UVa 10772

From Algorithmist
Jump to navigation Jump to search

10772 - Rose Windows[edit]

Summary[edit]

A regular polygon with vertices ( is an even number) is inscribed in a circle of radius . Then every two vertices of the polygon are joined with straight lines, and the resulting regions are colored in different colors as shown in figures.

10772-rose8.png   10772-rose12.png
n=8 n=12

Determine the area of the -th region. Regions are numbered from 1, starting from the innermost region.

Explanation[edit]

If you study the figures carefully, you'll see that the boundary of every regions (except the outermost one) is a regular polygon with vertices, and its vertices are intersections of chords of the polygon.

More formally, let be the i-th vertex of the polygon: . And let be the number of region. Then the -th vertex of the boundary of -th region is the intersection of line segments and .

The area of -th region is the difference between inner areas of boundaries of -th and -st regions.

Area of regular polygon can be computed, if you know coordinates of its center O and one of vertices V. It equals

Input[edit]

14
10 8 1
10 8 2
10 8 3
10 8 4
50 12 1
50 12 2
50 12 6
100 40 1
100 40 2
100 40 3
100 40 10
100 40 15
100 40 19
100 40 20

Output[edit]

48.5281
117.1573
117.1573
31.3166
538.4758
1471.1432
353.9816
193.7897
576.5974
945.2074
2462.3319
1878.1616
576.5974
129.0335