2 edition of **Composite double curves on rational ruled surfaces.** found in the catalog.

Composite double curves on rational ruled surfaces.

L.H Bowen

- 20 Want to read
- 32 Currently reading

Published
**1936** in [n.p.] .

Written in English

**Edition Notes**

Abstract of a thesis, Ph.D. Cornell University.

The Physical Object | |
---|---|

Pagination | [6] p. |

ID Numbers | |

Open Library | OL20573857M |

The rational method is a simple technique for estimating a design discharge from a small watershed. It was developed by Kuichling () for small drainage basins in urban areas. The rational method is the basis for design of many small structures. In particular, theCited by: To find the domain of composite functions, some care is required. Some instructors tell you to find the expression of the composite function and determine the domain from that. However, sometimes information is lost in the algebra and it is easy to get the incorrect answer. The Arc Length of a Plane Curve The Area of a Surface of Revolution Problems in Physics ImproperIntegrals The Mean Value of a Function Trapezoid Rule and Simpson's Formula Chapter Curvature of Plane and Space Curves Curvature of a Plane Curve. The Centre and Radius of Curvature. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. $\begingroup$ The first Google result for "triply ruled surface" gives a book excerpt: " There are no non-planar triply ruled surfaces.".

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A trimmed surface is an implicitly defined surface, which consists of one surface or composite surfaces that are trimmed by one or more boundaries. The chapter presents an assumption that there is one outer boundary, which may enclose zero or more inner boundaries.

To compute the self-intersection curves of a rational ruled surface, we ﬁrst consider the singular points on the surface, since all the pointson the self-intersection curvesarenecessarily singular points. Deﬁnition Let F(x,y,z,w) = 0 be the implicit equation of the rational ruled surface P(s t) in homogeneous form.

THE GAUSSIAN MAP FOR RATIONAL RULED SURFACES JEANNE DUFLOT AND RICK MIRANDA Abstract. In this paper the Gaussian map curve C lying on a minimal rational ruled surface is computed.

It is shown that the corank of O is determined for almost all such curves by the rational surface in which it lies. Composite Bezier Curves 71´ Chapter 13 Rational Bezier and B-Spline Curves ´ Rational Bezier Curves ´ Ruled Surfaces Coons Patches Translational Surfaces Tensor Product Interpolation Bicubic Hermite Patches [4] For a curve ψ(s) lying on a ruled surface, the following statements are well-known: (1): The base curve ψ(s) of the ruled surface Ψ is a geodesic curve if and only if the geodesic curvature κg vanishes.

(2): The base curve ψ(s) of the ruled surface Ψ is an nasymptotic line. geometric ruled surfaces, and the converse is true (Chapter 0). Nagata [10] gave a complete classification of rational ruled surfaces by a method of. elementary transformation, and we studied in [9] a partial classification *) In general, we denote by Pr the projective space of dim r.

degree has inﬁnitely many rational curves (see Theorem 9 and [7]). The idea is to specialize the K3 surface S to a K3 surface S0 with Picard group of rank 2, where some multiple of the polarization can be expressed as a sum of linearly independent classes of smooth rational curves.

The union of these rational curves deforms to an irreducible. Rational Curves (algebraic ratio of two polynomials) NURB (non-uniform rational B-spline) curve combines all features of •a composite curve of degree k polynomials with joining knots in [u j,u j+k+1] •n+1 control points, P 0,P Ruled (lofted) surface (surface created by two curves being blended) Surface of RevolutionFile Size: 2MB.

Automatic and High-Quality Surface Mesh Generation for CAD Models. Guo, F. Ding, X. Jia and D. Yan. Computer Aided Design, Vol.Quaternion Rational Surfaces. Hoffman, X. Jia and H. Wang. Journal of Commutative Algebra,to appear. Survey on the Theory and Applications of µ-Bases for Rational Curves and Surfaces.

(Advanced) Curves and curved surfaces provide a convenient mathematical means of describing a geometric model. Instead of using drawings, metal strips, or clay models, designers can use these mathematical expressions to represent the surfaces used on airplane wings, automobile bodies, machine parts, or other smooth curves and surfaces.

Output: the message “ V is not a rational ruled surface ” or a proper parametrization P of “ the rational ruled surface V in the standard reduced form ”.

If deg x 3 (f) = 0 (similarly if deg x 1 (f) = 0 or deg x 2 (f) = 0), compute (p (t 1), q (t 1)) a parametrization of the curve defined by the polynomial f (x 1, x 2) = 0, and Return P (t ¯) = (p (t 1), q (t 1), t 2) ∈ K (t ¯) 3 “ is a proper parametrization ”.Cited by: Conversion of a Composite Trimmed Bézier Surface into Composite Bézier Surfaces Multivariate Model Building with Additive, Interaction and Tensor Product Thin Plate Splines Recursion Relations for 4x4 Determinants Related to Rational Cubic Bézier Curves Book Edition: 1.

