1929-Two Cylinders.sagews

Two Cylinders

Tim McLarnan, Earlham College

This is just a small worksheet I produced for a class who were working on the classic calculus problem of computing the volume shared by two cylinders of equal radius which meet at right angles.  Students were having trouble visualizing the intersection, and my drawings were not helping; so I went to Sage for backup.  Notice that all the figures here can be grabbed and rotated.

var('theta t')
tube1 = parametric_plot3d([t, cos(theta), sin(theta)], (theta,0,2*pi), (t, -2, 2), color = 'brown', opacity=1)
tube2 = parametric_plot3d([cos(theta), t, sin(theta)], (theta,0,2*pi), (t, -2, 2), color = 'blue', opacity=1)
show(tube1+tube2, aspect_ratio=[1,1,1])

curve1 = parametric_plot3d([cos(theta), cos(theta), sin(theta)], [theta, 0, 2*pi], color='black', thickness=2)
curve2 = parametric_plot3d([cos(theta), -cos(theta), sin(theta)], [theta, 0, 2*pi], color='black', thickness=2)
tube1 = parametric_plot3d([t, cos(theta), sin(theta)], (theta,0,2*pi), (t, -2, 2), color = 'brown', opacity=0.5)
tube2 = parametric_plot3d([cos(theta), t, sin(theta)], (theta,0,2*pi), (t, -2, 2), color = 'blue', opacity=1.0)
show(tube1+tube2+curve1+curve2, aspect_ratio=[1,1,1])

curve1 = parametric_plot3d([cos(theta), cos(theta), sin(theta)], [theta, 0, 2*pi], color='black', thickness=2)
curve2 = parametric_plot3d([cos(theta), -cos(theta), sin(theta)], [theta, 0, 2*pi], color='black', thickness=2)
tube1 = parametric_plot3d([t, cos(theta), sin(theta)], (theta,0,2*pi), (t, -2, 2), color = 'brown', opacity=0.5)
tube2 = parametric_plot3d([cos(theta), t, sin(theta)], (theta,0,2*pi), (t, -2, 2), color = 'blue', opacity=0.5)
show(tube1+tube2+curve1+curve2, aspect_ratio=[1,1,1])

curve1 = parametric_plot3d([cos(theta), cos(theta), sin(theta)], [theta, 0, 2*pi], color='black', thickness=2)
curve2 = parametric_plot3d([cos(theta), -cos(theta), sin(theta)], [theta, 0, 2*pi], color='black', thickness=2)
tube1 = parametric_plot3d([t, cos(theta), sin(theta)], (theta,0,2*pi), (t, -2, 2), color = 'brown', opacity=0.2)
tube2 = parametric_plot3d([cos(theta), t, sin(theta)], (theta,0,2*pi), (t, -2, 2), color = 'blue', opacity=0.2)
face1 = parametric_plot3d([t*cos(theta), cos(theta), sin(theta)], (theta, 0, 2*pi), (t, -1, 1), color='brown')
face2 = parametric_plot3d([cos(theta), t*cos(theta), sin(theta)], (theta, -pi, pi), (t, -1, 1), color='blue')
show(curve1+curve2+face1+face2+tube1+tube2, aspect_ratio=[1,1,1])

curve1 = parametric_plot3d([cos(theta), cos(theta), sin(theta)], [theta, 0, 2*pi], color='black', thickness=2)
curve2 = parametric_plot3d([cos(theta), -cos(theta), sin(theta)], [theta, 0, 2*pi], color='black', thickness=2)
tube1 = parametric_plot3d([t, cos(theta), sin(theta)], (theta,0,2*pi), (t, -2, 2), color = 'brown', opacity=0)
tube2 = parametric_plot3d([cos(theta), t, sin(theta)], (theta,0,2*pi), (t, -2, 2), color = 'blue', opacity=0)
face1 = parametric_plot3d([t*cos(theta), cos(theta), sin(theta)], (theta, 0, 2*pi), (t, -1, 1), color='brown')
face2 = parametric_plot3d([cos(theta), t*cos(theta), sin(theta)], (theta, -pi, pi), (t, -1, 1), color='blue')
show(tube1+tube2+curve1+curve2+face1+face2, aspect_ratio=[1,1,1])

