二維幾何基礎 （模板）
阿新 • • 發佈：2019-02-05
這裡總結一下二維幾何基礎知識！
常用定義：
//定義點的型別 struct Point { double x, y; Point(double x = 0, double y = 0) : x(x) , y(y) { } //建構函式，方便程式碼編寫 }; typedef Point Vector; //從程式實現上，Vector只是Point的別名 //向量 + 向量 = 向量 ，點 + 向量 = 點 Vector operator + (Vector A, Vector B) { return Vector(A.x+B.x, A.y+B.y); } //點 - 點 = 向量 Vector operator - (Vector A, Vector B) { return Vector(A.x-B.x, A.y-B.y); } //向量 * 數 = 向量 Vector operator * (Vector A, double p) { return Vector(A.x*p, A.y*p); } //向量 / 數 = 向量 Vector operator / (Vector A, double p) { return Vector(A.x/p, A.y/p); } bool operator < (const Point& a, const Point& b) { return a.x < b.x || (a.x == b.x && a.y < b.y); } const double eps = 1e-10; int dcmp(double x) { if(fabs(x) < eps) return 0; else return x < 0 ? -1 : 1; } bool operator == (const Point& a, const Point& b) { return dcmp(a.x - b.x) == 0 && dcmp(a.y - b.y) == 0; }
點積：
//點積
double Dot(Vector A, Vector B) { return A.x*B.x + A.y*B.y; } //求點積
double Length(Vector A) { return sqrt(Dot(A, A)); } //求向量長度
double Angle(Vector A, Vector B) { return acos(Dot(A, B) / Length(A) / Length(B)); }//求向量之間的夾角 叉積：
//叉積 double Cross(Vector A, Vector B) { return A.x*B.y - A.y*B.x; }//求叉積 double Area2(Point A, Point B, Point C) { return Cross(B-A, C-A); }//根據叉積求三角形面積的兩倍
旋轉：
//旋轉
Vector Rotate(Vector A, double rad) {//rad是弧度
return Vector(A.x*cos(rad) - A.y*sin(rad), A.x*sin(rad)+A.y*cos(rad) );
} 向量的單位法線：
//向量單位法向量，呼叫前請確保A不是零向量
Vector Normal(Vector A) {
double L = Length(A);
return Vector(-A.y/L, A.x/L);
}二直線交點：
//二直線交點（引數式） Point GetLineIntersection(Point P, Vector v, Point Q, Vector w) { Vector u = P - Q; double t = Cross(w, u) / Cross(v, w); return P + v * t; }
點到直線距離 ：
//點到直線距離
double DistanceToLine(Point P, Point A, Point B) {
Vector v1 = B-A, v2 = P - A;
return fabs(Cross(v1,v2) / Length(v1)); //如果不取絕對值，得到的是有向距離
} 點到線段距離 ：
//點到線段距離
double DistanceToSegment(Point P, Point A, Point B) {
if(A==B) return Length(P-A);
Vector v1 = B - A, v2 = P - A, v3 = P - B;
if(dcmp(Dot(v1, v2)) < 0) return Length(v2);
else if(dcmp(Dot(v1, v3)) > 0) return Length(v3);
else return fabs(Cross(v1, v2)) / Length(v1);
} 點在直線上的投影：
//點在直線上的投影
Point GetLineProjection(Point P, Point A, Point B) {
Vector v = B - A;
return A + v * ( Dot(v, P-A) / Dot(v, v) );
} 線段相交判定：
//線段相交判定
bool SegmentProperIntersection(Point a1, Point a2, Point b1, Point b2) {
double c1 = Cross(a2 - a1, b1 - a1), c2 = Cross(a2 - a1, b2 - a1),
c3 = Cross(b2 - b1, a1 - b1), c4 = Cross(b2 - b1, a2 - b1);
return dcmp(c1) * dcmp(c2) < 0 && dcmp(c3) * dcmp(c4) < 0;
} 判斷一個點是否在一條線段上：
//判斷一個點是否在一條線段上
bool OnSegment(Point p, Point a1, Point a2) {
return dcmp(Cross(a1 - p, a2 - p)) == 0 && dcmp(Dot(a1 - p, a2 - p)) < 0;