A rational curve on a rational surface such that the unit normal vector field of the surface along this curve is rational will be called a curve providing Pythagorean surface normals (or shortly a. the existence of rational curves in the generic K3 surface of degree 2g 2.

Proof. We exhibit a K3 surface S0 containing two smooth rational curves C1 and C2 meeting transversally at g+ 1 points, such that the class f = [C1 + C2] is primitive.

We then deform C1 [ C2 to an irreducible rational curve in a nearby polarized K3 surface. over Spec |, or over a smooth projective curve B. If the latter happens the surface Sis called ruled and, if S0= P2 or B˘=P1, it is called rational. A rational surface has p g = q= 0 since the latter are birational invariants.

InGuido Castelnuovo tried to prove that the. COUNTING CURVES OF ANY GENUS ON RATIONAL RULED SURFACES RAVI VAKIL Abstract. In this paper we study the geometry of the Severi va-rieties parametrizing curves on the rational ruled surface F n.

We compute the number of such curves through the appropriate num-ber of ﬁxed general points on F n (Theorem ), and the number of such curves which. They studied curves on ruled surfaces by choosing curves as cylindirical helices and Bertrand curves [6].

In their another paper [7], the notions of helices generalized to slant helices and. In geometry, a surface S is ruled if through every point of S there is a straight line that lies on S. Examples include the plane, the curved surface of a cylinder or cone, a conical surface with elliptical directrix, the right conoid, the helicoid, and the tangent developable of a smooth curve in space.

A ruled surface can be described as the set of points swept by a moving straight line. For example, a cone is. Volume 2 (Technical Handbook) Georgia Stormwater Management Manual CHAPTER 3 STORMWATER HYDROLOGY Table of Contents SECTION METHODS FOR ESTIMATING STORMWATER RUNOFFFile Size: 1MB.

roughly speaking, we can divide them in rational curves (i. e., genus 0 Riemann surfaces), elliptic curves (i. genus 1 Riemann surfaces) and curves of general type (i. Riemann surfaces with genus at least 2). There are some crucial facts in the theory of Riemann surfaces: rst ofFile Size: KB.

Evolute, Involute, Parallel Curves The Rotation Number Theorem Convex Curves. Then if the two curves are take a rational normal curve of order n placed in (1, 1) correspondence the ruled surface formed by joining pairs of corresponding points is of order n%\ it is rational and no two of its ruled surfaces of order distinct.

image of the ruled surface X= P(f K C) over Bunder the linear system jO X(1)j. (3) Conversely, if C is a double cover of a curve of genus +3 2, then C can is contained in a surface of degree g+. Remark When = 1, we recover the theorem of del Pezzo [D] and Fujita [F1] that a curve of g>10 which lies on a surface of degree g 1 is Size: KB.

Devoted to the control of all rational curves and surfaces in view of their applications to computer science, CAD, CAGD and CAM. Introduces and analyzes the new concept of massic vectors to dominate all rational curves and surfaces, including Bezier curves and surfaces. The algorithmic and geometrical aspects are examined and methods for Cited by: Developable surfaces are surfaces in Euclidean space which ‘can be made of a piece of paper’, i.e., are isometric to part of the Euclidean plane, at least locally.

If we do not assume sufficient smoothness, the class of such surfaces is too large to be useful — if includes all possible aways of arranging crumpled paper in space. and (2) Implicit algebraic (IA).

Non-Uniform Rational B-Spline (NURBS) curves and surfaces can be subdivided into RPP curves and surfaces, and analyzed in a similar manner. A detailed treatment of intersection problems including general procedural curves and surfaces can be found in [31,30].

Among all types of intersections, the surface to surfaceFile Size: KB. Compatible complex structures on rational ruled surfaces In this section, we describe the stratiﬁcation on the space J ω of compatible almost complex structures on a rational ruled surface, previously studied in [A], [AM], and [Mc1], and we use results from Section 2 to show that it induces an analogous stratiﬁcation on the.

Preliminary Mathematics The B-Spline Curve The Bézier Curve Rational Curves Interpolation Surfaces Two " diskettes with accompanying paperback book ISBN / Price: $ 25% DISCOUNT COUPON. Present this coupon to Morgan Kaufmann Publishers at Booth # and receive a 25% discount on your copy.

Interactive Curves and Surfaces:File Size: KB. Before we jump into the NURBS programming, I like to emphasize one more time, a NURBS curve is actually one or many pieces of Bezier curve(s) that connected with each other in one by one order and under certain math bonding rule.

