10885 - Martin the Gardener
Moderator: Board moderators
..?
i think that misof's papper is TOO DIFFICULT to understand
and that i don't need it (because i can't make an understanding
)
my idea is following :
1. all points are on a circle.
2. if coords are rational, multiplying appropriate large number, we can make them integer.
3. we can express some distances between two points on a circle with an arbitrary angle.
and that i don't need it (because i can't make an understanding
)my idea is following :
1. all points are on a circle.
2. if coords are rational, multiplying appropriate large number, we can make them integer.
3. we can express some distances between two points on a circle with an arbitrary angle.
Sorry For My Poor English..
There's really a much simpler solution than that described in misof's reference. Here's my method:
First, I'm looking for points with rational coordinates and rational distances between any two of them, of the form:
(x_i, y_i) = (cos t_i, sin t_i), for i=1..13 (i.e. all points are lying on the unit circle.)
The distance between the i-th and j-th points is given by (after trig simplifications):
2 sin(t_i/2) cos(t_j/2) - 2 cos(t_i/2) sin(t_j/2).
Now, if I demand that all sin(t_i/2), cos(t_i/2) are rationals, then all distances and coordinates will become rational, too.
Let sin(t_i/2)=p_i/q_i (for p_i,q_i \in Z.) Then cos(t_i/2)=sqrt(q_i^2 - p_i^2)/q_i.
For cos(t_i/2) to be rational, q_i^2 - p_i^2 must be a square.
The integers p_i, q_i can be found by checking first few pythagorean triples.
The coordinates then can be easily computed by trig formulas.
To get integer coordinates and distances, I multiply each x_i, y_i by the lcm of denominators of all x_i, y_i, and then shift all points such that all coordinates become non-negative.
First, I'm looking for points with rational coordinates and rational distances between any two of them, of the form:
(x_i, y_i) = (cos t_i, sin t_i), for i=1..13 (i.e. all points are lying on the unit circle.)
The distance between the i-th and j-th points is given by (after trig simplifications):
2 sin(t_i/2) cos(t_j/2) - 2 cos(t_i/2) sin(t_j/2).
Now, if I demand that all sin(t_i/2), cos(t_i/2) are rationals, then all distances and coordinates will become rational, too.
Let sin(t_i/2)=p_i/q_i (for p_i,q_i \in Z.) Then cos(t_i/2)=sqrt(q_i^2 - p_i^2)/q_i.
For cos(t_i/2) to be rational, q_i^2 - p_i^2 must be a square.
The integers p_i, q_i can be found by checking first few pythagorean triples.
The coordinates then can be easily computed by trig formulas.
To get integer coordinates and distances, I multiply each x_i, y_i by the lcm of denominators of all x_i, y_i, and then shift all points such that all coordinates become non-negative.
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Antonio Ocampo
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- Joined: Sat Jul 17, 2004 4:09 am
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