- #1
Loren Booda
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What is the average number of intersections for two infinite curves confined to a plane?
How long is a piece of string?Loren Booda said:What is the average number of intersections for two infinite curves confined to a plane?
Dragonfall said:How do you define a random curve? A random walk is discrete, but random curve?
Dragonfall said:No, when you just "choose" an element out of the set of all curves it isn't "random" in the probabilistic sense. First of all can we even define a probability measure on that set? How big is it? It might be bigger than the set of reals.
Loren Booda said:It may turn out that infinite sets cannot be averaged or proportioned, but if they can, I believe one of your answers is the correct number.
Dragonfall said:No, when you just "choose" an element out of the set of all curves it isn't "random" in the probabilistic sense. First of all can we even define a probability measure on that set? How big is it? It might be bigger than the set of reals.
Dragonfall said:The problem with the step size approach is that the limit might not be a curve, so there's no sense of talking about "intersecting" itself.
Russell Berty said:The set of all curves in R^2 is probably larger than the set of all reals. And the upper bound is the size of the set of all functions from R to R.
junglebeast said:The average of any (finite or infinite) set of numbers that are non-negative and contains at least 1 infinity is equal to infinity.
The number of intersections for 2 curves on a plane can be calculated using the intersection formula, which is given by (m1 - m2)/(1 + m1m2), where m1 and m2 are the slopes of the two curves. This formula can be derived from the equation of a line, y = mx + b, by setting the two equations equal to each other and solving for x.
Yes, two curves can intersect at more than one point on a plane. This can happen when the two curves have more than one point in common, or when the two curves intersect at a tangent point.
A negative value from the intersection formula indicates that the two curves do not intersect on the plane. This can happen when the two curves are parallel or when they are the same curve.
Yes, the number of intersections for 2 curves on a plane can change if the curves are moved or transformed in some way. For example, if one curve is rotated, its slope will change and therefore the number of intersections may change as well.
No, there is no limit to the number of intersections for 2 curves on a plane. The number of intersections can be any whole number, including zero, depending on the slopes and positions of the two curves.