Algebraic Manipulation Is Overrated
An intuition question.
Look at the function below. It may surprise you that it is a constant! For any value of 
Can you find some geometric way to see that?
It took me somewhere between a half hour and an hour of playing around to get it. I can post my solution in a couple days, but right now I don’t have the energy to find a program to draw my solution [1]. But let me just tell you: it’s beautiful. You’ll be stunned when you first do it. Yeah, the calculus way tells you it is a constant, but seeing the “why” is still a mystery. The geometric way takes a bit, but whoa nellie, you won’t regret spending the time!
[1] Or maybe I should claim there is no room in the margin! (JK)
Update: I did finally write up my solution. I quickly did something I never have done before: do my work in powerpoint. It worked fine.
Update: Mr. K solved the problem in 3 minutes and found a way to show the geometric solution. Head over to his very excellent blog to see it in all it’s glory.
Update: Besides mine and Mr. Ks, a third and perhaps more elegant solution is up at 11011110.
Of the three, I think I like Mr. K’s visualization best, even though it might not be a proof in the formal sense.
Mr. K’s solution is different from mine, in which (sin x + sin x+a)/2 and (cos x – cos x+a) are the height and base length (respectively) of a family of similar isosceles triangles with apex angle a: http://11011110.livejournal.com/139750.html
(in the diagram in that post, let AD be a horizontal line, and let B and C be points on the unit circle at angles x and x+a from horizontal).
Is there an animation of this?
(I remember seeing one before…)
Hi Pierre,
Actually Mr. K at MathStories created a great visualization for this problem. For some reason, at this moment, I can’t access the page, but it’s here: http://blog.mathsage.com/?p=202
Best,
Sam
Yes, the animation at
http://blog.mathsage.com/?p=202
shows well was is going on.
The straight line segment trace by the terminal point is a hypocycloid, corresponding to the case of a circle of radius 1/2 rolling without slipping inside a circle of radius 1.
See the case a/b=2 at
http://mathworld.wolfram.com/Hypocycloid.html
http://mathworld.wolfram.com/TusiCouple.html