WOLFRAM NOTEBOOK

WOLFRAM|DEMONSTRATIONS PROJECT

Country Curves

​
country
Belgium
number of Fourier series terms
50
rationalization tolerance
0.1
0.01
0.001
trace country border
parametric plot ofFourier series approximation
polygon of approximatecountry border coordinates
parametric equation: {x(t),y(t)} =
1
24
cos3-
2463t
8
+
1
7
cos
19
8
-
1508t
5
+
1
18
cos
26
9
-
2953t
10
+
1
22
cos
37
13
-
1131t
4
+
1
11
cos
28
11
-
3041t
11
+
1
10
cos
17
8
-
2972t
11
+
2
13
cos
21
10
-
2375t
9
+
1
9
cos
20
7
-
3349t
13
+
1
19
cos
6
5
-
3104t
13
+
1
16
cos
13
9
-
3952t
17
+
1
11
cos
11
5
-
1131t
5
+
1
6
cos
43
16
-
2419t
11
+
1
4
cos
25
8
-
1709t
8
+
2
7
cos
130
43
-
754t
5
+
1
4
cos
47
19
-
4191t
29
+
1
15
cos
11
5
-
1244t
9
+
1
9
cos
119
40
-
2111t
16
+
3
10
cos
38
15
-
2199t
25
+
3
10
cos
23
12
-
1552t
19
+
3
5
cos
14
5
-
509t
9
+
5
13
cos
14
9
-
377t
12
+
53
21
cos
70
23
-
132t
7
+
33
2
cos
44t
7
+
9
5
+
197
28
cos
88t
7
+
57
23
+
35
34
cos
201t
8
+
11
4
+
3
11
cos
377t
10
+
19
13
+
19
12
cos
1627t
37
+
5
4
+
5
8
cos
553t
11
+
13
10
+
5
6
cos
377t
6
+
19
10
+
5
3
cos
553t
8
+
18
11
+
7
9
cos
377t
5
+
16
9
+
2
5
cos
377t
4
+
16
9
+
10
13
cos
1307t
13
+
6
5
+
2
9
cos
1175t
11
+
11
8
+
1
14
cos
1131t
10
+
17
12
+
5
12
cos
955t
8
+
11
8
+
1
4
cos
377t
3
+
5
3
+
1
7
cos
1885t
12
+
29
11
+
1
9
cos
1797t
11
+
26
9
+
2
9
cos
1866t
11
+
31
12
+
1
5
cos
2287t
13
+
21
11
+
1
8
cos
1640t
9
+
20
11
+
1
6
cos
377t
2
+
7
4
+
1
7
cos
1753t
9
+
35
18
+
1
7
cos
3217t
16
+
17
8
+
1
6
cos
3525t
17
+
13
5
+
1
7
cos
5391t
22
+
19
11
+
1
10
cos
754t
3
+
13
10
+
1
13
cos
10694t
37
+
11
10
+
25077
16
,
1
17
cos
13
9
-
2463t
8
+
1
22
cos
16
7
-
1508t
5
+
1
9
cos
4
9
-
2972t
11
+
1
9
cos
3
5
-
2375t
9
+
1
11
cos
11
10
-
3349t
13
+
1
33
cos
109
36
-
754t
3
+
1
15
cos
4
5
-
5391t
22
+
1
25
cos
63
31
-
2419t
11
+
1
10
cos
1
11
-
377t
2
+
1
12
cos
1
7
-
1640t
9
+
1
15
cos1-
1797t
11
+
1
11
cos
5
9
-
1885t
12
+
1
5
cos
10
19
-
754t
5
+
3
13
cos
41
40
-
4191t
29
+
1
7
cos
5
2
-
1244t
9
+
1
8
cos
6
11
-
1131t
10
+
2
5
cos
10
7
-
1175t
11
+
1
5
cos
17
11
-
1307t
13
+
1
5
cos1-
377t
5
+
2
9
cos
11
7
-
553t
8
+
9
7
cos
2
7
-
1891t
43
+
5
7
cos
7
13
-
377t
12
+
5
2
cos
1
12
-
201t
8
+
88
3
cos
44t
7
+
1
9
+
94
13
cos
88t
7
+
1
8
+
29
10
cos
132t
7
+
17
10
+
7
15
cos
377t
10
+
17
10
+
5
8
cos
553t
11
+
2
9
+
8
9
cos
509t
9
+
16
7
+
5
14
cos
377t
6
+
9
4
+
1
4
cos
1307t
16
+
1
8
+
1
8
cos
2023t
23
+
15
14
+
1
7
cos
377t
4
+
1
6
+
2
9
cos
955t
8
+
11
12
+
2
13
cos
377t
3
+
9
11
+
1
11
cos
2111t
16
+
8
15
+
1
9
cos
1866t
11
+
33
17
+
2
13
cos
2287t
13
+
20
11
+
1
15
cos
1753t
9
+
18
11
+
1
14
cos
3217t
16
+
5
2
+
1
9
cos
3525t
17
+
13
10
+
1
9
cos
1709t
8
+
8
9
+
1
9
cos
1131t
5
+
82
27
+
1
7
cos
3952t
17
+
16
11
+
1
11
cos
3104t
13
+
6
7
+
1
10
cos
3041t
11
+
1
4
+
1
18
cos
1131t
4
+
16
13
+
1
16
cos
10694t
37
+
5
4
+
1
73
cos
2953t
10
+3+
14555
11
The border of a country can be represented by a parametric curve derived from a Fourier series approximation. In this Demonstration, the Fourier series is computed using the border coordinates stored in CountryData. To form an elegant parametric equation representing the curve, the coefficients are rationalized.
Increasing the number of terms of the series or decreasing the tolerance of the rationalization gives a more accurate curve.
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