Hi all.
Simple curious thing:
Am I understanding this wrong?
I would never have guessed this correctly…
I wonder why is this so.
Also this make me think that while using involutes (like in gears) it would be best practice to avoid contact near that point, too much wear…
Strange!
In most C languages, if you divide a positive floating point number by zero, you will get positive infinity as a result! Not a ZeroDivisionError like in Python. It should be seen as undefined instead.
There are couple reasons why you can get a zero division here.
If you have a curvature circle with an radius of 0 for instance:
then the curvature radius is 0 you divide a K = 1.0/0.0. K can also be determined by
If both derivatives are 0, then you have a problem…
Basically it could indicate that there you can’t simply compute curvature here.
Other than that, an interpolation will always diverge from the actual curve. This is also-error prone. Not sure if this answers the question, but this would be my guess.
Yeah, ok.
But by increasing the point count we can see how generally the shape and the curvature graph tends to stabilize … and at the start it really seems the curvature simply go nuts.
I was trying to make a clean curve of involute + dedendum, but it seems that at the start of the involute there is an equivalent of a G0 continuity.
Like, having that shape manufactured you would always feel by hand a small “dent” in that position, an edge.
If you start normal from the circle and then become spiral, I would expect the initial curvature to be very high at the beginning, and then getting less and less. The interpolation will just prevent you from having a nice flow of curvature, especially at the beginning. But this is due to the problems of interpolation. But this does not mean that the curvature is positive infinity at some point. I think the curvature graph itself makes sense. If you get positive infinity at t=0, then its rather a problem of zero division. Btw, you might also get this problem if you divide by a number smaller than epsilon of double!
bool isInf = double.IsPositiveInfinity(1.0 / 0.1e-500); //yields true in C#
The involute of a circle has zero radius / infinite curvature where it intersects the circle. The rate of change of curvature is inversely proportional to the distance along the curve from the intersection with the circle.
I find it very confusing to speak of infinite curvature. You divide a pizza into 0 pieces, so you get infinite pieces? Or a singular beer belly is infinitely curved? Problem solved 
What if there is no curvature. If there is no derivate, there is no way you can compute curvature. Its not entirely wrong to speak of infinite curvature, you see this often, but I think undefined curvature feels more right…
That’s why I wanted to create this “meta” discussion: it’s incredibly odd to me, but the more i increase the detail, the more is confirmed:
Clearly it “tends” or “suggest” infinite curvature. Radius = 0 at start.
Extremely odd for me, because the involute also have to be perpendicular to the primitive circle.
My insight failed me this time.
A better example: If you divide a pizza into pieces with zero area you get an infinite number of pieces.
Infinity is not a number but a concept. Infinity is larger than any finite number. Back to the example of dividing a pizza. A pizza can be divided into any number of pieces if each piece is small enough. For more pieces just divide into smaller pieces. The pizza can be divided into any arbitrarily large number of pieces.
The curvature of the involute of a circle doubles each time the distance to the intersection point is halved. Pick a very large curvature. There is some distance from the intersection point where the curve has that curvature. Pick a larger curvature. There is a point closer to the intersection poiint where the curvature has that value. And so forth.
But this sounds rather like pieces = area of pizza / positive-infinite cuts. Unfortunately this equals 0 on my PC. I would have expected to get positive-infinite pieces.
I mean in this case, if you get closer to t=0, you converge to infinity. That makes sense somehow. But I still find it strange to imagine a purpose of having infinite curvature…
Anyway… I guess one day, maybe someone comes up with a better concept.
The moral of the story, never ask someone to divide your pizza infinitely. Sounds like I should go for lunch.
Yeah, this wanted to be that kind of meta discussion.
more than “purpose”, a real life scenario, a real life case where that happens. What does that mean?
Now i’m thinking of a rectangle rolling around the circle. Due its mass-center being away from the surface of the circle, the motion is able to keep inertia/momentum.
The smaller you make the rectangle, the more violent is the rolling over the circle…
… until > tends to infinitely small > the rectangle no longer “roll” but simply “slam flat” on the circle, losing its kinetic energy.
Maybe this is a brain fart…
I thought of the area thing as well, but I’m pretty positive the comparision doesn’t work like that. The area should have no influence. if you divide a pizza by 0 people, you can indeed serve infinitely many slices of pizza to noone and still have much pizza left
i would not say that this is correct necessarily. how big of a big number can you imagine, if we settle on being able to imagine infinitely long the answer becomes equal 
if the slices are infinitely small you can serve them to infinite amount of people and still almost (minus the infinite small slices which start at an infinitely small piece) have a whole pizza left.
yeah, but wouldn’t that rather represent:
infinity / infinity = 1
instead of
1 / 0 = infinity (argh, that division by 0 severely hurts my eyes)
if our universe is infinite, it means we all come back to the beginning of this discussion at some point. With infinite permutation, there are infinite copies of us. Not sure if I would like this, but it certainly is a strange thing. If there is no infinity in this universe, then there is no infinite curvature either. Nobody has observed infinity yet. Only reptoloids told you so.
if we see it from a binary point, it sure does, anything that exists divided by itself is true (1) anything that does not exist is zero.
the question is does infinity exist. probably existence is infinity, one big bang, another, maybe another, it all keeps banging… or maybe at some point all energy is lost and the universe ceases to exist, reality becomes flattened and fades into an infinity of nothingness. then infinity can become zero. 
I’m afraid I’m way too stupid to think of the existence of infinity and my brain’s structrure not adapted to that at all.
At least in math infinity exists. Catmull and Clarke got a nobel price for calculating the new vertex points location if you were to smooth it infinitely often.
Also Hawking reviewed his big bang theory and set up another theory that the big bang does not exist, but rather a pulsation where the universe would by gravity get back to one huge block before re-exploding into some kind of pseudo-big bang.
hopefully no physicist joins this discussion 
they all are just theorists anyway energy conversion bla bla. no seriously, if people believe they get infinite chances they never stop messing everything up.
I’d give you infinitely many 
if I could for this response 
Another way to look at this: if you had a weight (point mass) on a string wrapping around and spiraling in toward a circle/cylinder under its own momentum, would the tension on the string increase without bound as the weight completed its approach?
