann (see, ). Focal surfaces are also used as a surface interrogation tool to analyses the "quality" of the surface before further processing of the surface, for example in a NC-milling operation (see ).
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The Focal Surfaces of the Congruence Formed by the Tangents to the Lines of Curvature of a Given Surface. [Taliaferro, Thomas Hardy ] on *FREE* shipping on qualifying offers. The Focal Surfaces of the Congruence Formed by the Tangents to the Lines of Curvature of a Given : Thomas Hardy Taliaferro.
Congruences of Tangents to a Surface andi Derived Congruences. BY L. EISENHART. Given any famiily of curves upon a surface S; the tangents to these curves form a rectilinear congruence for which S is one of the focal sheets.
The other focal sheet is a determinate surface Si, and upon it there is a family of curves to. Our consideration of the focal surfaces of the congruences of the tangents to the lines of curvature of the surface (X), is based on the following two theorems given by Darboux* and Koenigs.t.
focal surface all surfaces of a congruence are tangent to each other. A surface of a congruence r¡ on which the curves Ct have an envelope is called a principal surface of Tt.
It is known that each envelope curve lies on a focal surface, and that each envelope curve is. A congruence is a surface in the Grassmannian [equation] of lines in projective 3-space. To a space curve C, we associate the Chow hypersurface in [equation] consisting of all lines which intersect C Cited by: 7.
developable which is formed by the tangents of the curves v = const, on S„ constructed at the various points of the same curve u = const. The locus of Pp, the surface S, is therefore the second sheet of the focal surface of the congruence which is composed of the tangents to the curves v = const, on S„.
surfaces X. In the generic case each family consists of the tangent lines to a surface, and these two surfaces Σ,Σ are called the¯ focal surfaces of the congruence. The congruence gives a mapping f: Σ → Σ with the property¯ that the congruence consists of lines which are tangent to both Σ and Σ and¯ join x∈ Σ to ¯x= f(x) ∈ Σ.¯ 2.
In section §0, we give the basic deﬁnitions about congruences and their focal surfaces. In section §1, we obtain the classical invariants of the focal surface. The key new technique in this section is to use the construction given in  of varieties parametrizing inﬁnitely close points of a given variety.
In general, a congruence contains two families of developable surfaces (a developable surface is formed by the tangent to a space curve). Eisenhart recognized that in all known cases, the intersections of these surfaces with the given surface and its image form a net of curves with special properties.
Moutard quadrics of the focal surfaces Sv, S. Segre-Darboux nets on the focal surfaces Sy, Sz. W congruences. Curves of the focal nets iV„, N. Correspondences associated with the focal nets Ny, Nt. Axis congruences and ray congruences.
The congruences aij, yf and the principal congruence rçf. totality of tangents to a surface from points on a given curve is a rectilinear congruence having an infinity of lines through each point of the curve. If we establish a relation between u and v, say (2) v = 0(w) the curves of the congruence whose parameters satisfy this relation depend upon.
Discrete Line Congruences for Shading and Lighting J. Wang1, C. Jiang1, P. Bompas, ing focal surfaces of meshes with known or estimated nor-mals, have been presented by [YYG07]. There are some it is the tangent surface of a parabola).
We are going to make use of these congruences in. On the focal surfaces of the congruences of tangents to a given surface / gaev, serguei. Countries and Regions of Publication (1) View the list below for more details. SOME THEOREMS ON RECTILINEAR CONGRUENCES AND TRANSFORMATIONS OF SURFACES BY CHENKUO PA 1.
Let S be a non-developable analytic surface in ordinary space and L a rectilinear congruence with each of its generators passing through a cor-responding point of S but not tangent to S. For any given surface these con. KoG•13 (),27–S. Gorjanc: Pedal Surfaces of First Order Congruences Proposition4 An (n+1)-ple point exists on Pn+2 n iff a pole P lies on d.
The highest number of such points on Pn+2 n is two only if c n lies in the plane perpendicularto d. PROOF: If a pole P lies on d, then every circle c passing through P, and it is the (n+1)-ple point of Pn+2 n because. The focal net of lines on every focal surface is conjugate.
In a hyperbolic domain, a congruence is the set of common tangents to two focal surfaces; in the elliptic case, a congruence is formed by real common tangents to two conjugate imaginary surfaces; in the parabolic case, a congruence is formed by the tangents to a family of asymptotic lines of a unique focal surface.
Visualization and computation of freeform curves and surfaces Surfaces A paramctnc surface is a mapping X from U c into R 3 of class C'(rè 1). X is called regular if, for all u = (u, u) E U, is an invertible linear mapping.
The two partial derivatives of X in u are denoted by and xo(u). On the Line Congruences In this paper, we found general equations to determine the surfaces of a line congruence. For the convenience of some long term equations, abbreviations have been used for trigonometric functions such as: cos(u):= c(u);sin(u):= s(u) 2.
Fundamentals Line geometry In this section, we introduce some elements of the Cited by: 1. Line congruences can be viewed as surfaces in the four dimensional space of lines, a homo- geneous space of the group of Euclidean motions in R”.
Using the method of moving frames. we calculate first order invariants for line congruences with two real focal surfaces. Introduction Line congruences can be viewed as surfaces in the four dimensional space of lines, a homo- geneous space of the group of Euclidean motions in R3.
Using the method of moving frames. we calculate first order invariants for line congruences with two real focal surfaces.
In Section by: 6. contains n-ple straight line. According to , the pedal surfaces of the ﬁrst type congruence C1 n with respect to a pole P is the image of the plane at inﬁnity given by the (n + 2)-degree inversion with respect to C1 n and any sphere with the center P. Thus, it is an (n + 2)-order surface with n-ple straight line d and contains the.Given conjugate net on a surface, it defines two new conjugate nets called the Laplace transforms of the old net; the transformations are provided by tangents to the parametric lines (see Fig.
1). The discrete version of conjugate net on a surface is given by two-dimensional quadrilateral lattice Cited by: where is a unit principal normal vector and is a nonzero curvature of the curve.
Thus the focal curve or evolute of a curve is the locus of its centers of curvature. Focal surfaces [,] can be constructed in a similar way by using the principal curvature functions of the given surface.