scholarly journals XI. On a graphical representation of the twenty-seven lines on a cubic surface

1894 ◽  
Vol 54 (326-330) ◽  
pp. 148-149
Keyword(s):  
Graphical Representation ◽  
Three Dimensions ◽  
Cubic Surface ◽  
Straight Lines

The converse of Pascal’s well-known theorem may be stated thus: if two triangles be in perspective, their non-corresponding sides intersect in six points lying on a conic. An extension of this theorem to three dimensions may be stated thus: if two tetrahedrons be in perspective, their non-corresponding faces intersect in twelve straight lines lying on a cubic surface. This theorem may be deduced from the equation xyzu = ( x + a T) ( y + b T) ( z + c T) ( u + d T), where T = αx + βy + γz + δu ; and a, b, c, d, α, β, γ ,δ are constants. The equations of twelve lines on the surface are evident.

scholarly journals On Helical Projection and Its Application in Screw Modeling

2014 ◽  
Vol 6 ◽  
pp. 901047
Author(s):  
Riliang Liu ◽  
Haiguang Zhu
Keyword(s):  
Mathematical Model ◽  
Projection Method ◽  
Jordan Curve ◽  
Graphical Representation ◽  
Practical Approach ◽  
Efficient Design ◽  
Space Object ◽  
Axial Profile ◽  
Straight Lines ◽  
Design And Manufacture

As helical surfaces, in their many and varied forms, are finding more and more applications in engineering, new approaches to their efficient design and manufacture are desired. To that end, the helical projection method that uses curvilinear projection lines to map a space object to a plane is examined in this paper, focusing on its mathematical model and characteristics in terms of graphical representation of helical objects. A number of interesting projective properties are identified in regard to straight lines, curves, and planes, and then the method is further investigated with respect to screws. The result shows that the helical projection of a cylindrical screw turns out to be a Jordan curve, which is determined by the screw's axial profile and number of flights. Based on the projection theory, a practical approach to the modeling of screws and helical surfaces is proposed and illustrated with examples, and its possible application in screw manufacturing is discussed.

scholarly journals The Veronesean of quadrics and associated loci

1936 ◽  
Vol 5 (1) ◽  
pp. 14-25
Author(s):  
J. W. Head
Keyword(s):  
Three Dimensions ◽  
Cubic Surface ◽  
Polar System ◽  
Twisted Cubic ◽  
Pencils Of Quadrics

In this paper we consider the correspondence between tangential quadrics of [3] and points of [9]. Godeaux has considered this geometrically, with the object of obtaining a representation for a twisted cubic of three dimensions. We have considered it from a standpoint more algebraic than that of Godeaux, with particular reference to the types of pencils of quadrics that correspond to special lines of [9], and to the interpretation in [9] of the fact that the condition for a net of quadrics to be part of the polar system of a cubic surface is poristic.

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).

scholarly journals A general vector analysis, with applications to electrodynamical theory

1925 ◽  
Vol 108 (747) ◽  
pp. 418-455 ◽  
Keyword(s):  
Three Dimensions ◽  
Vector Analysis ◽  
Cartesian Coordinates ◽  
Straight Line ◽  
Straight Lines ◽  
General Vector

The vector analyses in use up to the present, as a rule, are concerned with quantities which are represented by straight lines, and the space to which they are applicable is Euclidean in its properties. The straight line, AB, in space of three dimensions, is represented by a vector a, and if B has Cartesian coordinates ( x, y, z ) with respect to A, we write: a = i x + j y + k z , where i, j, k, are fundamental vectors. An account will be given of a vector analysis in which a vector is represented by δa' = Σ n i n δx n . The vector is of infinitesimal length and represents a component measured in any system of co-ordinates.

Line geometry in three dimensions over GF (3), and the allied geometry of quadrics in four and five dimensions

1955 ◽  
Vol 228 (1172) ◽  
pp. 129-146 ◽  
Keyword(s):  
Canonical Form ◽  
Linear Subspace ◽  
Galois Field ◽  
Three Dimensions ◽  
Line Geometry ◽  
Five Dimensions ◽  
Cubic Surface

