Lines
and Curves| 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 
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 Rn);
Given w Rn, the
ray emanating from w is
Rw:= {x Rn | 
p Rn \
{0} such that x = w + tp for some t R+}
.
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 
is
x cos + y sin – p =
0 .
* Length of a curve: For a closed curve,
L = 
02pi d p()
,
where p() is the
support function for the compact set bounded by the curve wrt an interior point O.
* Writhing number: The expression
* Envelope of a family of
curves: Given the family of curves F(x, y; 
)
= 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 from F =
0 and 
F/ =
0.
@ References: Balakrishnan & Satija mp/05 [linking number, twist and writhe].
In Curved Spaces > s.a. geodesics;
spacetime subsets; Worldline.
$ Curvature: The vector Ca:= 
m 
m a, where a is
the
unit tangent to the line; Ca is
always perpendicular to a,
and vanishes iff the line is a geodesic; In
general relativity, one identifies Ca for
world-lines with Aa,
its acceleration.
@ References: Ehlers & Köhler JMP(77)
[congruences of curves on manifolds].
Related Concepts > s.a. Congruence; Fiber;
Field Line; Writhe.
$ Line Bundle: An R-bundle; > s.a. fiber
bundle;
Quillen Determinant.
Main page – Abbreviations – Journals – Comments – Other
sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified
20 jun 2008