XVIII.—The Invariant Theory of Forms in Six Variables relating to the Line Complex

1927 ◽  
Vol 46 ◽  
pp. 210-222 ◽  
Author(s):  
H. W. Turnbull
Keyword(s):  
Invariant Theory ◽  
Line Geometry ◽  
Five Dimensions ◽  
Ordinary Space ◽  
Homogeneous Coordinates ◽  
Plücker Coordinates ◽  
Algebraic Theories ◽  
Straight Line ◽  
Image Position ◽  
Theory Of Forms

It is well known that the Plücker coordinates of a straight line in ordinary space satisfy a quadratic identitywhich may also be considered as the equation of a point-quadric in five dimensions, if the six coordinates Pij are treated as six homogeneous coordinates of a point. Projective properties of line geometry may therefore be treated as projective properties of point geometry in five dimensions. This suggests that certain algebraic theories of quaternary forms (corresponding to the geometry of ordinary space) can best be treated as algebraic theories of senary forms: that is, forms in six homogeneous variables.

A Geometrical treatment of the Correspondence between Lines in Threefold Space and Points of a Quadric in Fivefold space

1925 ◽  
Vol 22 (5) ◽  
pp. 694-699 ◽  
Author(s):  
H. W. Turnbull
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Quadratic Relation ◽  
Four Dimensions ◽  
Homogeneous Coordinates ◽  
Plücker Coordinates ◽  
Straight Line ◽  
Set Up ◽  
Image Position ◽  
Three Dimensional Space

§ 1. The six Plücker coordinates of a straight line in three dimensional space satisfy an identical quadratic relationwhich immediately shows that a one-one correspondence may be set up between lines in three dimensional space, λ, and points on a quadric manifold of four dimensions in five dimensional space, S5. For these six numbers pij may be considered to be six homogeneous coordinates of such a point.

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.

Rational normal octavic surfaces with a double line, in space of five dimensions

1933 ◽  
Vol 29 (1) ◽  
pp. 95-102 ◽  
Author(s):  
D. W. Babbage
Keyword(s):  
Rational Surface ◽  
Simple Rule ◽  
Plane Curves ◽  
Double Line ◽  
Five Dimensions ◽  
Multiple Line ◽  
Image Position

The following paper arises from a remark in a recent paper by Professor Baker. In that paper he gives a simple rule, under which a rational surface has a multiple line, expressed in terms of the system of plane curves which represent the prime sections of the surface. The rule is that, if one system of representing curves is given by an equation of the formthe surface being given, in space (x0, x1,…, xr), by the equationsthen the surface contains the linecorresponding to the curve φ = 0; and if the curve φ = 0 has genus q, this line is of multiplicity q + 1.

A Specialised Net of Quadrics Having Selfpolar Polyhedra, with Details of the Fivedimensional Example

1981 ◽  
Vol 33 (4) ◽  
pp. 885-892
Author(s):  
W. L. Edge
Keyword(s):  
Projective Space ◽  
Complex Number ◽  
Roots Of Unity ◽  
Parametric Form ◽  
Normal Curve ◽  
Homogeneous Coordinates ◽  
Image Position ◽  
Rational Normal Curve

If x0,x1, … xn are homogeneous coordinates in [n], projective space of n dimensions, the prime (to use the standard name for a hyperplane)osculates, as θ varies, the rational normal curve C whose parametric form is [2, p. 347]Take a set of n + 2 points on C for which θ = ηjζ where ζ is any complex number andso that the ηj, for 0 ≦ j < n + 2, are the (n + 2)th roots of unity. The n + 2 primes osculating C at these points bound an (n + 2)-hedron H which varies with η, and H is polar for all the quadrics(1.1)in the sense that the polar of any vertex, common to n of its n + 2 bounding primes, contains the opposite [n + 2] common to the residual pair.

