计算几何基础（入土）知识

施工中。。。

目录

笔记板子向随笔

暂时只有2维，有时间再去弄个三维的

1、一些基础运算函数与结构体定义（部分板子）

2、一些基础的变换公式

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一、一些基础运算函数与结构体定义

double pi = acos(-1);
const double eps = 1e-6;

inline int dcmp(double x)//cmp x with 0
{
    if (fabs(x) <= eps)return 0;
    return x < 0 ? -1 : 1;
}
inline int cmp(double x, double y)
{
    //x>y return 1
    //x<y reutrn -1
    //x==y return 0
    return dcmp(x - y);
}

double max(double x, double y)
{
    return dcmp(x - y) > 0 ? x : y;
}
double min(double x, double y)
{
    return dcmp(x - y) < 0 ? x : y;
}

int sgn(double x)
{
    if (fabs(x) < eps)return 0;
    else return x < 0 ? -1 : 1;
}

struct Point {
    double x, y;
    Point() {}
    Point(double xx, double yy) { x = xx; y = yy; }
    Point operator -(Point s) { return Point(x - s.x, y - s.y); }
    Point operator +(Point s) { return Point(x + s.x, y + s.y); }
	
    double operator *(Point s) { return x * s.x + y * s.y; }
    Point operator * (double p) { return Point(x * p, y * p); }//向量乘实数
	
    double operator ^(Point s) { return x * s.y - y * s.x; }
    Point operator /(double k) { return Point(x / k, y / k); }
	
    bool operator ==(Point s) { return sgn(x - s.x) == 0 && sgn(y - s.y) == 0; }
}p[maxn];

typedef Point Vector;//向量也可以用点表示

double len(Point a) { return sqrt(a * a); }
double dis(Point a, Point b) { return len(b - a); }//两点之间的距离

double Cross(Point a, Point b, Point c)//叉乘,a为公共点
{
    return (b - a) ^ (c - a);
}
double Cross(Vector A, Vector B) {
    return A.x * B.y - A.y * B.x;
}

double Dot(Point a, Point b, Point c)//点乘 ,a为公共点
{
    return (b - a) * (c - a);
}
double Dot(Vector a, Vector b)//点乘 ,a为公共点
{
    return a * b;
}

double Angle(Vector A, Vector B) //向量夹角
{
    return acos(Dot(A, B) / len(A) / len(B));
}

Vector Rotate(Vector A, double rad) //向量旋转，rad为弧度 且为逆时针旋转的角
{
    return Vector(A.x * cos(rad) - A.y * sin(rad), A.x * sin(rad) + A.y * cos(rad));
}

Vector Normal(Vector A) //向量A左转90°的单位法向量
{
    double L = len(A);
    return Vector(-A.y / L, A.x / L);
}

---

struct Line //直线定义,v为方向向量
{
    Point p;
    Vector v;
    Line(Point p, Vector v) :p(p), v(v) {}
    Point point(double t) {//返回直线上一点P = v + (p - v)*t
        return v + (p - v) * t;
    }
};
//计算两直线交点，调用前需保证 Cross(v, w) != 0
Point GetLineIntersection(Point P, Vector v, Point Q, Vector w) //POINT AND VECTOR VERSION
{
    Vector u = P - Q;
    double t = Cross(w, u) / Cross(v, w);
    return P + v * t;
}
Point GetLineIntersection(Line A, Line B) //LINE VERSION
{
    Point P = A.p, Q = B.p;
    Vector u = P - Q;
    Vector v = A.v, w = B.v;
    double t = Cross(w, u) / Cross(v, w);
    return P + v * t;
}



//点P到直线AB距离公式
double DistanceToLine(Point P, Point A, Point B) {
    Vector v1 = B - A, v2 = P - A;
    return fabs(Cross(v1, v2) / len(v1));
}//不去绝对值，得到的是有向距离

//点P到线段AB距离公式
double DistanceToSegment(Point P, Point A, Point B)
{
    if (A == B)
        return len(P - A);
    Vector v1 = B - A, v2 = P - A, v3 = P - B;
    if (dcmp(Dot(v1, v2)) < 0)
        return len(v2);
    if (dcmp(Dot(v1, v3)) > 0)
        return len(v3);
    return DistanceToLine(P, A, B);
}

//点P在直线AB上的投影点
Point GetLineProjection(Point P, Point A, Point B)
{
    Vector v = B - A;
    return A + v * (Dot(v, P - A) / Dot(v, v));
}

