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.

@ __Timelike 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 princieple]; > s.a. causality violations [closed
timelike curves]; 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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