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The original formulation of the Schoenflies problem states that not only does every simple closed curve in the plane separate the plane into two regions, one (the "inside") bounded and the other (the "outside") unbounded; but also that these two regions are homeomorphic to the inside and outside of a standard circle in the plane.
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 ...
For the purpose of this discussion, let a tangent of a closed plane curve be a line that intersects the curve, but not the interior of the region enclosed by the curve. So the line given by the equation x + y = 0 {\displaystyle x+y=0} in the Cartesian plane is a tangent of an axes-oriented square in the first quadrant with the origin as one of ...
A smooth plane curve is a curve in a real Euclidean plane and is a one-dimensional smooth manifold.This means that a smooth plane curve is a plane curve which "locally looks like a line", in the sense that near every point, it may be mapped to a line by a smooth function.
The Getty Villa art museum is threatened by the flames of the wind-driven Palisades Fire in Pacific Palisades, California, Jan. 7, 2025. A fast-moving brushfire in a Los Angeles suburb burned ...
In geometry, a curve of constant width is a simple closed curve in the plane whose width (the distance between parallel supporting lines) is the same in all directions. The shape bounded by a curve of constant width is a body of constant width or an orbiform , the name given to these shapes by Leonhard Euler . [ 1 ]
Below are the primetime rankers for broadcast, cable and premium cable networks in 2024, among total viewers (as well as the top 50 list in adults 18-49).