//点是否在线段上
bool OnSegment(Point p, Point a1, Point a2) {
    return dcmp(Cross(a1 - p, a2 - p)) == 0 && dcmp(Dot(a1 - p, a2 - p)) < 0;
}

//线段是否相交（不包括端点）
bool Segmentxj(Point a1, Point a2, Point b1, Point b2) {
    double c1 = Cross(a2 - a1, b1 - a1), c2 = Cross(a2 - a1, b2 - a1);
    double c3 = Cross(b2 - b1, a1 - b1), c4 = Cross(b2 - b1, a2 - b1);
    return (sgn(c1) * sgn(c2) < 0 && sgn(c3) * sgn(c4) < 0);
}
//不包括端点
bool Segmentxjdd(Point a1, Point a2, Point b1, Point b2) {
    double c1 = Cross(a2 - a1, b1 - a1), c2 = Cross(a2 - a1, b2 - a1);
    double c3 = Cross(b2 - b1, a1 - b1), c4 = Cross(b2 - b1, a2 - b1);
    //if判断控制是否允许线段在端点处相交，根据需要添加
    if (!sgn(c1) || !sgn(c2) || !sgn(c3) || !sgn(c4)) {
        bool f1 = OnSegment(b1, a1, a2);
        bool f2 = OnSegment(b2, a1, a2);
        bool f3 = OnSegment(a1, b1, b2);
        bool f4 = OnSegment(a2, b1, b2);
        bool f = (f1 | f2 | f3 | f4);
        return f;
    }
    return (sgn(c1) * sgn(c2) < 0 && sgn(c3) * sgn(c4) < 0);
}

//多边形有向面积
double PolygonArea(Point* p, int n) {//p为端点集合，n为端点个数
    double s = 0;
    for (int i = 1; i < n - 1; ++i)
        s += Cross(p[i] - p[0], p[i + 1] - p[0]);
    return s / 2.0;
}
//判断点是否在多边形内，若点在多边形内返回1，在多边形外部返回0，在多边形上返回-1
int isPointInPolygon(Point p, vector<Point> poly) {
    int wn = 0;
    int n = poly.size();
    for (int i = 0; i < n; ++i) {
        if (OnSegment(p, poly[i], poly[(i + 1) % n])) return -1;
        int k = sgn(Cross(poly[(i + 1) % n] - poly[i], p - poly[i]));
        int d1 = sgn(poly[i].y - p.y);
        int d2 = sgn(poly[(i + 1) % n].y - p.y);
        if (k > 0 && d1 <= 0 && d2 > 0) wn++;
        if (k < 0 && d2 <= 0 && d1 > 0) wn--;
    }
    if (wn != 0)
        return 1;
    return 0;
}

struct Circle
{
    Point c;//圆心
    double r;//半径
    Circle(Point c, double r) :c(c), r(r) {}
    Point point(double a) {//给出弧度，通过圆心角求坐标
        return Point(c.x + cos(a) * r, c.y + sin(a) * r);
    }
};

//求圆与直线交点
int getLineCircleIntersection(Line L, Circle C, double& t1, double& t2, vector<Point>& sol) {
    double a = L.v.x, b = L.p.x - C.c.x, c = L.v.y, d = L.p.y - C.c.y;
    double e = a * a + c * c, f = 2 * (a * b + c * d), g = b * b + d * d - C.r * C.r;
    double delta = f * f - 4 * e * g;//判别式
    if (sgn(delta) < 0)//相离
        return 0;
    if (sgn(delta) == 0) {//相切
        t1 = -f / (2 * e);
        t2 = -f / (2 * e);
        sol.push_back(L.point(t1));//sol存放交点本身
        return 1;
    }
    //相交
    t1 = (-f - sqrt(delta)) / (2 * e);
    sol.push_back(L.point(t1));
    t2 = (-f + sqrt(delta)) / (2 * e);
    sol.push_back(L.point(t2));
    return 2;
}

二、一些基础的（相对于我()）变换公式

1、点积和叉积都满足分配率，点积满足交换律但叉积不满足。

2、向量((x,y))逆时针旋转( heta)得到((xcos heta - ysin heta,xsin heta + ycos heta))

3、多边形面积：把多边形顶点按逆时针排序后，(S = sumlimits_{i=0}^{n-1}{overrightarrow{OA_i} imes overrightarrow{OA}_{(i+1)mod n}})

