On elliptic quartic curves with assigned points and chords

1931 ◽  
Vol 27 (1) ◽  
pp. 20-23 ◽  
Author(s):  
W. G. Welchman
Keyword(s):  
Finite Number ◽  
Three Dimensions ◽  
Number Of Solutions ◽  
Quartic Curve ◽  
Pass Through ◽  
Quartic Curves

1. The problem which I propose to solve is that of finding the number of quartic curves of intersection of two quadrics which pass through p points and have q lines as chords, where p + q = 8. There are ∞16 elliptic quartics in space; to contain a line as a chord is two conditions, and to pass through a point is two conditions, so we should expect a finite number of solutions. Throughout this paper I shall refer to an elliptic quartic curve in space of three dimensions simply as a “quartic.” I shall denote the number of solutions for a particular value of p by np.

On Five Lines, in Space of Four Dimensions, which lie upon a, Quadric Threefold and the Normal Quartic Curves of which they are Chords

1924 ◽  
Vol 22 (2) ◽  
pp. 189-199
Author(s):  
F. Bath
Keyword(s):  
Three Dimensions ◽  
Apparent Contradiction ◽  
Four Dimensions ◽  
Quartic Curve ◽  
Lines In Space ◽  
Quadric Threefold ◽  
Quartic Curves

The connexion between the conditions for five lines of S4(i) to lie upon a quadric threefold,and (ii) to be chords of a normal quartic curve,leads to an apparent contradiction. This difficulty is explained in the first paragraph below and, subsequently, two investigations are given of which the first uses, mainly, properties of space of three dimensions.

Three related quartic curves in four dimensions

1932 ◽  
Vol 28 (4) ◽  
pp. 403-415 ◽  
Author(s):  
H. G. Telling
Keyword(s):  
Linear System ◽  
Simple Proof ◽  
Differential Method ◽  
Four Dimensions ◽  
Quartic Curve ◽  
Pass Through ◽  
Quartic Curves

1. It has been shown by Pieri, and independently by James, that the trisecant planes of a quartic curve in [4] which meet a line meet also another quartic curve intersecting the former in six points. James generalises the theorem and shows that trisecant planes of a quartic curve C, which meet a quartic curve C1 having six points in common with C, also meet another quartic curve C2, and that the relation between the three curves is symmetrical. The object of this note is to give a simple proof of this theorem and to discuss the representation by which this proof is effected. The method used is analogous to that of Pieri; an interesting differential method is adopted by C. Segre who shows that the foci of the first and second orders, of any linear system of ∞2 planes in [4] of which two planes pass through a point, determine a conic and five points respectively, in any plane of the system: the trisecant planes of C meeting C1 give a particular case of such a system.

scholarly journals A plane quartic curve with twelve undulations

1945 ◽  
Vol 35 ◽  
pp. 10-13 ◽  
Author(s):  
W. L. Edge
Keyword(s):  
Homogeneous Coordinates ◽  
Quartic Curve ◽  
Base Points ◽  
Octahedral Group ◽  
Quartic Form ◽  
Image Position ◽  
Quartic Curves ◽  
Set Of Points

The pencil of quartic curveswhere x, y, z are homogeneous coordinates in a plane, was encountered by Ciani [Palermo Rendiconli, Vol. 13, 1899] in his search for plane quartic curves that were invariant under harmonic inversions. If x, y, z undergo any permutation the ternary quartic form on the left of (1) is not altered; nor is it altered if any, or all, of x, y, z be multiplied by −1. There thus arises an octahedral group G of ternary collineations for which every curve of the pencil is invariant.Since (1) may also be writtenthe four linesare, as Ciani pointed out, bitangents, at their intersections with the conic C whose equation is x2 + y2 + z2 = 0, to every quartic of the pencil. The 16 base points of the pencil are thus all accounted for—they consist of these eight contacts counted twice—and this set of points must of course be invariant under G. Indeed the 4! collineations of G are precisely those which give rise to the different permutations of the four lines (2), a collineation in a plane being determined when any four non-concurrent lines and the four lines which are to correspond to them are given. The quadrilateral formed by the lines (2) will be called q.

A series of rational loci with one apparent double point

1931 ◽  
Vol 27 (3) ◽  
pp. 399-403 ◽  
Author(s):  
D. W. Babbage
Keyword(s):  
Finite Number ◽  
Double Point ◽  
General Point ◽  
Pass Through

1. A locus Vn, of dimension n, in [2n + 1], for example a curve in [3] or a surface in [5], has ∞ 2n chords, of which a finite number pass through a general point of the space.

Stable Lattices

1953 ◽  
Vol 5 ◽  
pp. 261-270 ◽  
Author(s):  
Harvey Cohn
Keyword(s):  
Convex Body ◽  
Finite Number ◽  
Quadratic Forms ◽  
Convex Bodies ◽  
Lattice Points ◽  
Three Dimensions ◽  
General Convex ◽  
Critical Lattice ◽  
The Absolute

The consideration of relative extrema to correspond to the absolute extremum which is the critical lattice has been going on for some time. As far back as 1873, Korkine and Zolotareff [6] worked with the ellipsoid in hyperspace (i.e., with quadratic forms), and later Minkowski [8] worked with a general convex body in two or three dimensions. They showed how to find critical lattices by selection from among a finite number of relative extrema. They were aided by the long-recognized premise that only a finite number of lattice points can enter into consideration [1] when one deals with lattices “admissible to convex bodies.”

scholarly journals Construction of Cooling Corridors with Multiscenarios on Urban Scale: A Case Study of Shenzhen

