I have no idea.
I would guess … “yes”?
(a=V*V/r , so as r decreases a increases … V is unchanged… ?)
But how is this related to the original topic about the involute? There is a geometrical correlation/analogy?
Infinite tension and infinite curvature imply each other I think. As the point mass spirals in, the distance between it and the tangent contact point of the string goes to zero, hence the radius of curvature goes to zero and the curvature goes to infinity.
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I think the misconception many people have is that infinity is similar to a very large number. As David already said, its not a number but a concept of dealing with edge cases from a pure mathematical standpoint.
There is a good reason they teach you in school not to divide by zero. This is what is happening here. To curve something infinitely is not only a weird statement, you could also say you can not compute the curvature or the tension or whatever gets divided by zero. That doesn’t necessarily mean that there is something like infinite curvature.It only means that for specific programming languages, a division by zero yields a special case of double called positive infinity.
At least this would my personal interpretation of what’s going on here.
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I do believe that the existance of this universe is a de-facto admission that either infinity exists. Regardless of how you view it, infinity must exist.
Either this place existed forever, or something that created it existed forever in a endless cycle.
I do firmly believe we will perform all permutations and end up here again, makes me quite happy actually. I try to be on the good choices permutation.
Fascinating truly !
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I mean I don’t want to say that there is no infinity. We will never know. I’m just pointing to the fact that we divide by zero here, which is usually “undefined” for real numbers.
Another view is, how often can you fold a paper? I guess it ends when there are no particles there anymore…
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The curvature of an involute curve cannot be calculated directly at the intersection with the base curve. But it can be calculated any point not at the intersection. And the closer to the intersection the curvature is calculated the larger it will be. Pick a point close to but not at the intersection and calculate the curvature. Move to a point closer to the intersection and calculate the curvature. It will be larger than at the previous point. And so forth. This is what is meant by infinite curvature at the intersection. No need to invoke cosmology or similar.
But division by zero does not always mean “infinity”. It depends on the situation. For example consider sin(x) / x as x approaches 0. The result will asymptote to 1, not infinity. Pick a value of x close to zero and calculate sin(x) / x. The result will be close to 1. Next pick a value of x closer to zero and the result will be closer to 1. And so forth. So the value of sin (x) / x with x=1 is considered to be 1. Another way to express this is “The limit of sin(x) / x = 1”.
The concept of “infinity” was around long before digital computers were conceived.
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Well… my whole premise is that I was looking at the curvature graph.
Mine is a “visual” interpretation, I were not doing any math here.
We can evaluate the involute a “bit” away from the originating circle, and the curvature graph (imo) still suggests it “tends to” infinity curvature.
I really like this.
You solved this virtual, abstract problem with a real and simple point of view.
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