# Mathematical Experiment 10

exp10.mws

1. Construct the astroid.
2. Construct the deltoid.
3. Construct the nephroid.
4. Construct the cardioid.
5. Construct the inversion of the cardioid with respect to a circle.
6. Construct the inversion of the nephroid with respect to a circle.
7. Construct the inversion of the deltoid with respect to a circle.
8. Construct the inversion of the astroid with respect to a circle.
9. Construct the tangent to each of the curves in problems 1-4.
10. Construct the tangent to each of the inverse curves in problems 5-8.
11. Construct the osculating circle to an ellipse.
12. The center of curvature (xc,yc) of a plane curve (x,y) is given by
13.  xc = x- (x¢)2+(y¢)2 | x¢y¢¢-y¢x¢¢| y¢
 yc = y+ (x¢)2+(y¢)2 | x¢y¢¢-y¢x¢¢| x¢.
These are the steps to draw the line segments connecting points of the the curve
 (cos2t+2cost,sin2t+2sint),0 £t£ 2p
to the center of curvature with Maple:
 restart: x:=cos(2*t)+2*cos(t): y:=sin(2*t)+2*sin(t): x1:=diff(x,t): y1:=diff(y,t): x2:=diff(x1,t): y2:=diff(y1,t): r:=(x12+y12)/abs(x1*y2-y1*x2): xx:=x-r*y1: yy:=y+r*x1: m:= x,y , xx,yy : t:=n*2*Pi/100: w:=evalf(m): plot( w\$n=0..100,color=black,axes=none);

To avoid the problem of ``Dividing by Zero'', replace the line

 >  t: = n*2*Pi/100:
with the line
 >  t: = 0.001+n*2*Pi/100:

This is the drawing of the line segments connecting points of the the curve

 (cos3t+3cost,sin3t+3sint),0 £t£ 2p
to the center of curvature:

This is the drawing of the line segments connecting points of the the curve

 (cos4t+4cost,sin4t+4sint),0 £t£ 2p
to the center of curvature:

This is the drawing of the line segments connecting points of the the curve

 (cos3t+3cost,sin3t-3sint),0 £t£ 2p
to the center of curvature:

This is the drawing of the line segments connecting points of the the curve

 (cos2t+2cost,sin2t-2sint),0 £t£ 2p
to the center of curvature:
14. Construct the animation of the osculating circle of the ellipse with Maple.
 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:=(x12+y12)(3/2)/abs(x1*y2-y1*x2); r:=(x12+y12)/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);

exp10.tex