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 R ^{n}**

*

*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* cos*φ* + *y* sin*φ* − *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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send feedback and suggestions to bombelli at olemiss.edu – modified 13 mar 2021