Secant Lines Research Articles

A characterization of the family of secant or external lines of an ovoid of PG$(3,q)$

In this paper we characterize the family of secant lines of an ovoid of PG$(3,q)$ and the family of external lines to an ovoid of PG$(3,q)$.

Focal Loci in G(1,n)

We introduce the different focal loci (focal points, planes and hyperplanes) of (n - 1)-dimensional families (congruences) of lines in P-n and study their invariants, geometry and the relation among them. We also study some particular congruences whose focal loci have special behaviour, namely (n - 1)-secant lines to an (n - 2)-fold and (n - 1)-tangent lines to a hypersurface. In case n = 4 we also give, under some smoothness assumptions, a classification result for these congruences.

Smooth projective varieties with extremal or next to extremal curvilinear secant subspaces

We intend to give a classification of smooth nondegenerate projective varieties admitting extremal or next to extremal curvilinear secant subspaces. Gruson, Lazarsfeld and Peskine classified all projective integral curves with extremal secant lines. On the other hand, if a locally Cohen-Macaulay variety X ⊂ P n+e of degree d meets with a linear subspace L of dimension β at finite points, then length (X ∩ L) < d - e + β as a finite scheme. A linear subspace L for which the above length attains maximal possible value is called an extremal secant subspace and such L for which length (X ∩ L) = d-e+β-1 is called a next to extremal secant subspace. In this paper, we show that if a smooth variety X of degree d ≥ 6 has extremal or next to extremal curvilinear secant subspaces, then it is either Del Pezzo or a scroll over a curve of genus g ≤ 1. This generalizes the results of Gruson, Lazarsfeld and Peskine (1983) for curves and the work of M-A. Bertin (2002) who classified smooth higher dimensional varieties with extremal secant lines. This is also motivated and closely related to establishing an upper bound for the Castelnuovo-Mumford regularity and giving a classification of the varieties on the boundary.

Secant lines in a preliminary study of phonetic reduction of / ɪ / and /ε /

Two subjects from the X-Ray Microbeam Speech Production Database were examined in their production of the vowels /I/ and /epsilon/ in alveolar and dental consonant contexts. Secant lines, or first-order splines, between the three most anterior pellets were examined at vowel critical times. These critical times were zero crossings in the tangential acceleration of the midpoints of the secant lines. We expected and found, in general, that vowel reduction occurred as a function of vowel duration in measures of the secant line midpoint-to-palate distance and secant line orientation at vowel critical times. The shorter the vowel, the smaller the distance of the secant line midpoints to the palate and the less downward the orientation of the secant lines at the vowel critical times. Phonetic reduction was also apparent in the formant frequencies. There were differences between the speakers in terms of the range of vowel duration and degree of reduction. The subjects differed in the functional parts of the tongue spanned by the secant lines and the shape of their palates. These differences were factors in the observed relations between formant frequencies and the articulatory, secant line measures for each subject.

Varieties with one apparent double point

The number of apparent double points of a smooth, irreducible projective variety X of dimension n in P 2n+1 is the number of secant lines to X passing through the general point of P 2n+1 . This classical notion dates back to Severi. In the present paper we classify smooth varieties of dimension at most three having one apparent double point. The techniques developed for this purpose allow us to treat a wider class of projective varieties.

Threefolds in ℙ5 with One Apparent Quadruple Point

In this article we classify all the smooth threefolds in ℙ5 with an apparent quadruple point provided that the family of its 4-secant lines is an irreducible (first order) congruence. This is sufficient to conclude the classification of all the smooth codimension two varieties in ℙ n with one apparent (n − 1)-point and with irreducible family of (n − 1)-secant lines.

Tongue position and orientation for front vowels in the X-Ray Microbeam Speech Production DataBase

The positions and orientations of the secant lines between pellets will be examined for front vowels of four talkers in the X-Ray Microbeam Speech Production Database. The effect of vowel height and palate shape will be examined, as will the contextual effects of neighboring stop, fricative and nasal segments. This work is part of a project to describe tongue motion in terms of secant line kinematics. Preliminary results suggest that tongue blade orientation for high front vowels is determined largely by palate shape.

A description of midsagittal tongue movement in terms of secant line kinematics

Tongue shape is a major determiner of the shape of the air channel in the vocal tract, and, hence, the acoustic signal during speech production. Flesh-point movement is a way that midsagittal tongue motion in speech production is examined, such as with the EMMA system and the x-ray microbeam system. We propose to transform the Cartesian coordinates of the flesh-points into one that specifies the position of the centers of the secant lines between each neighboring pair of flesh points, as well as the length and orientation of each secant line in this coordinate system, motion is considered in terms of translational, rotational, and stretching motions of the secant lines, with the former specifying gross motion and the latter two specifying local motion. Utterances from the X-Ray Microbeam Speech Production Database involving consonants with alveolar place of articulation are examined for kinematic differences resulting from differences in manner, vowel context, and syllable position. A further refinement of the secant line coordinates will be offered in terms of midsagittal secant line position and orientation relative to the palate and pharynx. This coordinate system should be helpful for interpreting flesh-point kinematics with regard to acoustic output.

