Spacetime Algebra and Line Geometry

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.

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The Effect of Line Geometry on Void Growth in Thin, Narrow Aluminum Lines

MRS Proceedings ◽

10.1557/proc-226-407 ◽

1991 ◽

Vol 226 ◽

Cited By ~ 3

Author(s):

P. Borgesen ◽

J. K. Lee ◽

M. A. Korhonen ◽

C.-Y. Li

Keyword(s):

Stress Relaxation ◽

Void Growth ◽

Room Temperature ◽

Line Geometry

AbstractThe stress induced growth of individual voids in passivated Al-lines at room temperature was monitored in-situ without removing the passivation. The kinetics was strongly influenced by variations in line gec.etry, even over distances of many Am, indicating variations in the stress relaxation as well.

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Note on Projective Differential Line Geometry

Springer Collected Works in Mathematics - Selected Papers II ◽

10.1007/978-1-4614-9343-3_19 ◽

1989 ◽

pp. 191-193

Author(s):

Shing-Shen Chern

Keyword(s):

Line Geometry ◽

Differential Line

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Explicit Baker–Campbell–Hausdorff–Dynkin formula for spacetime via geometric algebra

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887821502261 ◽

2021 ◽

Author(s):

Joseph Wilson ◽

Matt Visser

Keyword(s):

Efficient Method ◽

Geometric Algebra ◽

Clifford Algebras ◽

Matrix Representation ◽

Spin Representation ◽

Lorentz Transformations ◽

Rodrigues Formula ◽

Dimension Formula ◽

Spacetime Algebra ◽

Geometric Algebras

We present a compact Baker–Campbell–Hausdorff–Dynkin formula for the composition of Lorentz transformations [Formula: see text] in the spin representation (a.k.a. Lorentz rotors) in terms of their generators [Formula: see text]: [Formula: see text] This formula is general to geometric algebras (a.k.a. real Clifford algebras) of dimension [Formula: see text], naturally generalizing Rodrigues’ formula for rotations in [Formula: see text]. In particular, it applies to Lorentz rotors within the framework of Hestenes’ spacetime algebra, and provides an efficient method for composing Lorentz generators. Computer implementations are possible with a complex [Formula: see text] matrix representation realized by the Pauli spin matrices. The formula is applied to the composition of relativistic 3-velocities yielding simple expressions for the resulting boost and the concomitant Wigner angle.

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Characterizations of the $G_2(4)$ and $L_3(4)$ Near Octagons

The Electronic Journal of Combinatorics ◽

10.37236/8476 ◽

2021 ◽

Vol 28 (4) ◽

Author(s):

Bart De Bruyn

Keyword(s):

Generalized Quadrangle ◽

Line Geometry ◽

Near Polygon ◽

Generalized Polygon ◽

Generalized Hexagon ◽

Split Cayley Hexagon ◽

Point Line ◽

Line Spread

A triple $(\mathcal{S},S,\mathcal{Q})$ consisting of a near polygon $\mathcal{S}$, a line spread $S$ of $\mathcal{S}$ and a set $\mathcal{Q}$ of quads of $\mathcal{S}$ is called a polygonal triple if certain nice properties are satisfied, among which there is the requirement that the point-line geometry $\mathcal{S}'$ formed by the lines of $S$ and the quads of $\mathcal{Q}$ is itself also a near polygon. This paper addresses the problem of classifying all near polygons $\mathcal{S}$ that admit a polygonal triple $(\mathcal{S},S,\mathcal{Q})$ for which a given generalized polygon $\mathcal{S}'$ is the associated near polygon. We obtain several nonexistence results and show that the $G_2(4)$ and $L_3(4)$ near octagons are the unique near octagons that admit polygonal triples whose quads are isomorphic to the generalized quadrangle $W(2)$ and whose associated near polygons are respectively isomorphic to the dual split Cayley hexagon $H^D(4)$ and the unique generalized hexagon of order $(4,1)$.

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