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An Euler spiral is a curve whose curvature changes ... The principle of linear variation of the curvature of the transition curve between a tangent and a circular ...
An Euler spiral, the theoretically optimum transition curve, linearly increases centripetal acceleration and results in constant jerk (see graphic). In real-world applications, the plane of the track is inclined ( cant ) along the curved sections.
A variation of Euler spiral, using sine integral and cosine integrals Polygonal spiral: Special case approximation of arithmetic or logarithmic spiral Fraser's Spiral: 1908 Optical illusion based on spirals Conchospiral =, =, =
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 ...
The spiral is a frequent symbol for spiritual purification, both within Christianity and beyond (one thinks of the spiral as the neo-Platonist symbol for prayer and contemplation, circling around a subject and ascending at the same time, and as a Buddhist symbol for the gradual process on the Path to Enlightenment).
The calculus of variations began with the work of Isaac Newton, such as with Newton's minimal resistance problem, which he formulated and solved in 1685, and later published in his Principia in 1687, [2] which was the first problem in the field to be formulated and correctly solved, [2] and was also one of the most difficult problems tackled by variational methods prior to the twentieth century.
Euler's second axiom or law (law of balance of angular momentum or balance of torques) states that in an inertial frame the time rate of change of angular momentum L of an arbitrary portion of a continuous body is equal to the total applied torque M acting on that portion, and it is expressed as
Further, Google search reveals that when the term "linear spiral" is used, it most often refers to the linear equation =, which is a different beast entirely. The poor curve has enough names in the literature (I've found Euler spiral, Cornu spiral, clothoid, Railroad Transition Spiral, Holbrook spiral, Glover's spiral, linarc, lince) - it doesn ...