2020 ◽  
Vol 12 (15) ◽  
pp. 5903
Author(s):  
Jiansheng Wu ◽  
Si Li ◽  
Nan Shen ◽  
Yuhao Zhao ◽  
Hongyi Cui
Keyword(s):  
Urban Heat Island ◽  
Three Dimensions ◽  
Cooling Effect ◽  
Control Line ◽  
Heat Island ◽  
Rapid Urbanization ◽  
Administrative Region ◽  
Total Length ◽  
Urban Heat ◽  
Pass Through

Under the background of rapid urbanization, the urban heat island (UHI) effect is becoming increasingly significant. It is very important for the sustainable development of cities to carry out quantitative research on the mitigation of the UHI effect at an urban scale. Taking Shenzhen as an example, this paper puts forward a method for building a cooling corridor for the city with multiscenarios based on the theory of ecological security pattern (ESP), which can realize quantitative planning of the spatial layout of urban green infrastructure (UGI) to alleviate the UHI effect. In this study, cooling sources are identified from the three dimensions of habitat quality, landscape connectivity, and the capacity to provide cooling ecosystem services. The cooling corridors that are superior at cooling, isolation, and ventilation are selected and optimized. The results show that the identified ecological cooling source area accounts for 33.18% of the total area of Shenzhen, and more than 85% of the area falls within the scope of the basic ecological control line of Shenzhen. There are 48 cooling corridors with a total length of 289.17 km in the cooling priority scenario, which mostly pass through the high-temperature and subhigh-temperature areas of each administrative region and city, providing a good cooling effect but poor feasibility. There are 48 corridors with a total length of 326.66 km in the isolation priority scenario, which mostly pass through the administrative region boundary and have a weak connection with the urban heat island, avoiding the built-up areas with strong human activities. As consequence, cooling is relatively achievable, but its effect is not ideal. There are 47 corridors with a total length of 368.06 km in the ventilation priority scenario, including many urban main roads and river systems that fully utilize the area’s strong natural wind conditions and realize various functions; however, the cooling effect is suboptimal. Corridors with great potential in cooling, isolation, ventilation, and noise reduction were determined after comprehensive optimization.

scholarly journals A proof of Aronhold's theorem upon quartic curves

1940 ◽  
Vol 6 (3) ◽  
pp. 190-191
Author(s):  
H. W. Richmond
Keyword(s):  
Finite Number ◽  
Simple Proof ◽  
Plane Curves ◽  
Cubic Surface ◽  
Quartic Curves ◽  
Order Four ◽  
The Given

It is to be expected that a finite number of plane curves of order four should have seven given lines as bitangents, because the number of conditions imposed is equal to the number of effective free constants in the equation of such a curve, viz., 14. Aronhold made the interesting discovery that one curve could be determined in which no three of the given lines have their six points of contact on a conic. The method, due to Geiser, of obtaining the bitangents as projections of the lines of a cubic surface leads to a simple proof of the existence of this quartic.

Geometrical Properties of Three Symmetric Matrices

1935 ◽  
Vol 31 (2) ◽  
pp. 174-182 ◽  
Author(s):  
H. W. Turnbull
Keyword(s):  
Explicit Form ◽  
Three Dimensions ◽  
Symmetric Matrices ◽  
Geometrical Properties ◽  
Cubic Surface ◽  
Quadric Surfaces ◽  
Quartic Curve ◽  
Polar Property ◽  
Linear Complexes ◽  
Image Position

In the early editions of the Geometry of Three Dimensions Salmon had stated that the equations of any three quadric surfaces could be simultaneously reduced to the sums of five squares. Such a reduction is not possible in general, but can be performed if and only if a certain combinant Λ, of the net of quadrics, vanishes. Algebraically the theory of such a net of quadrics is equivalent, as Hesse(2) showed, to that of a plane quartic curve: and the condition for the equation a quartic to be expressible to the sum of five fourth powers is equivalent to the condition Λ = 0(1). While Clebsch(3) was the first to establish this condition, Lüroth(4) gave it more explicit form by studying the quartic curvewhich satisfies the condition. Frahm(5) seems to have been the first to prove the impossibility of the above reduction of three general quadric surfaces, by remarking that the plane quartic curve obtained in Hesse's way from the locus of the vertices of cones of the net of quadrics would be a Lüroth quartic. Frahm further remarked that the three quadrics, so conditioned, could be regarded as the polar quadrics belonging to a cubic surface in ∞2 ways; but that for three general quadrics no such cubic surface exists. An explicit algebraical account of these properties was given by E. Toeplitz(6), who incidentally noticed that certain linear complexes associated with three general quadrics became special linear complexes when Λ = 0. This polar property of three quadrics in [3] was generalized to n dimensions by Anderson (7).

The No-Three-In-Line Problem

1968 ◽  
Vol 11 (4) ◽  
pp. 527-531 ◽  
Author(s):  
Richard K. Guy ◽  
Patrick A. Kelly
Keyword(s):  
Finite Number ◽  
The Other ◽  
Probabilistic Argument ◽  
Number Of Solutions ◽  
Other Hand ◽  
Image Position

Let Sn be the set of n2 points with integer coordinates n (x, y), 1 ≤ x, y <n. Let fn be the maximum cardinal of a subset T of Sn such that no three points of T are collinear. Clearly fn < 2n.For 2 ≤ n ≤ 10 it is known ([2], [3] for n = 8, [ 1] for n = 10, also [4], [6]) that fn = 2n, and that this bound is attained in 1, 1, 4, 5, 11, 22, 57, 51 and 156 distinct configurations for these nine values of n. On the other hand, P. Erdös [7] has pointed out that if n is prime, fn ≥ n, since the n points (x, x2) reduced modulo n have no three collinear. We give a probabilistic argument to support the conjecture that there is only a finite number of solutions to the no-three-in-line problem. More specifically, we conjecture that

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