2,490 research outputs found
Curves with rational chord-length parametrization
It has been recently proved that rational quadratic circles in standard Bezier form are parameterized by chord-length. If we consider that standard circles coincide with the isoparametric curves in a system of bipolar coordinates, this property comes as a straightforward consequence. General curves with chord-length parametrization are simply the analogue in bipolar coordinates of nonparametric curves. This interpretation furnishes a compact explicit expression for all planar curves with rational chord-length parametrization. In addition to straight lines and circles in standard form, they include remarkable curves, such as the equilateral hyperbola, Lemniscate of Bernoulli and Limacon of Pascal. The extension to 3D rational curves is also tackled
Multi-Dimensional Sigma-Functions
In 1997 the present authors published a review (Ref. BEL97 in the present
manuscript) that recapitulated and developed classical theory of Abelian
functions realized in terms of multi-dimensional sigma-functions. This approach
originated by K.Weierstrass and F.Klein was aimed to extend to higher genera
Weierstrass theory of elliptic functions based on the Weierstrass
-functions. Our development was motivated by the recent achievements of
mathematical physics and theory of integrable systems that were based of the
results of classical theory of multi-dimensional theta functions. Both theta
and sigma-functions are integer and quasi-periodic functions, but worth to
remark the fundamental difference between them. While theta-function are
defined in the terms of the Riemann period matrix, the sigma-function can be
constructed by coefficients of polynomial defining the curve. Note that the
relation between periods and coefficients of polynomials defining the curve is
transcendental.
Since the publication of our 1997-review a lot of new results in this area
appeared (see below the list of Recent References), that promoted us to submit
this draft to ArXiv without waiting publication a well-prepared book. We
complemented the review by the list of articles that were published after 1997
year to develop the theory of -functions presented here. Although the
main body of this review is devoted to hyperelliptic functions the method can
be extended to an arbitrary algebraic curve and new material that we added in
the cases when the opposite is not stated does not suppose hyperellipticity of
the curve considered.Comment: 267 pages, 4 figure
Twistor Theory and Differential Equations
This is an elementary and self--contained review of twistor theory as a
geometric tool for solving non-linear differential equations. Solutions to
soliton equations like KdV, Tzitzeica, integrable chiral model, BPS monopole or
Sine-Gordon arise from holomorphic vector bundles over T\CP^1. A different
framework is provided for the dispersionless analogues of soliton equations,
like dispersionless KP or Toda system in 2+1 dimensions. Their
solutions correspond to deformations of (parts of) T\CP^1, and ultimately to
Einstein--Weyl curved geometries generalising the flat Minkowski space. A
number of exercises is included and the necessary facts about vector bundles
over the Riemann sphere are summarised in the Appendix.Comment: 23 Pages, 9 Figure
- …