} 多邊形面積：
//多邊形面積
double ConvexPolygonArea(Point* p, int n) {
double area = 0;
for(int i = 1; i < n-1; i++)
area += Cross(p[i] - p[0], p[i + 1] - p[0]);
return area / 2;
} 總結：
//定義點的型別
struct Point {
double x, y;
Point(double x = 0, double y = 0) : x(x) , y(y) { } //建構函式，方便程式碼編寫
};
typedef Point Vector; //從程式實現上，Vector只是Point的別名
//向量 + 向量 = 向量 ，點 + 向量 = 點
Vector operator + (Vector A, Vector B) { return Vector(A.x+B.x, A.y+B.y); }
//點 - 點 = 向量
Vector operator - (Vector A, Vector B) { return Vector(A.x-B.x, A.y-B.y); }
//向量 * 數 = 向量
Vector operator * (Vector A, double p) { return Vector(A.x*p, A.y*p); }
//向量 / 數 = 向量
Vector operator / (Vector A, double p) { return Vector(A.x/p, A.y/p); }
bool operator < (const Point& a, const Point& b) {
return a.x < b.x || (a.x == b.x && a.y < b.y);
}
const double eps = 1e-10;
int dcmp(double x) {
if(fabs(x) < eps) return 0; else return x < 0 ? -1 : 1;
}
bool operator == (const Point& a, const Point& b) {
return dcmp(a.x - b.x) == 0 && dcmp(a.y - b.y) == 0;
}
//點積
double Dot(Vector A, Vector B) { return A.x*B.x + A.y*B.y; } //求點積
double Length(Vector A) { return sqrt(Dot(A, A)); } //求向量長度
double Angle(Vector A, Vector B) { return acos(Dot(A, B) / Length(A) / Length(B)); }//求向量之間的夾角
//叉積
double Cross(Vector A, Vector B) { return A.x*B.y - A.y*B.x; }//求叉積
double Area2(Point A, Point B, Point C) { return Cross(B-A, C-A); }//根據叉積求三角形面積的兩倍
//旋轉
Vector Rotate(Vector A, double rad) {//rad是弧度
return Vector(A.x*cos(rad) - A.y*sin(rad), A.x*sin(rad)+A.y*cos(rad) );
}
//向量單位法向量，呼叫前請確保A不是零向量
Vector Normal(Vector A) {
double L = Length(A);
return Vector(-A.y/L, A.x/L);
}
//二直線交點（引數式）
Point GetLineIntersection(Point P, Vector v, Point Q, Vector w) {
Vector u = P - Q;
double t = Cross(w, u) / Cross(v, w);
return P + v * t;
}
//點到直線距離
double DistanceToLine(Point P, Point A, Point B) {
Vector v1 = B-A, v2 = P - A;
return fabs(Cross(v1,v2) / Length(v1)); //如果不取絕對值，得到的是有向距離
}
//點到線段距離
double DistanceToSegment(Point P, Point A, Point B) {
if(A==B) return Length(P-A);
Vector v1 = B - A, v2 = P - A, v3 = P - B;
if(dcmp(Dot(v1, v2)) < 0) return Length(v2);
else if(dcmp(Dot(v1, v3)) > 0) return Length(v3);
else return fabs(Cross(v1, v2)) / Length(v1);
}
//點在直線上的投影
Point GetLineProjection(Point P, Point A, Point B) {
Vector v = B - A;
return A + v * ( Dot(v, P-A) / Dot(v, v) );
}
//線段相交判定
bool SegmentProperIntersection(Point a1, Point a2, Point b1, Point b2) {
double c1 = Cross(a2 - a1, b1 - a1), c2 = Cross(a2 - a1, b2 - a1),
c3 = Cross(b2 - b1, a1 - b1), c4 = Cross(b2 - b1, a2 - b1);
return dcmp(c1) * dcmp(c2) < 0 && dcmp(c3) * dcmp(c4) < 0;
}
//判斷一個點是否在一條線段上
bool OnSegment(Point p, Point a1, Point a2) {
