4109.sagews

This is the trajectory followed by a point on the surface of a disk while the disk rolls on the floor, without slipping, with a uniform linear velocity. The point is chosen at a distance 'r' from the center of the disk.

R = 4.
r = 2.

var('wt')
parametric_plot(( r*sin(wt) + R*wt, r*cos(wt)), (wt, 0, 4*pi))

This is the trajectory of a point the lies on the perimeter of the disk as it rolls on a flat surface.

R = 4.
r = 4.

var('wt')
parametric_plot(( r*sin(wt) + R*wt, r*cos(wt)), (wt, 0, 4*pi))

Spirograph: Inner perimeter

R = 100.
r = 50.
r_ = 40.
n = 100
var('wt')
parametric_plot( (R*cos(wt) + r_*cos((R+r)*wt/r) , R*sin(wt) - r_*sin((R+r)*wt/r)), (wt, -n*pi, n*pi))

R = 100.
r = 70.
r_ = 60.
n = 100
var('wt')
parametric_plot( (R*cos(wt) + r_*cos((R+r)*wt/r) , R*sin(wt) - r_*sin((R+r)*wt/r)), (wt, -n*pi, n*pi))

R = 100.
r = 80.
r_ = 50.
n = 100
var('wt')
parametric_plot( (R*cos(wt) + r_*cos((R+r)*wt/r) , R*sin(wt) - r_*sin((R+r)*wt/r)), (wt, -n*pi, n*pi))

R = 100.
r = 70.
r_ = 50.
n = 200
var('wt')
parametric_plot( (R*cos(wt) + r_*cos((R+r)*wt/r) , R*sin(wt) - r_*sin((R+r)*wt/r)), (wt, -n*pi, n*pi))

R = 100.
r = 90.
r_ = 20.
n = 100
var('wt')
parametric_plot( (R*cos(wt) + r_*cos((R+r)*wt/r) , R*sin(wt) - r_*sin((R+r)*wt/r)), (wt, -n*pi, n*pi))

R = 100.
n = 10

@interact
def f(r=(26..R/2), r_=(14..R/2)):
    var('wt,t')
    pt = parametric_plot( (R*cos(wt) + r_*cos((R+r)*wt/r) , R*sin(wt) - r_*sin((R+r)*wt/r)), (wt, -n*pi, n*pi)) + parametric_plot(((R+r)*cos(t), (R+r)*sin(t)), (t,0,2*pi), color='red')
    show(pt)

If the smaller circle (the one with radius = r) is made to 'roll' along the outer perimeter of the bigger circle (whose width is negligible in this case), the following differences are to be taken care of:

- the radius of the bigger circle now will be equal to R

- the radius of the circle traced by the smaller circle will be R+r

- both rotations are positive.

R = 500.

@interact
def f(r=(550..2*R), r_=(420..2*R), n=30):
    print("R="+str(R))
    var('wt,t')
    pt = parametric_plot( ((R + r)*cos(wt) - r_*cos((R+r)*wt/r) , (R+r)*sin(wt) - r_*sin((R+r)*wt/r)), (wt, -n*pi, n*pi)) + parametric_plot((R*cos(t), R*sin(t)), (t,0,2*pi), color='red')
    show(pt)

r  r_  n  [removed] [removed] [removed] [removed]

R = 500.

@interact
def f(r=(R/2..2*R), r_=(R/4..2*R), n=10):
    #r = 100; r_ = 100
    var('wt')
    pt = parametric_plot( ((R + r)*cos(wt) - r_*cos((R+r)*wt/r) , (R+r)*sin(wt) - r_*sin((R+r)*wt/r)), (wt, -n*pi, n*pi)) + parametric_plot((R*cos(t), R*sin(t)), (t,0,2*pi), color='red')
    show(pt)