计算几何——交点、面积的计算
交点、面积的计算
一·、基本操作
0、准备
1 const int maxn = 10005; 2 typedef pair<double, double> _pair; 3 _pair point[maxn];
1、两点间距离的计算
Dis= sqrt[ (x2- x1)2+ (y2- y1)2 ];
1 double Get_Dis (_pair point1, _pair point2) 2 { 3 //计算两点间距离 4 return sqrt(((point1.first- point2.first)* (point1.first- point2.first) ) 5 + ((point1.second- point2.second)* (point1.second- point2.second) ) ); 6 }
2、计算叉乘（向量积/外积）
PAxQB= { [ (A.x- P.x)* (B.y- Q.y) ]- [ (B.x- Q.x)* (A.y- P.y) ] };
1 double Get_axb (_pair a_point1, _pair a_point2, _pair b_point1, _pair b_point2) 2 { 3 //计算两条向量的叉积 4 //向量a= a_point1 --> a_point2= a_point2- a_point1; 5 //向量b= b_point1 --> b_point2= b_point2- b_point1; 6 //叉积axb= (a.x* b.y)- (b.x* a.y); 7 //a.x= a_point2.x- a_point1.x; a.y= a_point2.y- a_point1.y; 8 return (((a_point2.first- a_point1.first)* (b_point2.second- b_point1.second) ) 9 - ((b_point2.first- b_point1.first)* (a_point2.second- a_point1.second) ) ); 10 }
Ps：求面积或判断相对位置。
3、计算点乘（内积）
PA·QB= { [ (A.x- P.x)* (B.x- Q.x) ]- [ (A.y- P.y)* (B.y- Q.y) ] };
1 double Get_a_xb (_pair a_point1, _pair a_point2, _pair b_point1, _pair b_point2) 2 { 3 //计算两条向量的点乘 4 //向量a= a_point1 --> a_point2= a_point2- a_point1; 5 //向量b= b_point1 --> b_point2= b_point2- b_point1; 6 //点乘a·b= (x1* x2)- (y1* y2)= (a.x* b.x)- (a.y* b.y); 7 //a.x= a_point2.x- a_point1.x; a.y= a_point2.y- a_point1.y; 8 return (((a_point2.first- a_point1.first)* (b_point2.first- b_point1.first) ) 9 - ((a_point2.second- a_point1.second)* (b_point2.second- b_point1.second) ) ); 10 }
Ps：判断直线是否正交（垂直）。
4、余弦定理
cos A= (b²+ c²- a² )/ 2bc；
用法：配合各种反三角函数（如acos(), atan();）求各种角度，配合点旋转求交点。
Ps：给出一个向量a求与x 轴所成角的余弦值。
1 double Get_Cos (_pair point1, _pair point2) 2 { 3 //计算向量a(point1-->point2) 的余弦值; 4 point2.first-= point1.first; //把point1看作坐标原点(0, 0); 5 point2.second-= point1.second; //则point2的坐标为(P2.x- P1.x, P2.y- P1.y); 6 point1.first= 0; 7 point1.second= 0; 8 _pair point3; //在X轴上找一点P3,做直角三角形; 9 point3.first= point2.first; //P3.x= P2.x; 10 point3.second= 0; //P3.y= P1.y= 0; 11 double Dis_P1_P2= Get_Dis(point1, point2); //计算直角三角形的斜边长,即P1P2之间的距离; 12 return point3.first/ Dis_P1_P2; //邻边/ 斜边; 13 }
end;
