For a plane curve
![$[x(t),y(t)]$](curvature__1.png)
its curvature is given by
The quantity
is called the radius of curvature. The point
with coordinates given by
is called the center of curvature. The circle with center
radius
is called the osculating circle or the circle of
curvature. The center of curvature
lies on the normal of the curve at the point
the line segment joining
with
is perpendicular to the tangent at
The following procedure draws the line segment joining
with the corresponding center of curvature, where
,
are equally spaced points in the interval
![$[0,2\pi ]$](curvature__16.png)
in the domain of the ellipse
,
3
![$\ \sin \ t]$](curvature__18.png)
,
![$t\in [0,2\pi ].$](curvature__19.png)
> restart;
> x:=5*cos(t);
> y:=3*sin(t);
> x1:=diff(x,t);
> y1:=diff(y,t);
> x2:=diff(x1,t);
> y2:=diff(y1,t);
> rr:=(x1^2+y1^2)/(x1*y2-y1*x2);
> xc:=x-rr*y1;
> yc:=y+rr*x1;
> m:=[[x,y],[xc,yc]];
> t:=n*2*Pi/100;
> w:=evalf(m);
> plot([w$n=1..100],color=black,axes=none);
The evolute of a curve is the locus of its center of curvature. All tangents to the evolute are normal to the given curve. The evolute is the envelope of normals to the given curve.
Draw the line segments joining
with the corresponding center of curvature
for each of the following curves:
(a) the cycloid:
(b) the epicycloid:
for the cases
(c) the hypocycloid:
for the cases
(d) the parabola:
(e)
(f)
(g)
(h)
An involute of a curve is the path of a point of a string tautly unwound from the curve. The curve itself is the evolute of its involute.
Show that the following parametric equation describes the involute of the
circle with center
radius
:
> restart;
> x:=cos(t);
> y:=sin(t);
> xx:=x+t*sin(t);
> yy:=y-t*cos(t);
> t:=n*2*Pi/100;
> m:=[[x,y],[xx,yy]];
> w:=evalf(m);
> plot([w$n=1..100],color=black,axes=none);
>
To construct an animation of the circle of curvature of the ellipse
the following steps are taken:
> restart;
> with(plots):
> x:=5*cos(t);
> y:=3*sin(t);
> x1:=diff(x,t);
> y1:=diff(y,t);
> x2:=diff(x1,t);
> y2:=diff(y1,t);
> rr:=(x1^2+y1^2)^(3/2)/abs(x1*y2-y1*x2);
> r:=(x1^2+y1^2)/abs(x1*y2-y1*x2);
> xx:=x-r*y1;
> yy:=y+r*x1;
>
animate({[5*cos(s),3*sin(s),s=0..2*Pi],[xx+rr*cos(s),yy+rr*sin(s),s=
0..2*Pi]},t=0..2*Pi,color=black,axes=none);