Graphics Reference
In-Depth Information
How algebraic geometry can be used to tile a planar curve defined by
f (x,y) = 0
Step 1:
Let p be a nonsingular point of f.
Step 2:
Trace f one step starting at p in the tangent direction to the next point p ¢.
Step 3:
If we are done, then quit.
Step 4:
If p ¢ is a nonsingular point, then let p be p ¢ and go to Step 2.
Step 5:
Transform the curve f to a curve f 1 so that the singular point p ¢ gets sent to the
origin.
Step 6:
Use a birational map F to transform the curve f 1 to a curve g, so that
(a) There is a neighborhood U of the curve f 1 about the origin and U - 0 gets
mapped in a bijective way onto F( U )-F( 0 ) by F.
(b) Each place of f 1 centered at 0 gets mapped to a regular place of g.
(c) The center of these places of g is a regular point of g.
Step 7:
Trace g to get past the singularity of f and to a new nonsingular point p .
Step 8:
Go to Step 2.
Algorithm 14.5.1.1.
Incremental curve tiling algorithm.
() =+ +
01 2 2
xt
a
at a t
+
...
...
() =+ +
2
(14.9)
yt
b
bt bt
+
0
1
2
corresponds to a place of f with center p 0 . We can solve for these power series by
substituting them into the equation (14.8) and setting the coefficients to 0.
14.5.1.1
Example.
To find the parameterization (14.9) at the origin for f(x,y) =
x 3
- x 2
+ y 2 .
Solution.
In this case (a 0 ,b 0 ) = (0,0) and we need to solve
2
2
ba
aaabb
-=
0
1
1
3
-
2
-
2
=
0
12
12
2
2
2
3
aa
-
2
aa
-
a
+
2
bb
+
b
=
0
2
1
3
13
(14.10)
...
...
Two solutions to (14.10) are
() =
() =- () - () -
xt
t
2
3
(14.11)
yt
t
12
t
18
t
...
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