Forgive the recent apparent obsession (I'd call it a fascination) with the #cycloid but I've just discovered something I'd not heard of before. It is also called a #TautochroneCurve or #Isochrone curve, which means that a particle starting from any location on the curve will get to the #MinimumPoint at precisely the same time as a particle starting at any other point.

Here's an #animation I wrote today in #Maxima which illustrates the property.

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A couple of weeks ago, I posted an #animation of a point on a circle generating a #cycloid.

If you turn the curve "upside down", you get the #BrachistochroneCurve. This curve provides the shortest travel time starting from one cusp to any other point on the curve for a ball rolling under uniform #gravity. It is always faster than the straight-line travel time. This is an interesting problem in #ClassicalMechanics and exercised luminaries like #Newton and #Euler. I think the latter's use of the #CalculusOfVariations is a stroke of genius.

Anyway, the #animation took a bit of thought as it requires a bit of #Mechanics, some #Integration and is made a bit more tricky as the curve is multi-valued and so you need to treat different branches separately. The #AnimatedGif was produce with #WxMaxima.

For some reason that I can't fathom, I'm not finding it possible to upload the graphic and so if you want to see it, please use the link below.
https://drive.google.com/file/d/1edcZnZ_uiaQOQrFB03XHslNUVCmxcSut/view?usp=drive_link

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Imagine a circular wheel rolling, without skidding, on a flat, horizontal surface. The #locus of any given point on its #circumference is called a #cycloid. It is a #periodic #curve over a length equivalent to the #circle's circumference and has #cusps whenever the point is in contact with the surface (i.e. the two sides of the curve are tangentially vertical at that point).

Interestingly, it is also the curve that solves the #Brachistochrone problem, which means that starting at a cusp on the inverted curve (maximum height), a frictionless ball will roll under uniform gravity in minimum time from the start to any other point on the curve, even beating the straight line path.

#Mathematics #Geometry #Maths #AppliedMathematics #Mechanics #Kinematics #Dynamics #Physics #MyWork #CCBYSA #WxMaxima