Trochoid demo

\theta

r\theta

C

P

a

$P = C + \overrightarrow{CP}= C + ||\overrightarrow{CP}||\frac{\overrightarrow{CP}}{||\overrightarrow{CP}||}=(r\theta, r) + \underbrace{a}_{\textsf{length}}\overbrace{(\cos{\left(\frac{3\pi}{2}-\theta\right)}, \sin{\left(\frac{3\pi}{2}-\theta\right)})}^{\textsf{direction}} = (r\theta - a \sin{\theta}, r - a \cos{\theta})$

r

Compare the above three.js implementation with the following Sage interact from my older Calc2 TechCompanion. Click the Interact button, then drag the slider to determine how far to move the wheel along the $x$ -axis.

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a = 2;  b1 = 0.4; b2 = 1.7;
​
# Trochoid generated by a point close to the wheel's hub.
x1(t) = a*t - b1*sin(t)
y1(t) = a - b1*cos(t)
​
# Trochoid generated by a point close to the wheel's rim.
x2(t) = a*t - b2*sin(t)
y2(t) = a - b2*cos(t)
​
@interact
def _(h=(0.01, 4*pi)):
        p = line( [(-a,0),(4*pi+a,0)] );
        p += line( [(0,0), (0,2*a)] );
        p += parametric_plot( (x1(t), y1(t)), (t, 0, h) )
        p += parametric_plot( (x2(t), y2(t)), (t, 0, h), color='red' )
        p += circle( (a*h, a), a, color='green' )
        p += line( [(a*h,a),(x1(h),y1(h))], color='green' )
        p += line( [(x2(h),y2(h)),(x1(h),y1(h))], color='green' )
        show(p, figsize=10)