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