Two or more Bezier curves connected one by one in Microsoft’s terminology is called Poly-Bezier/5(34). Ruled Surfaces and Rational Fibrations. Ask Question Asked 6 years, 11 months ago. S' \rightarrow C$ is a birationally ruled surface, but one of its fibres is a reducible curve, so it is not geometrically ruled.

Actually, this last example also highlights a problem with your idea in the final paragraph: blowing up points unfortunately can't. APPROXIMATING CURVES ON REAL RATIONAL SURFACES 3 isotopy classes, hence the end result (S 1,L 1)#(RP2,L) is unique.

This is the only case that we use. Deﬁnition 5 (Intersection numbers). The intersection number of two algebraic curves C 1 C 2 on a smooth, projective surface is the intersection number of the underlying complex curves.

Overview of the talk I The problem: existence of rational curves on a K3 surface I The conjecture: in nitely many rational curves on an algebraic K3 I The technique: lifting from a special K3 to a general K3 I The new approach: of Bogomolov-Hassett-Tschinkel, using the jumping of Picard ranks I Theorem (L-Liedtke): When ˆ(X) is odd, then X contains in nitely.

4)-CURVES ON RATIONAL SURFACE MARÍA MARTÍ SÁNCHEZ (Received Augrevised Febru ) Abstract We study rational surfaces having an even set of disjoint (erties of the surface 4)-curves. The prop-S obtained by considering the double cover branched on the even set are studied.

It is shown, that contrarily to what happens for. The aim of the present monograph is to give a systematic exposition of the theory of algebraic surfaces emphasizing the interrelations between the various aspects of the theory: algebro-geometric, topological and transcendental.

To achieve this aim, and still remain inside the limits of the allotted space, it was necessary to confine the exposition to topics which are absolutely fundamental. This book offers a wide-ranging introduction to algebraic geometry along classical lines.

It consists of lectures on topics in classical algebraic geometry, including the basic properties of projective algebraic varieties, linear systems of hypersurfaces, algebraic curves (with special emphasis on rational curves), linear series on algebraic curves, Cremona transformations, rational surfaces.

An~isophote on a surface is a curve consisting of surface points whose normals form a constant angle with some fixed vector.

Choosing an angle equal to $\pi/2$ we obtain a special instance of a~isophote -- the so called contour curve. While contours on rational ruled surfaces are rational curves, this is no longer true for the isophotes.

The whole curve, that is the set of its complex point is, from the topological point of view a surface. In particular, the nonsingular complex projective algebraic curves are called Riemann surfaces. The points of a curve C with coordinates in a field G are said to be rational over G and can be denoted C(G)).

Abstract. The μ-bases of rational curves and surfaces are newly developed tools which play an important role in connecting parametric forms and implicit forms of curves and r, exact μ-bases may have high degree with complicated rational coefficients and are often hard to compute (especially for surfaces), and sometimes they are not easy to use in geometric modeling and Cited by: 1.

group of a rational ruled surface, extending results of Abreu and McDuff. INTRODUCTION The work of Gromov on J-holomorphic curves [Gr] has provided tools for un-derstanding the topology of the group of symplectomorphisms of certain four-dimensional symplectic manifolds.

Gromov began the study of. Any birational morphism between smooth projective surfaces is a composite of blow-downs to points. Any birational map between smooth projective surfaces is a composite of blow-ups and blow-downs. Theorem 2. There are 3 species of “pure-bred” surfaces: (Rational): For these surfaces the internal birational geometry is very compli-File Size: KB.

IV Surfaces with MAPLE.- 19 Surfaces in Space.- What Is a Surface?.- Regular Parametrized Surface.- Methods of Generating Surfaces.- Tangent Planes and Normal Vectors.- The Osculating Paraboloid and a Type of Smooth Point.- Singular Points on Surfaces.- 20 Some Classes of Surfaces.- Algebraic Surfaces.- Author: Vladimir Rovenski.Cayley nodal cubic surface, a certain cubic surface with 4 nodes; Cayley's ruled cubic surface; Clebsch surface or Klein icosahedral surface; Fermat cubic; Monkey saddle; Parabolic conoid; Plücker's conoid; Whitney umbrella; Rational quartic surfaces.

Châtelet surfaces; Dupin cyclides, inversions of a cylinder, torus, or double cone in a sphere; Gabriel's horn.Zaslow [Y-Z] to express the number of rational curves on a K3 surface. Projective K3 surfaces fall into countably many families (F g) g≥1; a surface in F g admits a g-dimensional linear system of curves of genus g.

A na¨ıve count of constants suggests that such a system will contain a positive number, say n(g), of rational (highly singular.