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

5/24 something getting active now :/ unsure where that leaves workable areas

we’re mostly nonfunctional today, looking for any activity. gear bearings were exciting, pretty hard to find exciting things

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.

i’m thinking another trick might be that when pulling on the thread anything nearer to the spool than the tangent point will pull away, so this length would extend in the direction of the tangent. maybe not the most intuitive explanation but describes conceptual dots.

ok so: tangent and arclength as function of angle maybe?

arclength not hard. C=pi*r*2 i think, so arclength = radians * radius i think

then the tangent is the perpendicular to the normal, that is, the vertical component is the cosine and the horizontal the sine, with one of them negative and the other not.

ok. so we’d move from the point on the circumference, in the direction of the tangent, by the arclength, for each angle of the curve. noting that there’s only 1 curve shape for a given radius?
- (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))

( :D )

i wonder if there is a way to write it in vector notation, i guess i’d cross the angle vector with Z to take perpendicular :s

oh i could use imaginary exponents

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

i wonder if wolfram alpha will plot it on range like 0..2pi, r=1

…

i removed the r because it scales all points the same. i didnt find how to plot the exponential version, but did plot it parametrically. the curve is tighter than i expected thinking about profiles

> parametric plot (cos(t) - t * sin(t)), (sin(t) + t * cos(t)), t=0..2pi/4

i guess looking at it, i’m wondering if there’s a mistake. ohhhh i see, i think, it starts at the right with a thread length of 0, then once it unwinds 90 degrees it is actually pulling from much farther along the spool

i’m considering the result will be incorrect if the tangent is taken in the opposite direction as the winding. i guess i should figure whether i got that right.

looks like its unwinding counterclockwise because sin rises and cosine drops as angle rises

i think it’s possibly right? with the sine negative, it will start aiming to the left and up, whereas if the cosine were negative it would be aiming to the right and down. is that right? hmmm …

thinking of unwinding thread. you start with end matching radius, but as you pull the tangent point moves away from the end. it’s actually pulling in the opposite direction! the tangent point travels in the opposite direction as the line to the end. oops.

so if it’s unwinding counterclockwise, then the line should be drawn along the clockwise tangent, i believe. better fix that.

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

i doubled the domain to see the origin

this looks more correct!