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?
At d=1, the curve seems to start to intersect itself near the minimum and build a concave-downward arc connecting endpoints that extend beyond the point of intersection, and growing as d gets larger.
Wolfram Alpha gives the focus of this parabola as being at (0,1/2). Thus it appears that the cusp (dividing later into 2 cusps) starts at d equal to twice the distance of apex to focus of the parabola.
These stages can be viewed at:
The figures were zoomed manually after the program below created them. I attempted to make the scales almost equal, but the last one needed toinclude more of the cartesian space.
for dist=[.8 1 1.4 2.5 5.2 7]
x = linspace(-12,12,500);
y1 = x.^2/2;
figure
plot(x,y1 )
t=title(['dist is ',num2str(dist) ]);
t.FontSize=16;
grid
hold on
for i=1:length(x)
x0=x(i);
dx=x0;
slope=-1/dx; angle=atan(slope);
if x0<0
x2(i)=x0+dist*cos(angle);
y2(i)=y1(i)+dist*sin(angle);
else
x2(i)=x0-dist*cos(angle);
y2(i)=y1(i)-dist*sin(angle);
end
end
plot(x2,y2 )
axis equal
end
|
|
Posted by Charlie
on 2023-10-05 09:08:44 |
graphical computer exploration
