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A similar concept related to elliptical constants and proportional areas, is the curve formed by the cycloid [1] of a circle, which is the curve formed by traci

by espeed 8y ago

A similar concept related to elliptical constants and proportional areas, is the curve formed by the cycloid [1] of a circle, which is the curve formed by tracing a point on the circumference of a circle as it rolls across a surface from point A to point B. For example, the fastest distance between two points is not a straight line, it's the cycloid, specifically the Brachistochrone curve [2]. This is the path light follows. One common misconception of the cycloid is related to its arc-length. At first glance many assume the arc-length of the cycloid is equal to circumference of the circle, but this is not the case. The line-of-sight distance between the cycloid starting point A and ending point B is equal to 2πr, the circumference of the circle -- however the arc-length of the cycloid curve is 8r, which is an integer value given an integer radius. The cycloid curve is full of interesting properties, many yet to be discovered and all its implications are not yet fully understood. Another interesting aspect of the cycloid is not only is it the fastest path, but no matter where two objects begin on the curve, they'll both traverse the curve at maximal/optimal speed and both will arrive at the bottom of the curve at the same time, regardless of the delta between their starting positions. This aspect of the cycloid is referred to as the Tautochrone curve [3]. So if you're looking for ways to distribute partitions or encode invariants in your models, data or otherwise, the geometrical aspects of elliptical and cycloidal curves are a good place to explore. Grant did a 3Blue1Brown video with with Steven Strogatz on the Brachistochrone a few years back: https://www.youtube.com/watch?v=Cld0p3a43fU https://www.youtube.com/watch?v=Cld0p3a43fU And Vsauce did one with Adam Savage on the Brachistochrone where they build a mechanical model of one that shows it's the fastest/optimal path among different curves, and their experiment also shows the cycloid Tautochrone invariant property where objects begin up the curve at different distances apart and yet all arrive together simultaneously in constant time. https://www.youtube.com/watch?v=skvnj67YGmw https://www.youtube.com/watch?v=skvnj67YGmw NB: Consider this, two seperate impulses of light beginning at different distances away from the observer, both impulses of light traveling along the optimal path at the optimal speed, and both arriving at the observer simultaneously, without bending time. And as shown above, on a cycloidal curve, this phenomenon is not unique to light. [1] https://en.wikipedia.org/wiki/Cycloid https://en.wikipedia.org/wiki/Cycloid [2] https://en.wikipedia.org/wiki/Brachistochrone_curve https://en.wikipedia.org/wiki/Brachistochrone_curve [3] https://en.wikipedia.org/wiki/Tautochrone_curve https://en.wikipedia.org/wiki/Tautochrone_curve

speleo 8y ago
>For example, the fastest distance between two points is not a straight line, it's the cycloid, specifically the Brachistochrone curve [2]. This is the path light follows. You may be conflating [Bernoulli's solution to the Brachistochrone curve](http://www.math.rug.nl/~broer/pdf/ws-ijbc.pdf http://www.math.rug.nl/~broer/pdf/ws-ijbc.pdf) with the optimal path for light. [Fermat's Principle](https://en.wikipedia.org/wiki/Fermat%27s_principle https://en.wikipedia.org/wiki/Fermat%27s_principle) states that, when traveling between two points, light will always take the path that minimizes the time taken from the first point to the last. In a medium of constant refractive index (which includes free space), this results in a line. >So if you're looking for ways to distribute partitions or encode invariants in your models, data or otherwise, the geometrical aspects of elliptical and cycloidal curves are a good place to explore. How do you mean? What sort of data can be encoded this way and how? > NB: Consider this, two seperate impulses of light beginning at different distances away from the observer, both impulses of light traveling along the optimal path at the optimal speed, and both arriving at the observer simultaneously, without bending time. And as shown above, on a cycloidal curve, this phenomenon is not unique to light. Two separate impulses of light beginning at different distances away from a stationary observer will necessarily arrive at different times, otherwise you violate the basis of special relativity: the speed of light is constant and invariant of reference frame.
- taejo 8y ago
  You can solve the brachistochrone problem using Fermat's principle, as shown in the 3blue1brown video GP is referring to. Since you're looking for the shortest time, if you can construct a lens where the speed of light is proportional to the speed of the bead on a wire, then the shortest-time wire is the path that light would take by Fermat's principle, and then you can use (an infinitesimal version of) Snell's law to find the direction of the wire at each height.
  - speleo 8y ago
    Yes, but they were saying: >For example, the fastest distance between two points is not a straight line, it's the cycloid, specifically the Brachistochrone curve [2]. This is the path light follows. Which is not true for free space, or any space with a constant index of refraction.

by espeed 8y ago

speleo 8y ago

>For example, the fastest distance between two points is not a straight line, it's the cycloid, specifically the Brachistochrone curve [2]. This is the path light follows. You may be conflating [Bernoulli's solution to the Brachistochrone curve](http://www.math.rug.nl/~broer/pdf/ws-ijbc.pdf http://www.math.rug.nl/~broer/pdf/ws-ijbc.pdf) with the optimal path for light. [Fermat's Principle](https://en.wikipedia.org/wiki/Fermat%27s_principle https://en.wikipedia.org/wiki/Fermat%27s_principle) states that, when traveling between two points, light will always take the path that minimizes the time taken from the first point to the last. In a medium of constant refractive index (which includes free space), this results in a line. >So if you're looking for ways to distribute partitions or encode invariants in your models, data or otherwise, the geometrical aspects of elliptical and cycloidal curves are a good place to explore. How do you mean? What sort of data can be encoded this way and how? > NB: Consider this, two seperate impulses of light beginning at different distances away from the observer, both impulses of light traveling along the optimal path at the optimal speed, and both arriving at the observer simultaneously, without bending time. And as shown above, on a cycloidal curve, this phenomenon is not unique to light. Two separate impulses of light beginning at different distances away from a stationary observer will necessarily arrive at different times, otherwise you violate the basis of special relativity: the speed of light is constant and invariant of reference frame.

taejo 8y ago
You can solve the brachistochrone problem using Fermat's principle, as shown in the 3blue1brown video GP is referring to. Since you're looking for the shortest time, if you can construct a lens where the speed of light is proportional to the speed of the bead on a wire, then the shortest-time wire is the path that light would take by Fermat's principle, and then you can use (an infinitesimal version of) Snell's law to find the direction of the wire at each height.
- speleo 8y ago
  Yes, but they were saying: >For example, the fastest distance between two points is not a straight line, it's the cycloid, specifically the Brachistochrone curve [2]. This is the path light follows. Which is not true for free space, or any space with a constant index of refraction.