Consider the parabola y=0.5*(x^2), and specifically that portion of the x-y plane above the curve. In this area, construct a new curve, defined as being the locus of points, each of which is fixed normal distance, d (d>=0), from each point on the original parabola. For d=0, the new curve is the original parabola. For relatively small d, (e.g. d=0.1), the new curve is similar to but not a parabola.
a) Is there a name for this constructed curve?
Something interesting happens to the constructed curve when d becomes equal to and then exceeds a specific finite value.
b) What is this value for d and what happens?
c) Can you generalize what happens at this transition to any parabola by relating the critical value of d to a mathematical property of said parabola?
This problem requires just a bit of calculus: The slope of a tangent curve to y=0.5x^2 is given by slope=x.
If we define the parabola parametrically by (t,t^2) then the slope at t is t and the perpendicular slope is -1/t.
The parametric equations for the curve we seek are found by moving d units from the curve along this slope:
(t-dt/(sqrt(1+t^2)) , t^2+d/(sqrt(1+t^2)))
https://www.desmos.com/calculator/i9vivxwrcz
I made some other tools to play with. A animated slider for a point on the curve (a,0.5a^2). If you make d negative you get the outer curve.
Answers: a) I don't know if this curve has a name but I'd call it an equal length border curve.
b) At d=1 the curve has a cusp at (0,1). This point is very close to 1 unit away from points near the parabola vertex.
For d>1 the curve has two cusps and a central curvy triangle region. These points are d units away from the parabola across the y-axis.
I didn't yet find the coordinates of these points for a given value of d.
I did add an arc of circle (yellow) to show when the curvature of the border curve is tight enough for the cusps to occur.
c) For the parabola y=ax^2 the cusps will occur for smaller d as a increases. It wouldn't be hard to edit the graphs to show this. One could easily edit the graph to give the border curve for any parametric graph (x(t),y(t))
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Posted by Jer
on 2023-10-06 16:42:25 |
Some solutions and fun graph
