it said an involute curve is like unwinding a spool of thread …

so if unwinding thread, basically the length of your line is equal to the arc length it was unwound from.

we could describe that as a function of the angle of unwinding. for each point on the circumference, there’d be an associated line length.

- (point on circumference) + (vector of tangent) * arclength

function of angle

r = radius of spool

t = time/theta

x(t) = r * (cos(t) + t * sin(t))

y(t) = r * (sin(t) - t * cos(t))

p(t) = r * (exp(i t) - i t * exp(i t))

> parametric plot (cos(t) + t * sin(t)), (sin(t) - t * cos(t)), t=0..2pi/2