Fat conic section and fat conic spline are defined. With well established properties of fat conic splines, the problem of approximating a ruled surface by a tangent smooth cone spline can then be changed as the problem of fitting a plane fat curve by a fat conic spline. Moreover, the fitting error between the ruled surface and the cone spline can be estimated explicitly via fat conic spline fit...

node curve is fundamental in this theory. A number of Cremona's theorems on ruled surfaces with straight line directrices are generalized to apply to all ruled surfaces. Br. Wilczynski gives both analytic and synthetic proofs of these theorems. The third covariant furnishes another congruence associated with a given surface, and in particular a third ruled surface associated with the original o...

For a Riemannian submersion from a simple compact Lie group with a bi-invariant metric, we prove the action of its holonomy group on the fibers is transitive. As a step towards classifying Riemannian submersions with totally geodesic fibers, we consider the parameterized surface induced by lifting a base geodesic to points along a geodesic in a fiber. Such a surface is “doubly ruled” (it is rul...

Chen, Sederberg, and Zheng introduced the notion of a μ-basis for a rational ruled surface in Chen et al. (2001) and showed that its resultant is the implicit equation of the surface, if the parametrization is generically injective. We generalize this result to the case of an arbitrary parametrization. We also give a new proof for the corresponding theorem in the curve case and treat the repara...

We study higher syzygies of a ruled surface X over a curve of genus g with the numerical invariant e. Let L ∈ PicX be a line bundle in the numerical class of aC0 + bf . We prove that for 0 ≤ e ≤ g − 2, L satisfies Property Np if a ≥ p + 2 and b − ae ≥ 3g − e + p and for e ≥ g − 1, L satisfies Property Np if a ≥ p + 2 and b − ae ≥ 2g + 1 + p. By using these facts, we obtain Mukai type results. F...

In this thesis we are concerned with the geometric matching of shapes. We consider the local and rigid alignment of point clouds and the reconstruction of ruled surfaces from point clouds within an active shape framework. The emerging nonlinear and occasionally non-smooth optimization problems center around minimizations of the squared and unsigned distance function, respectively. Several appli...

Introduction: Through this paper, a geometrically ruled surface, or simply a ruled surface, will be a P-bundle over a smooth curve X of genus g. It will be denoted by π : S = P(E0)−→X and we will follow the notation and terminology of R. Hartshorne’s book [3], V, Section 2. We will suppose that E0 is a normalized sheaf and X0 is the section of minimum self-intersection that corresponds to the s...

This paper presents a positioning strategy for flank milling ruled surfaces. It is a modification of a positioning method developed by Bedi et al. [1]. A cylindrical cutting tool is initially positioned tangential to the two boundary curves on a ruled surface. Optimization is used to move these tangential points to different curves on the ruled surface to reduce the error. A second optimization...

We present Wide Ruled, an authoring tool for the creation of generative stories. It is based on the Universe author-goal-based model of story generation, and extends this model by providing support for reader interactivity, episodic memory elements, and an extensive graphical interface that is aimed at authors with little or no experience with computer programming and artificial intelligence pl...