This paper, a sequel to another, describes the line geometry of a [3] S over the Galois field K of 3 marks; this involves some investigation of the geometry, also over K , of quadrics in [4] and [5]. Sections 2 to 4 introduce th e Plücker co-ordinates and explain how reguli in S can be positive or negative. Section 6 introduces K lein’s m apping of the lines of S on a quadric Q in [5]. The lines of [5] fall into four categories relative to Q, and the num ber in each category is found. The points off Q fall into two oppositely signed batches of 117 each, and it is helpful to use X 2 + Y 2 + Z 2 = U 2 + V 2 + W 2 as a canonical form for th e equation of Q. Section 10 digresses to calculate th e number of quadrics in the [5] which ad m it such an equation. Section 11 introduces the screws of S , each of which consists of 40 lines that can be arranged as 36 decades, and explains how a screw is signed. Section 12 shows that the 40 lines fall into 270 reguli signed oppositely to and 540 reguli signed similarly to the screw. It is in section 12 that the number 27, so familiar in the geometry of the cubic surface, arises. A screw is mapped by a prime section w of Q, and the lines of the prime fall, when regard is had to the signs of tangents, into five categories relative to w . Sections 13 to 15 describe the resulting geometry in [4], account for every linear subspace and display a table of incidences. The numbers in this table are essential to a proper study of the group w (5, 3) of order 51840. Section 16 compares this figure in [4] with that which Baker and Todd elaborated from Burkhardt’s quartic primal. Sections 18 and 19 give the order of PO 2 (6, 3) and describe its involutions, w (5, 3) is a subgroup of index 117 of PO 2 (6, 3) and sections 20 to 22 discuss briefly the involutions in w (5, 3) and mention certain of its permutation representations.

scholarly journals Classification of Geometries with Projective Metric

1909 ◽  
Vol 28 ◽  
pp. 25-41 ◽  
Author(s):  
D. M. Y. Sommerville
Keyword(s):  
Dihedral Angle ◽  
Plane Geometry ◽  
Three Dimensions ◽  
Angular Measurement ◽  
Plane Angle ◽  
Degenerate Form ◽  
Projective Metric ◽  
The Absolute ◽  
Straight Lines

In the Cayley-Klein projective metric it is ordinarily assumed that the measure of angles, plane and dihedral, is always elliptic, i.e. in a sheaf of planes or lines there is no actual plane or line which makes an infinite angle with the others. With this restriction there are only three kinds of geometry—Parabolic, Hyperbolic and Elliptic, i.e. the geometries of Euclid, Lobachevskij and Riemann ; and the form of the absolute is also limited. Thus in plane geometry the only degenerate form of the absolute which is possible is two coincident straight lines and a pair of imaginary points ; in three dimensions the absolute cannot be a ruled quadric, other than two coincident planes. If, however, this restriction as to angular measurement is removed, there are 9 systems of plane geometry and 27 in three dimensions; for the measure of distance, plane angle and dihedral angle may be parabolic, hyperbolic, or elliptic.

scholarly journals A generalized vector analysis of four dimensions

1923 ◽  
Vol 103 (723) ◽  
pp. 644-663 ◽  
Keyword(s):  
Three Dimensions ◽  
Vector Analysis ◽  
Four Dimensions ◽  
Straight Line ◽  
Straight Lines

The vector analysis in use up to the present, as a rule, are concerned with quantities which are represented by straight lines. The straight line AB, in space of three dimensions, is represented by a vector a, and if B has Cartesian co-ordinates ( x, y, z ) with respect to A, we write a = i x + j y + k z , where i, j, k are fundamental vectors. An account will be given of a vector analysis in which the vector is represented by δa = Ʃ n i n δ x n .

Estimation of length for digitized straight lines in three dimensions

1990 ◽  
Vol 11 (3) ◽  
pp. 207-213 ◽  
Author(s):  
T.M. Amarunnishad ◽  
P.P. Das
Keyword(s):  
Three Dimensions ◽  
Straight Lines

Assessment of Heterogeneity of Welded Joint of Two Different Metals by Modeling of Diffusion of Substitution Elements

2008 ◽  
Vol 138 ◽  
pp. 193-200 ◽  
Author(s):  
Lenka Řeháčková ◽  
Jaroslav Kalousek ◽  
Jana Dobrovská ◽  
Zdenek Dostál ◽  
Karel Stránský
Keyword(s):  
Weld Joint ◽  
Graphical Representation ◽  
Welded Joint ◽  
Weld Interface ◽  
Mutual Position ◽  
Concentration Profiles ◽  
Linear Regression Modeling ◽  
Straight Lines ◽  
Narrow Area

The paper presents original methodology for assessment of heterogeneity of welded joint by measurement of concentration profiles of substitution elements in three straight lines in a plane perpendicular to the weld interface. The weld joint is formed by low-alloyed CrNiMoV steel and low-alloy silicon steel. Three concentration profiles are evaluated by modeling of diffusion based on solution of the 2nd Fick’s law. Primary goal of evaluation consists in assessment of chemical heterogeneity in the narrow area on both sides of the weld joint on the basis of their mutual position. Experimental data are optimized by original adaptation of the Levenberg–Marquardt’s algorithm of non-linear regression. Modeling comprises also determination of diffusivities and graphical representation of their temperature dependencies. The paper deals with diffusion of silicon as an example of many other investigated elements.

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