Lines and Curves

In Affine Space > s.a. affine structures.
$ Line: Given an affine space, the line through two points P and Q is the set of all points S with S−Q = t (P−Q), for some t ∈ \(\mathbb R\); Equivalently, with an abuse of notation, S = tP + (1−t) Q.
* Line segment between two points: For points P and Q, the subset of the line through P and Q with 0 < t < 1.

In Rⁿ
* Ray emanating from a point w: The straight line passing by w (the notion of straight line is given by the linear structure of \(\mathbb R^n\)); Given w ∈ \(\mathbb R^n\), the ray emanating from w is

R_w:= {x ∈ \(\mathbb R\)ⁿ | ∃ p ∈ \(\mathbb R\)ⁿ \ {0} such that x = w + tp for some t ∈ \(\mathbb R\)⁺} .

@ References: Darst et al 09 [curious curves].

In Euclidean Geometry > s.a. differential and euclidean geometry.
$ Line: The line whose distance from the origin is p and angle of the normal with the x axis φ is

x cosφ + y sinφ − p = 0 .

* Length of a curve: For a closed curve, \( L = \int_0^{2\pi}{\rm d}\phi\,p(\phi) \), where p(ψ) is the support function for the compact set bounded by the curve with respect to an interior point O.
* Writhing number: The expression

\[ W[C] = {1\over4\pi} \oint_C {\rm d}x_\alpha \oint_C {\rm d}x_\beta\, \epsilon^{\alpha\beta\gamma}
{(x-y)_\gamma\over|x-y|^3}\; .\]

* Envelope of a family of curves: Given the family of curves F(x, y; λ) = 0 in the plane, the envelope is the curve every point of which is a point of contact with a curve in the family; Its equation can be obtained by eliminating λ from F = 0 and ∂F/∂λ = 0.
@ References: Toponogov & Rovenski 05; Balakrishnan & Satija mp/05 [linking number, twist and writhe].
> Online resources: see Wikipedia page on Geodesic Curvature.

In Lorentzian Geometry > s.a. spacetime subsets.
* Result: Maximal causal curves in Lipschitz continuous Lorentzian manifolds are either everywhere lightlike or everywhere timelike.
@ Timelike / causal curves: Ehrlich & Galloway CQG(90) [and Lorentzian splitting theorem]; Low CQG(90) [topology of the space of causal geodesics]; Pourkhandani & Bahrampour CQG(12) [the space of causal curves and separation axioms]; Pienaar et al PRL(13) [open timelike curves and violation of the uncertainty principle]; Miller JGP(17)-a1609 [Polish space of causal curves]; Lange et al a2009 [maximal causal curves in Lipschitz continuous Lorentzian manifolds]; > s.a. causality violations [closed timelike curves]; Simon Tensor; Worldline.
@ Null curves: Duggal & Jin 07.

In Curved and Generalized Spaces > s.a. geodesics [and geodesic circles]; spacetime subsets.
$ Curvature: The vector C^a:= ξ^m ∇_m ξ^a, where ξ^a is the unit tangent to the line; C^a is always perpendicular to ξ^a, and vanishes iff the line is a geodesic; In general relativity, one identifies C^a for world-lines with A^a, its acceleration.
@ General references: Ehlers & Köhler JMP(77) [congruences of curves on manifolds].
@ In Riemannian manifolds, curvature: Castrillón et al DG&A(10) [total curvature]; Gutkin JGP(11) [in terms of invariants].
> Frenet curvature / Frenet-Serret formulas: see coordinates on a manifold; relativistic particles; MathWorld page; Wikipedia page.
> More general settings: see manifolds [curves in supermanifolds].

Related Concepts > s.a. Congruence; Fiber; Field Line; Vorticity; Writhe.
$ Line bundle: An \(\mathbb R\)-bundle; > s.a. fiber bundle; Quillen Determinant.
$ Quantum curve: A solution of the equation [P, Q] = h/2π, where P, Q are ordinary differential operators.
@ And physics: Delphenich a1309, a1404 [electromagnetism and line geometry, projective geometry]; Shaikh et al EPJP(14)-a1312 [families of classical trajectories]; Liu & Schwarz a1403 [relation between quantum and classical curves]; Adler a1402 [quantum theory of distance along a curve].

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