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
yayyy involute curve yayy

so now how is this related to gears?

after plotting an involute curve i felt more comfortable understanding material a little when a websearch led me back to

https://khkgears.net/new/gear_knowledge/introduction_to_gears/involute_tooth_profile.html .

they draw a circle partway up the teeth of a gear and call it the base circle. if two gears move at the same velocity and are continuously touching teeth, the contact points follow a shared tangent of the two base circles as if they were connected via a strap with opposite wrapping on each.

so if a tooth contact point were at the radius of the base circle, at the endpoint of the shared tangent, then turning the gear toward the center would move it farther from its base circle, and toward the opposing gear. Meanwhile, the opposing gear has the opposite: its contact point progresses toward it.

looking a little more.

to make this contact line, an involute curve would be rotated as the thread is pulled. maybe like rotating a spool toward the thread pulled from it.

having some trouble engaging. the pitch point is the point of contact that is most distant from both gears and on a line between their centers. the pitch circle is a circle of this radius about a gear. the pressure angle is the angle of the tooth at this point — the angle of the tooth tangent from the circle normal or radius, or identically the angle of the tooth normal from the circle tangent or a line between the gears.

So a 0 degree pressure angle would mean the gears push in the direction of motion like stepping on somebody’s foot, and a high degree would mean the gears push against each other a lot, exerting pressure to spread apart, maybe.

1909