| Exact Coordinates |
Robert P. Munafo, 2003 Sep 22.
Using the Brown method, the boundary of many small period cardioids and circular mu-atoms can be found. For example, the following formulas express the relation between a point c on the boundary of a mu-atom and a point D on the unit disk:
| R2a | D2/4 - D/2 + c = 0 | |
| R2.1/2a | D/4 - 1 - c = 0 | |
| R2F(1/2B1)Sa, R2.1/3a and R2.2/3a | c3 + 2 c2 - (D/8-1) c + (D/8-1)2 = 0 | |
| Period-4 mu-atoms | c6 + 3 c5 + (D/16+3) c4 + (D/16+3) c3 - (D/16+2) (D/16-1) c2 - (D/16-1)3 = 0 | |
| Period-5 mu-atoms |
c15 + 8 c14 + 28 c13 + (mu + 60) c12 + (7 mu + 94) c11
+ (3 mu2 + 20 mu + 116) c10 + (11 mu2 + 33 mu + 114) c9
+ (6 mu2 + 40 mu + 94) c8
+ (2 mu3 - 20 mu2 + 37 mu + 69) c7
+ (3 mu - 11) (3 mu2 - 3 mu - 4) c6
+ (mu - 1) (3 mu3 + 20 mu2 - 33 mu - 26) c5
+ (3 mu2 + 27 mu + 14) (mu - 1)2 c4
- (6 mu + 5) (mu - 1)3 c3 + (mu + 2) (mu - 1)4 c2
- (mu - 1)5 c + (mu - 1)6 = 0
where mu = D/32 |
The formulas for period-1 through period-3 mu-atoms can be solved explicitly; the others are evaluated numerically using Newton's method or a similar technique (see derivative).
See also rational coordinates.
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