return dcmp(Cross(a1 - p, a2 - p)) == 0 && dcmp(Dot(a1 - p, a2 - p)) < 0;
}
//多邊形面積
double ConvexPolygonArea(Point* p, int n) {
double area = 0;
for(int i = 1; i < n-1; i++)
area += Cross(p[i] - p[0], p[i + 1] - p[0]);
return area / 2;
} 無註釋純淨版
：
struct Point {
double x, y;
Point(double x = 0, double y = 0) : x(x) , y(y) { }
};
typedef Point Vector;
Vector operator + (Vector A, Vector B) { return Vector(A.x+B.x, A.y+B.y); }
Vector operator - (Vector A, Vector B) { return Vector(A.x-B.x, A.y-B.y); }
Vector operator * (Vector A, double p) { return Vector(A.x*p, A.y*p); }
Vector operator / (Vector A, double p) { return Vector(A.x/p, A.y/p); }
bool operator < (const Point& a, const Point& b) {
return a.x < b.x || (a.x == b.x && a.y < b.y);
}
const double eps = 1e-10;
int dcmp(double x) {
if(fabs(x) < eps) return 0; else return x < 0 ? -1 : 1;
}
bool operator == (const Point& a, const Point& b) {
return dcmp(a.x - b.x) == 0 && dcmp(a.y - b.y) == 0;
}
double Dot(Vector A, Vector B) { return A.x*B.x + A.y*B.y; }
double Length(Vector A) { return sqrt(Dot(A, A)); }
double Angle(Vector A, Vector B) { return acos(Dot(A, B) / Length(A) / Length(B)); }
double Cross(Vector A, Vector B) { return A.x*B.y - A.y*B.x; }
double Area2(Point A, Point B, Point C) { return Cross(B-A, C-A); }
Vector Rotate(Vector A, double rad) {
return Vector(A.x*cos(rad) - A.y*sin(rad), A.x*sin(rad)+A.y*cos(rad) );
}
Vector Normal(Vector A) {
double L = Length(A);
return Vector(-A.y/L, A.x/L);
}
Point GetLineIntersection(Point P, Vector v, Point Q, Vector w) {
Vector u = P - Q;
double t = Cross(w, u) / Cross(v, w);
return P + v * t;
}
double DistanceToLine(Point P, Point A, Point B) {
Vector v1 = B-A, v2 = P - A;
return fabs(Cross(v1,v2) / Length(v1));
}
double DistanceToSegment(Point P, Point A, Point B) {
if(A==B) return Length(P-A);
Vector v1 = B - A, v2 = P - A, v3 = P - B;
if(dcmp(Dot(v1, v2)) < 0) return Length(v2);
else if(dcmp(Dot(v1, v3)) > 0) return Length(v3);
else return fabs(Cross(v1, v2)) / Length(v1);
}
Point GetLineProjection(Point P, Point A, Point B) {
Vector v = B - A;
return A + v * ( Dot(v, P-A) / Dot(v, v) );
}
bool SegmentProperIntersection(Point a1, Point a2, Point b1, Point b2) {
double c1 = Cross(a2 - a1, b1 - a1), c2 = Cross(a2 - a1, b2 - a1),
c3 = Cross(b2 - b1, a1 - b1), c4 = Cross(b2 - b1, a2 - b1);
return dcmp(c1) * dcmp(c2) < 0 && dcmp(c3) * dcmp(c4) < 0;
}
bool OnSegment(Point p, Point a1, Point a2) {
return dcmp(Cross(a1 - p, a2 - p)) == 0 && dcmp(Dot(a1 - p, a2 - p)) < 0;
}
double ConvexPolygonArea(Point* p, int n) {
double area = 0;
for(int i = 1; i < n-1; i++)
area += Cross(p[i] - p[0], p[i + 1] - p[0]);
return area / 2;
}