Re: Bouncing Path in Circle
- From: quasi <quasi@xxxxxxxx>
- Date: Sat, 28 Jul 2007 14:28:31 -0400
On Sat, 28 Jul 2007 10:53:09 -0700, Narasimham <mathma18@xxxxxxxxxxx>
wrote:
On Jul 28, 2:11 pm, Costas Vlachos <c-X-vlac...@xxxxxxxxxxxxxx> wrote:
Hi all,
Here is a problem I came up with, which I don't know how to approach.
Suppose we have a circle and we pick a starting point anywhere on the
circumference of that circle. We then choose an angle Theta (with
respect to the tangent of the circle at the starting point) and we
"depart" from that point towards the inside of the circle, moving in a
straight line. As the path reaches the circumference at another point,
the path is "reflected" according to the tangent of the circle at that
second point, and continues in a straight line towards another point in
the circumference, where it is reflected again and so on. This continues
for infinite time. You can think of this as a beam of light starting
from a point on the circumference towards the inside of the circle, with
a given angle Theta, and being reflected as it hits the circumference at
various points in its infinitely long path.
When Theta = 90 degrees, the path will simply "bounce" between only two
points on the circumference, passing through the centre of the circle
and travelling along its diameter. When Theta = 60 degrees, the path
will visit only 3 points on the circumference and will form an
equilateral triangle enclosed by the circle. Similarly, when Theta = 45
degrees, the path will visit only 4 points on the circumference and will
form a square enclosed by the circle. Other angles may result in the
points being "shifted" at each iteration, resulting in more complex
shapes and many more points visited.
Question 1: Is there an angle Theta for which the path will eventually
visit *all* points on the circumference of the circle? If yes, what is
the value (or values) of this angle?
Question 2: Is there an angle Theta for which the path will eventually
visit *all* points in the disk that is defined by the circle (i.e., it
will "fill" the entire disk)? If yes, what is the value (or values) of
this angle?
Many thanks for any insights into this problem. This is purely out of my
own curiosity, I just want to know if there is an answer and the method
needed to approach it.
[PS: I posted this message again a few days ago using another news
server, but it never appeared in the group. Apologies if this appears
twice to some of you. Thanks.]
--
Regards,
Costas
Well posed questions. First addressed this problem when winding
surfaces of revolution with a basket weave using filaments.
First, a fractional polygon definition. n/k (n > k > 1) is a
rational number where n and k are integers. A n/k polygon is an 'n'
sided regular polygon if k =1 and an n*k sided self-intersecting star
of 'n' spikes (or vertices) if k > 1. E.g., a 5/1 polygon is a
regular pentagon, 5/2 is the common pentagonal star with 5 spikes
formed in two revolutions of a tracing point along segments/sides of
the polygon around the circle center (radius R) during repeated
bouncing. It is thus a 2.5 cornered star, so to say, with Theta = 72
degrees.
If Theta satisfies n/k (no common factors for n and k) = pi / Theta
(radians) then the star is self-repeating after k rotations. If pi/
Theta is irrational then even after an infinite number of rotations
there is no closure star pattern.The smallest circle to which the
sides are tangent is simply r/R = cos(Theta)= cos(k pi/n).
When n and k are integers, there is periodic bouncing after n points
are impacted at reflection.
Right, but possibly less. More precisely, the number of points hit
(counting the initial point) before repeating is n/d where d=(k,n).
When n/k is irrational then eventually all points of the circle are impacted.
That's false. As several posters have pointed out, the set of points
hit is at most countably infinite.
quasi
.
- References:
- Bouncing Path in Circle
- From: Costas Vlachos
- Re: Bouncing Path in Circle
- From: Narasimham
- Bouncing Path in Circle
- Prev by Date: Re: type of diophantine equations
- Next by Date: Re: JSH: Stolen dreams and the distributive property
- Previous by thread: Re: Bouncing Path in Circle
- Next by thread: Re: Bouncing Path in Circle
- Index(es):
Relevant Pages
|
