content
Question one
Circle A and circle B have the same radius. A circles around B without sliding. How many circles does A make in total?
The answer is 2 circles. There are many ways to prove it. Regular polygons can be used to approximate circles.
Question two
The radii of circle A and circle B are r1 and r2 respectively. A circles around B without sliding. How many circles does A make in total?
The answer is (r1+r2)/r1
Obviously problem one is a special case of problem two.
Question three
The radius of a circle A is r, and A rotates around the outside of a convex simple curve B. How many times does A rotate in total?
There are two ideas. The first idea is the perimeter of B / r + 1, and the second idea is the trajectory length of the center of A / r
Obviously, these two ideas are not equivalent, but the result is the same for the second problem.
When I think about whether the first idea is right or the second idea is right, there are two ideas. The first is to see which is more universal for different scenarios, and the second is to directly base on calculus. Calculate.
Let's first look at different scenarios to get a better understanding of the formula.
Question four
The radii of circle A and arc B are r1 and r2 respectively, the angle of the arc is d, and A goes from one end of B to the other end of B. How many times does A rotate in total?
According to the conclusion of question 2, it is easy to deduce that it is (r1+r2)/r1 * d / (2π)
When d=2π, B is a circle, the above formula=(r1+r2)/r1
With the conclusion of the arc, the convex simple curve can be understood as a lot of small arcs.
Question five
The radii of circle A and arc B are r1 and r2 respectively, the angle of the arc is d, and A goes from one end of B to the other end of B on the inside of B. How many times does A rotate in total?
The answer is (r2-r1)/r1 * d / (2π)
summary
After analyzing various scenarios, it is still difficult to analyze which of the ideas 1 and 2 of question 3 is correct.
However, after referring to the idea of relative motion mentioned by some netizens, it should be the first idea.
Generalizing to general curves:
The radius of circle A is r, the length of simple curve B is c, and A goes from one end of B to the other end of B. How many times does A rotate in total?
Divide B into convex and concave segments, and find radians for each segment. The total number of turns of A is c/r + (sum of convex radians - sum of concave radians) / (2π)