Curvey Weirdness (Posted on 2023-10-05)

	Submitted by Kenny M
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I have been thinking about curves like this and trying to formulate the thoughts into a problem for this site for quite a while, and my research has found interesting websites. Below is one that I think can explain the problem and solutions quite well. mathpages.com/home/kmath724/kmath724.htm a) As far as I can determine, there is no name for this curve. This question was sort of a red herring to be honest. b) For d<=1 the constructed curve is smooth and has a continuous first derivative. For d>1, the constructed curve has two separate singularities that are horizontally symmetric about x=0. The curve also crosses itself along the y-axis. I believe that there is still a 1-1 correspondence between points on the parabola and constructed curve but is it continuous? Not sure what is mathematically appropriate to say here. c) Here is the interesting part. The singularities appear when the radius of curvature, r (calculus definition), of the original parabola at a given point becomes less than the normal distance d. In effect, the original parabola is curving so quickly relative to the value d that, at the singularity, the next point on the constructed curve is created by using a point on the parabola that is a finite distance away from the previously used point on the parabola. This continues until the radius of curvature of the parabola again becomes greater than d, which defines the location of the second singularity. In my mind, this sort of blows-up any idea of a first derivative at those two points. Any of the math experts care to give an opinion? (Engineers don't usually get that deep into stuff like this). The singularities are always symmetric to the line that splits the parabola into two mirror image halves.

	Subject	Author	Date
	Some solutions and fun graph	Jer	2023-10-06 16:42:25
	graphical computer exploration	Charlie	2023-10-05 09:08:44