On a theorem of Severi

P 2r+1, on the projection we will obtain a finite number δ(X) of improper nodes, i.e. non-normal double points whose tangent cones break into two r-planes. The number δ(X) is called the number of apparent double points ofX and it is equal to the number of secant lines to X going through a general point of P2r+1. If X ⊂ P3 is a non-degenerate curve, then it has necessarily apparent double points because the secant variety of X fills the whole space and δ(X) is then the difference between the arithmetic genus of the projection of X from a general point and the geometric genus ofX; so ifX has degreed, then δ(X) = (d−1)(d−2) 2 − g(X) > 0. On the other hand it is an immediate consequence of Castelnuovo’ s bound for the genus of a curve that the only smooth curve in P3 having one apparent double point is the rational normal curve of degree 3. In 1901 Severi proved that the only surface in P5 having degenerate secant variety, i.e. the only surface that can be projected isomorphically in P4 (or equivalently without apparent double points), is the Veronese surface in P5. In the same paper Severi claims that the smooth surfaces in P5 having one apparent double point are only the rational normal scroll(s) of degree 4 and the Del Pezzo surface of degree 5 (see [19]). Unfortunately Severi gives a proof which works provided the surface S has not too many trisecants. This gap was remarked first by Ciliberto and Sernesi. Here we prove that the surfaces in P5 having one apparent double point are the rational normal scrolls S(2, 2) and S(1, 3) of degree 4 and the Del Pezzo surface of degree 5.

On Unitals ith Many Baer Sublines

We identify the points of \pg(2,q) ith the directions of lines in \gf(q^3), viewed as a 3-dimensional affine space over \gf(q). Within this frameork we associate to a unital in \pg(2,q) a certain polynomial in to variables, and show that the combinatorial properties of the unital force certain restrictions on the coefficients of this polynomial. In particular, if q=p^2 herep is prime then e show that a unital is classical if and only if at least (q-2)\sq secant lines meet it in the points of a Baer subline.

On the Hilbert Function of Fat Points on a Rational Normal Cubic

In this paper we find an algorithm which computes the Hilbert function of schemesZof “fat points” in P3whose support lies on a rational normal cubic curveC. The algorithm shows that the maximality of the Hilbert function in degreetis related to the existence of fixed curves (eitherCitself or one of its secant lines) for the linear system of surfaces of degreetcontainingZ.

Characterizations of Buekenhout-Metz unitals

We give a characterization of the Buekenhout-Metz unitals in PG(2, q 2), in the cases that q is even or q=3, in terms of the secant lines through a single point of the unital. With the addition of extra conditions, we obtain further characterizations of Buekenhout-Metz unitals in PG(2, q 2), for all q. As an application, we show that the dual of a Buekenhout-Metz unital in PG(2, q 2) is a Buekenhout-Metz unital.

Introducing the Derivative through the Iteration of Linear Functions

The traditional approach to introducing the study of the derivative is through the tangent line as the limiting behavior of secant lines. This geometric discussion leads to the definition of the derivative, after which the differentiation formulas are developed and tangent lines can finally be computed. Unfortunately, this approach seldom gives the students any idea of what tangent lines actually are and why they might be of interest. Today's computer-graphics capabilities, however, allow exciting new ways to introduce the derivative. Topics from dynamical systems serve as interesting applications that are accessible even to students in precalculus courses (see, e.g., Devaney [1990]). Coupled with modern graphing technology, the iteration of functions, especially linear functions, not only allows an instructor to illustrate ideas from an area of great current mathematical interest but also affords an excellent opportunity to introduce the derivative and tangent lines.

Secant Spaces to Curves

A classical question in algebraic geometry is whether a given projection of a projective space induces an isomorphism on a given closed subvariety. To answer it, one investigates secant lines to the subvariety. There has been a lot of recent activity in this field ([12], [14],[18], [21], [23]): see [14] and [12] for references).An obvious generalization of the secant lines is provided by the secant r-planes, which intersect a given closed subvariety in r + 1 linearly independent points. The closure of the set of these secant r-planes is the secant variety, and the aim of this paper is to determine its rational equivalence class in the case of curves. There is an extensive classical literature about this problem.

Finite Linear Spaces with a Regularity Condition on the Transversals of Two Secant Lines

Projective and affine planes are, besides the degenerate projective planes, the only known examples of finite linear spaces for which there is a non-negative integer t such that for any two intersecting lines L, L′ of S and any point p outside L ∪ L′ , there are exactly t transversals of L and L′ passing through p. We prove that all finite linear spaces satisfying the above condition are either degenerate projective planes or Steiner systems S (2, k , v ) in which t and k must satisfy some rather restrictive arithmetic conditions.

Fixed Point Iteration-An Interesting Way to Begin a Calculus Course

(1981). Fixed Point Iteration—An Interesting Way to Begin a Calculus Course. The Two-Year College Mathematics Journal: Vol. 12, No. 1, pp. 2-7.

Some properties and applications of Hermitian varieties in a finite projective space PG( N, q2) in the construction of strongly regular graphs (two-class association schemes) and block designs

The section of a non-degenerate Hermitian variety V N−1 by a polar hyperplane in PG( N, q 2) and the number of u-flats, 0 ≤ u ≤ [ (N − 1) 2 ] , contained in a V N−1 are derived. From the incidence of external points, tangent lines and secant lines with respect to non-degenerate Hermitian varieties in PG(2, q 2) and PG(3, q 2) and the incidence of generators and tangent planes of a nondegenerate Hermitian variety in PG(3, q 2), three different families of strongly regular graphs (two-class association schemes) and block designs are derived.