Cycloids, hipocycloids and epicycloids have an often forgotten common property: they are homothetic to their evolutes. But what if use convex symmetric polygons as unit balls, can we define evolutes and cycloids which are genuinely discrete? Indeed, we can! We define discrete cycloids as eigenvectors of a discrete doub…
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6 results for “cycloids”
Isoptic curves of cycloidsmath.DG
The study explores isoptic curves of cycloids and their applications.
problem Understanding isoptic curves of cycloids.
method Definition of isoptic curve, parametric equations, and support function.
result Determination of isoptic curves for hypocycloids and epicycloids.
Given a normed plane , we call -cycloids the planar curves which are homothetic to their double -evolutes. It turns out that the radius of curvature and the support function of a -cycloid satisfy a differential equation of Sturm-Liouville type. By studying this equati…
Paper analyzes EM algorithm's trajectory in 2MLR, revealing cycloid behavior.
problem Understanding the convergence and trajectory of EM algorithm in 2MLR.
method Explicit closed-form expressions for EM updates, recurrence relation derivation at population level.
result EM iterations lie on a cycloid trajectory, leading to theoretical estimate of convergence exponent.
A Kinematic and Geometric Analysis of Trochoidal Wavesphysics.flu-dyn
Analyzes Gerstner's trochoidal waves and their geometric properties.
problem Understanding the geometry and kinematics of trochoidal waves.
method Derives velocity and arc length conditions for cycloidal, curtate, and prolate trochoids using Galilean transformations.
result Conditions for arc lengths of prolate and curtate trochoids to coincide over a wave cycle.
Study of timelike surfaces with time-minimizing rulings in Newtonian and relativistic spacetimes.
problem Understanding time-minimizing paths in spacetime geometries.
method Constructing timelike surfaces ruled by geodesics of Finsler or Jacobi metrics.
result Explicit examples of brachistochrone-ruled timelike surfaces in Minkowski and Schwarzschild spacetimes.