curve1 = parametric_plot3d([cos(theta), cos(theta), sin(theta)], [theta, 0, 2*pi], color='black', thickness=2)
curve2 = parametric_plot3d([cos(theta), -cos(theta), sin(theta)], [theta, 0, 2*pi], color='black', thickness=2)
@interact
def _(stage = (j for j in range(29))):
    if stage<8:
        tube1 = parametric_plot3d([t, cos(theta), sin(theta)], (theta,0,2*pi), (t, -2, 2), color = 'brown', opacity=1-0.1*stage)
        tube2 = parametric_plot3d([cos(theta), t, sin(theta)], (theta,0,2*pi), (t, -2, 2), color = 'blue', opacity=1)
        show(tube1+tube2+curve1+curve2, aspect_ratio=[1,1,1])
    elif stage < 16:
        tube1 = parametric_plot3d([t, cos(theta), sin(theta)], (theta,0,2*pi), (t, -2, 2), color = 'brown', opacity=0.2)
        tube2 = parametric_plot3d([cos(theta), t, sin(theta)], (theta,0,2*pi), (t, -2, 2), color = 'blue', opacity=1-0.1*(stage-8))
        show(tube1+tube2+curve1+curve2, aspect_ratio=[1,1,1])
    elif stage <24:
        tube1 = parametric_plot3d([t, cos(theta), sin(theta)], (theta,0,2*pi), (t, -2, 2), color = 'brown', opacity=0.2)
        tube2 = parametric_plot3d([cos(theta), t, sin(theta)], (theta,0,2*pi), (t, -2, 2), color = 'blue', opacity=0.2)
        face1 = parametric_plot3d([t*cos(theta), cos(theta), sin(theta)], (theta, 0, 2*pi), (t, -1, 1), color='brown',opacity=0.125*(stage-16))
        face2 = parametric_plot3d([cos(theta), t*cos(theta), sin(theta)], (theta, -pi, pi), (t, -1, 1), color='blue',opacity=0.125*(stage-16))
        show(tube1+tube2+face1+face2+curve1+curve2, aspect_ratio=[1,1,1])
    else:
        tube1 = parametric_plot3d([t, cos(theta), sin(theta)], (theta,0,2*pi), (t, -2, 2), color = 'brown', opacity=0.2-0.05*(stage-24))
        tube2 = parametric_plot3d([cos(theta), t, sin(theta)], (theta,0,2*pi), (t, -2, 2), color = 'blue', opacity=0.2-0.05*(stage-24))
        face1 = parametric_plot3d([t*cos(theta), cos(theta), sin(theta)], (theta, 0, 2*pi), (t, -1, 1), color='brown',opacity=0.125*(stage-16))
        face2 = parametric_plot3d([cos(theta), t*cos(theta), sin(theta)], (theta, -pi, pi), (t, -1, 1), color='blue',opacity=0.125*(stage-16))
        show(tube1+tube2+face1+face2+curve1+curve2, aspect_ratio=[1,1,1])

var('theta phi')
outside = parametric_plot3d([cos(theta)*sin(phi), sin(theta)* sin(phi), cos(phi)], [theta,0,2*pi], [phi,pi/4,3*pi/4], color='brown', aspect_ratio=[1,1,1])
inside = parametric_plot3d([sqrt(2)*cos(theta)/2, sqrt(2)*sin(theta)/2, t], [theta, 0, 2*pi], [t, -sqrt(2)/2, sqrt(2)/2], color='brown', aspect_ratio=[1,1,1])
show(outside+inside)