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This equation, stated by Euler in 1758, [2] is known as Euler's polyhedron formula. [3] It corresponds to the Euler characteristic of the sphere (i.e. = ), and applies identically to spherical polyhedra. An illustration of the formula on all Platonic polyhedra is given below.
The Euler spiral provides the shortest transition subject to a given limit on the rate of change of the track superelevation (i.e. the twist of the track). However, as has been recognized for a long time, it has undesirable dynamic characteristics due to the large (conceptually infinite) roll acceleration and rate of change of centripetal ...
In mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ramified covering of the other. It therefore connects ramification with algebraic topology, in this case.
The third independent discovery occurred in the 1800's when various railway engineers sought a formula for gradual curvature in track shape. By 1880 Arthur Newell Talbot worked out the integral formulas and their solution, which he called the "railway transition spiral". The connection to Euler's work was not made until 1922. [2]
The non-orientable genus, demigenus, or Euler genus of a connected, non-orientable closed surface is a positive integer representing the number of cross-caps attached to a sphere. Alternatively, it can be defined for a closed surface in terms of the Euler characteristic χ, via the relationship χ = 2 − k , where k is the non-orientable genus.
the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ramified covering of the other. It therefore connects ramification with algebraic topology , in this case.
There are two main descriptions of motion: dynamics and kinematics.Dynamics is general, since the momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term dynamics refers to the differential equations that the system satisfies (e.g., Newton's second law or Euler–Lagrange equations), and sometimes to the solutions to those equations.
Euler's partition theorem relating the product and series representations of the Euler function Π(1 − x n) Goldbach–Euler theorem, stating that sum of 1/(k − 1), where k ranges over positive integers of the form m n for m ≥ 2 and n ≥ 2, equals 1; Gram–Euler theorem