Mathematical Experiment 10

exp10.mws
  1. Construct the astroid.

    		2.
  2. Construct the deltoid.

    		3.
  3. Construct the nephroid.

    		4.
  4. Construct the cardioid.

    		5.
  5. Construct the inversion of the cardioid with respect to a circle.

    		6.
  6. Construct the inversion of the nephroid with respect to a circle.

    		7.
  7. Construct the inversion of the deltoid with respect to a circle.

    		8.
  8. Construct the inversion of the astroid with respect to a circle.

    		9.
  9. Construct the tangent to each of the curves in problems 1-4.

    		10.
  10. Construct the tangent to each of the inverse curves in problems 5-8.

    		11.
  11. Construct the osculating circle to an ellipse.

    		12.
  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:
  13. 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