Polynomial Invariant Theory and Taylor Series

1991 ◽  
Vol 43 (6) ◽  
pp. 1243-1262 ◽  
Author(s):  
John E. Gilbert
Keyword(s):  
Irreducible Representation ◽  
Taylor Series ◽  
Invariant Theory ◽  
Regular Representation ◽  
Polynomial Functions ◽  
Polynomial Invariant ◽  
Finite Dimensional ◽  
Isotypic Component ◽  
Image Position ◽  
The Right

For any group K and finite-dimensional (right) K-module V let be the right regular representation of K on the algebra of polynomial functions on V. An Isotypic Component of is the sum of all k-submodules of on which π restricts to an irreducible representation can then be written as f = ΣƬ ƒƬ with ƒƬ in .

scholarly journals On the stationary eccentricity of a system of four-tangent conics

1909 ◽  
Vol 28 ◽  
pp. 2-5
Author(s):  
F. E. Edwards
Keyword(s):  
Variable Parameter ◽  
Convex Quadrilateral ◽  
Straight Line ◽  
Image Position

Let the convex quadrilateral formed by the four given tangents be ABA′B′, and O the intersection of the diagonals. Let OA and OB be taken as axes of x and y. Denote OA, OA′, OB and OB′ by a, a′, b and b′, a and b being positive, and a′ and b′ negative. The tangential equation of the system is thenwhere k is a variable parameter; for the equation is satisfied when the straight line lx + my + 1 = 0 passes through any two adjacent angular points of the quadrilateral.

Osculatory properties of a certain curve in [n]

1974 ◽  
Vol 75 (3) ◽  
pp. 331-344 ◽  
Author(s):  
W. L. Edge
Keyword(s):  
Projective Space ◽  
Homogeneous Coordinates ◽  
Linearly Independent ◽  
Image Position

1. When, as will be presumed henceforward, no two of a0, a1, …, an are equal the n + 1 equationsare linearly independent; x0, x1, …, xn are homogeneous coordinates in [n] projective space of n dimensions—and the simplex of reference S is self-polar for all the quadrics.

scholarly journals XIII.—The General Form of the Involutive 1-1 Quadric Transformation in a Plane

1905 ◽  
Vol 40 (2) ◽  
pp. 253-262
Author(s):  
Charles Tweedie
Keyword(s):  
Special Property ◽  
Cremona Transformation ◽  
Straight Line ◽  
Image Position

§ 1. In a communication read before the Society, 3rd December 1900, Dr Muir discusses the generalisation, for more than two pairs of variables, of the proposition that: IfthenIf we interpret (x, y) and (ξ, η) iis points in a plane, it is manifest that the transformation thereby obtained is a Cremona transformation. It has the special property of being reciprocal or involutive in character; i.e., if the point P is transformed into Q, then the repetition of the same transformation on Q transforms Q into P. Symbolically, if the transformation is denoted by T. T(P) = Q, and T(Q) = T2(P) = P; so that T2 = 1, and T = T−1. Moreover, if the locus of P (x, y) is a straight line, the locus of Q (ξ, η) is in general a conic.

The Problems of Random Flight and Conduction of Heat

1924 ◽  
Vol 22 (2) ◽  
pp. 167-168
Author(s):  
W. Burnside
Keyword(s):  
Constant Velocity ◽  
Time Interval ◽  
Straight Line ◽  
Random Flight ◽  
Image Position

In a paper on random flight Lord Rayleigh proved the following result: A number is formed by adding together n numbers each of which is equally likely to have any value from − a to + a. Then, if f (n, s) ds is the probability that the number so formed lies between s and s + ds, and if n is sufficiently great,This result may be stated as follows: A point moves discontinuously in a straight line. For a time τ it has a constant velocity. During the next time-interval τ it again has a constant velocity, and so on. Then if each of these velocities is equally likely to have any value from − v to + v, the probability that in the time nτ, the point moves a distance lying between s and s + ds is f (n, s) ds, with vτ written for a.

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