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
Rw:= {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
Ca:=
ξm
∇m
ξa, where
ξa is the unit
tangent to the line; Ca is
always perpendicular to ξa,
and vanishes iff the line is a geodesic; In general relativity, one identifies
Ca for world-lines with
Aa, 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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