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A Jordan curve or a simple closed curve in the plane R 2 is the image C of an injective continuous map of a circle into the plane, φ: S 1 → R 2. A Jordan arc in the plane is the image of an injective continuous map of a closed and bounded interval [a, b] into the plane. It is a plane curve that is not necessarily smooth nor algebraic.
The + and invariants keep track of how curves change under these transformations and deformations. The + invariant increases by 2 when a direct self-tangency move creates new self-intersection points (and decreases by 2 when such points are eliminated), while decreases by 2 when an inverse self-tangency move creates new intersections (and increases by 2 when they are eliminated).
Figure 1: Zindler curve. Any of the chords of equal length cuts the curve and the enclosed area into halves. Figure 2: Examples of Zindler curves with a = 8 (blue), a = 16 (green) and a = 24 (red). A Zindler curve is a simple closed plane curve with the defining property that: (L) All chords which cut the curve length into halves have the same ...
Quartic plane curves include Ampersand curve; Bean curve; Bicorn; Bow curve; ... (1901) online at Google Books This page was last edited on 2 December 2024, at ...
A plane curve can often be represented in Cartesian coordinates by an implicit equation of the form (,) = for some specific function f.If this equation can be solved explicitly for y or x – that is, rewritten as = or = for specific function g or h – then this provides an alternative, explicit, form of the representation.
We reflect across the plane through (), (), and the north pole, forming a closed curve containing antipodal points , with length () = (). A curve connecting ± p {\displaystyle \pm p} has length at least π {\displaystyle \pi } , which is the length of the great semicircle between ± p {\displaystyle \pm p} .
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According to Marcel Berger, Arnold revolutionized plane curves theory. [51] He developed the theory of smooth closed plane curves in the 1990s. [52] Among his contributions are the introduction of the three Arnold invariants of plane curves: J +, J-and St. [53] [54]