In this paper we study the isoperimetric-type equalities for rosettes, i.e. regular closed planar curves with non-vanishing curvature. We find the exact relations between the length and the oriented area of rosettes based on the oriented areas of the Wigner caustic, the Constant Width Measure Set and the Spherical Meas…
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8 results for “rosette”
Study on the geometry of secant caustic of planar curves.
problem Analyzing the secant caustic of planar curves.
method Investigating the geometrical properties of secant caustic, including number of branches, cusps, and inflexion points.
result Detailed investigation of secant caustic properties, especially for rosettes.
Billiard knots in a cylindermath.GT
We define cylinder knots as billiard knots in a cylinder. We present a necessary condition for cylinder knots: after dividing cylinder knots by possible rotational symmetries we obtain ribbon knots. We obtain an upper bound for the number of cylinder knots with two fixed parameters (out of three). In addition we prove …
Fourier Knotsmath.GT
We show that every knot has a checkerbord diagram and that every knot is the closure of a rosette braid. We define Fourier knots of type (n_1, n_2, n_3) as knots which have parametrizations where each coordinate function x_i(t) is a finite Fourier series of length n_i, and conclude that every knot is a Fourier knot of …
Study identifies prime strongly positive amphicheiral knots with double symmetry.
problem Characterizing prime strongly positive amphicheiral knots with specific symmetries.
method Examined knots up to 16 crossings, identified prime knots with double symmetry, and presented almost doubly symmetric diagrams.
result Found the first prime strongly positive amphicheiral knot not slice.
In this paper we study global properties of the Wigner caustic of parameterized closed planar curves. We find new results on its geometry and singular points. In particular, we consider the Wigner caustic of rosettes, i.e. regular closed parameterized curves with non-vanishing curvature. We present a decomposition of a…
Gauss-Bonnet theorem extended to surfaces with boundary and applied to map properties.
problem Extending Gauss-Bonnet theorem to surfaces with boundaries.
method Developed new formulas for coherent tangent bundles over compact oriented surfaces with boundaries.
result Established a link between total geodesic curvature on the boundary and total singular curvature of affine extended wave fronts.
Survey of Turk's head knots and links properties.
problem Characterize Turk's head knots and links.
method Discussion of various results in the mathematical literature.
result Turk's head links are alternating, fibered, hyperbolic, invertible, non-split, periodic, and prime.