Arc Length of Cycloid
Arc Length of Cycloid
A polygon rolls on a line . The positions of a vertex when has a side flush with form a polygonal path (orange). The orange segments are the base sides of green isosceles triangles. When you drag the "combine" slider, the green triangles combine to form a right triangle with height , the diameter of incircle of the polygon. Hence, the length of the orange path is the sum of the height and the hypotenuse of this triangle.
Q
m
Q
m
2r
As the number of sides goes to infinity, the orange path approaches a half cycloid and the hypotenuse of the triangle approaches . Hence the arc length of one arc of the cycloid is .
2r
8r
External Links
External Links
Permanent Citation
Permanent Citation
Okay Arik
"Arc Length of Cycloid"
http://demonstrations.wolfram.com/ArcLengthOfCycloid/
Wolfram Demonstrations Project
Published: March 7, 2011