Poj 1265 pick Theorem

Pick theorem, January 1899

If it is set as a plane, a single vertex has multiple shapes with a grid point as a vertex, and its surface is regarded

WhereBTo compare the number of grid points in the world,IIt is the number of internal lattice points. The formula (8) is called the pick formula.

/** Poj 1265 * Fuqiang * ry, pick theorem * 2013/8/7 */# include <iostream> # include <cstdio> # include <cstdlib> # include <cstring> # include <algorithm> # include <queue> using namespace STD; # define INF 0x3f3f3fconst int maxn = 100 + 3; int ABS (int n) {return n> 0? N:-N;} struct point {int X; int y;} p [maxn]; int gcd (int x, int y) {return y = 0? X: gcd (Y, X % Y);} int grid_onedge (int n) // point {int ret = 0; For (INT I = 0; I <n; I ++) {RET + = gcd (ABS (P [I]. x-P [(I + 1) % N]. x), ABS (P [I]. y-P [(I + 1) % N]. y);} return ret;} int grid_inside (int n) {int ret = 0; For (INT I = 0; I <n; I ++) {RET + = P [(I + 1) % N]. y * (p [I]. x-P [(I + 2) % N]. x);} return ABS (RET);} int main () {# ifndef online_judge freopen ("in", "r", stdin); # endif int t, case = 1; int n, a, B; CIN> T; while (t --) {CIN> N; P [0]. X = 0; P [0]. y = 0; For (INT I = 1; I <= N; I ++) {CIN> A> B; P [I % N]. X = P [I-1]. X + A; P [I % N]. y = P [I-1]. Y + B;} int e = grid_onedge (n); int I = (grid_inside (N)-E)/2 + 1; // number of internal cells double A = E/2.0 + I-1; printf ("Scenario # % d: \ n % d %. 1f \ n ", Case ++, I, e, a);} return 0 ;}

Evaluate the pick formula of the surface weight (
6 rows)

1 Other Calculation Formula The multi-plane shape on the plane has a variety of surface plane formulas, depending on the number given by the terminal plane. In addition to the pick formula introduced in this article, the heron formula and the regression formula form three important formulas for surface evaluation.

2 Heron formula and its push For a triangle, if we know the length of a triggerA,B,C, So that the surface is There are two situations in which I pushed the question to the four charts. cai yiming, "Four-dimensional facial recognition", "Data Broadcast", vol. 17th, stage 3, China National People's Republic of China) (I) When the four-dimensional Growth A, B, C, DAnd when it comes to the first hour, it is assumed that This is called the Brahmagupta formula. (Ii) for any four shapes, the surface is WhereBAndDTo the right corner of a four-character pair. This is called the bretschneider formula. It is worth noting that it will not work if you want to push the code to the fifth or above.

3 Formula for determining the number of samples (a surveyor's formula) One NBorder shape A1, A2 ,..., A NThe cursor is configured in the reverse direction and A K= ( X K, Y K), K= 1, 2 ,..., N, In principle RulesXN+ 1 =X1, andYN+ 1 =Y1. Among the above three formulas, the attention formula should be the most powerful, because it can be "simplified 」. For example: Fengqu in the reverse direction of the planeX=X(T),Y=Y(T), The plane of the completed domain is: If we continue to push the attention, we will get the famous Green theorem, and the solution will be the mainstream in analytics. It seems to be a great river. Along the way, the branches of the river are collected, and finally they flow into the sea.