4、以A为原点，B为单位点，求点P对于新坐标系的坐标(坐标系变换后的(overrightarrow{AB}=(1,0)))为(P(x,y)_{new}=(overrightarrow{AB} cdot overrightarrow{AP},overrightarrow{AB} imes overrightarrow{AP})*dfrac{1}{left|AB ight|^2})

5、点P在直线AB上的投影是：(A+frac{overrightarrow{AB}*(overrightarrow{AB}cdotoverrightarrow{AP})}{left|AB ight|^2})

6、判断点P与线段AB的关系：
①若(overrightarrow{AB}cdotoverrightarrow{AP}>0)并且(overrightarrow{AB}cdotoverrightarrow{BP}>0)，则P在线段AB右侧；
②若(overrightarrow{AB}cdotoverrightarrow{AP}<0)并且(overrightarrow{AB}cdotoverrightarrow{BP}<0)，则P在线段AB左侧；
③否则就在线段AB上。
④若(overrightarrow{BP}==0)，则点P在点B上；点A同理。

7、直线与直线的交点、线段与直线、线段与线段
①若两直线平行，则(overrightarrow{AB} imesoverrightarrow{CD}==0)；
②若不平行，则一定有交点，

(overrightarrow{AI}=overrightarrow{AB}*frac{left|AM ight|}{left|AM ight|+left|BN ight|}=overrightarrow{AB}*frac{S_{ riangle{ACD}}}{S_{ riangle ACD} ;+S_{ riangle{BCD}}})

同理

(I = A + overrightarrow{AB} * frac{overrightarrow{CD} imesoverrightarrow{CA}}{overrightarrow{CD} imesoverrightarrow{CA} + overrightarrow{CB} imesoverrightarrow{CD}} = A +overrightarrow{AB} * frac{overrightarrow{CD} imesoverrightarrow{CA}}{overrightarrow{AB} imesoverrightarrow{CD}})

③直线与线段的交点：先判断两直线交点，再判断线段的端点与交点的位置关系，同侧则没有交点，异侧则有交点；

④线段与线段是否相交：

跨立实验:对于线段(AB)和线段(CD)，若((overrightarrow{AB} imesoverrightarrow{AC}) * (overrightarrow{AB} imesoverrightarrow{AD}) < 0)并且((overrightarrow{CD} imesoverrightarrow{CA}) * (overrightarrow{CD} imesoverrightarrow{CB}) < 0)，则两线段严格相交

严格相交(交点不在端点上)

//线段是否相交（不包括端点）
bool Segmentxj(Point a1, Point a2, Point b1, Point b2) {
    double c1 = Cross(a2 - a1, b1 - a1), c2 = Cross(a2 - a1, b2 - a1);
    double c3 = Cross(b2 - b1, a1 - b1), c4 = Cross(b2 - b1, a2 - b1);
    return (sgn(c1) * sgn(c2) < 0 && sgn(c3) * sgn(c4) < 0);
}

不严格相交

bool OnSegment(Point p, Point a1, Point a2) {
    return dcmp(Cross(a1 - p, a2 - p)) == 0 && dcmp(Dot(a1 - p, a2 - p)) < 0;
}

bool Segmentxjdd(Point a1, Point a2, Point b1, Point b2) {
    double c1 = Cross(a2 - a1, b1 - a1), c2 = Cross(a2 - a1, b2 - a1);
    double c3 = Cross(b2 - b1, a1 - b1), c4 = Cross(b2 - b1, a2 - b1);
    //if判断控制是否允许线段在端点处相交，根据需要添加
    if (!sgn(c1) || !sgn(c2) || !sgn(c3) || !sgn(c4)) {
        bool f1 = OnSegment(b1, a1, a2);
        bool f2 = OnSegment(b2, a1, a2);
        bool f3 = OnSegment(a1, b1, b2);
        bool f4 = OnSegment(a2, b1, b2);
        bool f = (f1 | f2 | f3 | f4);
        return f;
    }
    return (sgn(c1) * sgn(c2) < 0 && sgn(c3) * sgn(c4) < 0);
}

8、直线与圆

点到直线的距离：（划掉，用向量做不用开方，误差小一些）

圆心到直线的距离：(d =frac{overrightarrow{AB} imes overrightarrow{AO}}{left|overrightarrow{AB} ight|})