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The curve of fastest descent is not a straight or polygonal line (blue) but a cycloid (red).. In physics and mathematics, a brachistochrone curve (from Ancient Greek βράχιστος χρόνος (brákhistos khrónos) 'shortest time'), [1] or curve of fastest descent, is the one lying on the plane between a point A and a lower point B, where B is not directly below A, on which a bead slides ...
3-point-form of a hyperbola's equation — The equation of the hyperbola determined by 3 points = (,), =,,, ,, is the solution of the equation () () = () () for . As an affine image of the unit hyperbola x 2 − y 2 = 1
A double-end Euler spiral. The curve continues to converge to the points marked, as t tends to positive or negative infinity. An Euler spiral is a curve whose curvature changes linearly with its curve length (the curvature of a circular curve is equal to the reciprocal of the radius). This curve is also referred to as a clothoid or Cornu spiral.
There are continuous curves on which every arc (other than a single-point arc) has infinite length. An example of such a curve is the Koch curve. Another example of a curve with infinite length is the graph of the function defined by f(x) = x sin(1/x) for any open set with 0 as one of its delimiters and f(0) = 0.
Important quantities in the Whewell equation. The Whewell equation of a plane curve is an equation that relates the tangential angle (φ) with arc length (s), where the tangential angle is the angle between the tangent to the curve at some point and the x-axis, and the arc length is the distance along the curve from a fixed point.
A tautochrone curve or isochrone curve (from Ancient Greek ταὐτό 'same' ἴσος 'equal' and χρόνος 'time') is the curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point is independent of its starting point on the curve.
From this equation, substituting φ by φ = r 2 / a 2 (a rearranged form of the polar equation for the spiral) and then substituting r by r = √ x 2 + y 2 (the conversion from Cartesian to polar) leaves an equation for the Fermat spiral in terms of only x and y: = (+).
The envelope of the normals of the tractrix (that is, the evolute of the tractrix) is the catenary (or chain curve) given by y = a cosh x / a . The surface of revolution created by revolving a tractrix about its asymptote is a pseudosphere. The tractrix is a transcendental curve; it cannot be defined by a polynomial equation.