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kuangbin计算几何模版


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// 计算几何模板
const double eps = 1e-8;
const double inf = 1e20;
const double pi = acos(-1.0);
const int maxp = 1010;
//Compares a double to zero
int sgn(double x){
if (fabs(x) < eps)return 0;
if (x < 0)return-1;
else return 1;
}
//square of a double
inline double sqr(double x){ return x*x; }
/*
* Point
* Point()-Empty constructor
* Point(double _x,double _y)-constructor
* input()-double input
* output()-%.2f output
* operator ==-compares x and y
* operator <-compares first by x, then by y
* operator-? return new Point after subtracting
curresponging x and y
* operator ^-cross product of 2d points
* operator *-dot product
* len()-gives length from origin
* len2()-gives square of length from origin
* distance(Point p)-gives distance from p
* operator + Point b-returns new Point after adding
curresponging x and y
* operator * double k-returns new Point after multiplieing x and
y by k
* operator / double k-returns new Point after divideing x and y
by k
* rad(Point a,Point b)-returns the angle of Point a and Point b
from this Point
* trunc(double r)-return Point that if truncated the
distance from center to r
* rotleft()-returns 90 degree ccw rotated point
* rotright()-returns 90 degree cw rotated point
* rotate(Point p,double angle)-returns Point after rotateing the
Point centering at p by angle radian ccw
*/
struct Point{
double x, y;
Point(){}
Point(double _x, double _y){
x = _x;
y = _y;
}
void input(){
scanf("%lf%lf", &x, &y);
}
void output(){
printf("%.2f?%.2f\n", x, y);
}
bool operator == (Point b)const{
return sgn(x - b.x) == 0 && sgn(y - b.y) == 0;
}
bool operator < (Point b)const{
return sgn(x - b.x) == 0 ? sgn(y - b.y) < 0 : x < b.x;
}
Point operator-(const Point &b)const{
return Point(x - b.x, y - b.y);
}
//叉积
double operator ^(const Point &b)const{
return x*b.y - y*b.x;
}
//点积
double operator *(const Point &b)const{
return x*b.x + y*b.y;
}
//返回长度
double len(){
return hypot(x, y);//库函数
}
//返回长度的平方
double len2(){
return x*x + y*y;
}
//返回两点的距离
double distance(Point p){
return hypot(x - p.x, y - p.y);
}
Point operator +(const Point &b)const{
return Point(x + b.x, y + b.y);
}
Point operator *(const double &k)const{
return Point(x*k, y*k);
}
Point operator /(const double &k)const{
return Point(x / k, y / k);
}
//计算 pa 和 pb 的夹角
//就是求这个点看 a,b 所成的夹角
//测试 LightOJ1203
double rad(Point a, Point b){
Point p = *this;
return fabs(atan2(fabs((a - p) ^ (b - p)), (a - p)*(b - p)));
}
//化为长度为 r 的向量
Point trunc(double r){
double l = len();
if (!sgn(l))return *this;
r /= l;
return Point(x*r, y*r);
}
//逆时针旋转 90 度
Point rotleft(){
return Point(-y, x);
}
//顺时针旋转 90 度
Point rotright(){
return Point(y, -x);
}
//绕着 p 点逆时针旋转 angle
Point rotate(Point p, double angle){
Point v = (*this) - p;
double c = cos(angle), s = sin(angle);
return Point(p.x + v.x*c - v.y*s, p.y + v.x*s + v.y*c);
}
};
/*
* Stores two points
* Line()-Empty constructor
* Line(Point _s,Point _e)-Line through _s and _e
* operator ==-checks if two points are same
* Line(Point p,double angle)-one end p , another end at
angle degree
* Line(double a,double b,double c)-Line of equation ax + by + c
= 0
* input()-inputs s and e
* adjust()-orders in such a way that s < e
* length()-distance of se
* angle()-return 0 <= angle < pi
* relation(Point p)-3 if point is on line
* 1 if point on the left of line
* 2 if point on the right of line
* pointonseg(double p)-return true if point on segment
* parallel(Line v)-return true if they are
parallel
* segcrossseg(Line v)-returns 0 if does not intersect
* returns 1 if non-standard
intersection
* returns 2 if intersects
* linecrossseg(Line v)-line and seg
* linecrossline(Line v)-0 if parallel
* 1 if coincides
* 2 if intersects
* crosspoint(Line v)-returns intersection point
* dispointtoline(Point p)-distance from point p to the
line
* dispointtoseg(Point p)-distance from p to the segment
* dissegtoseg(Line v)-distance of two segment
* lineprog(Point p)-returns projected point p on se
line
* symmetrypoint(Point p)-returns reflection point of p
over se
*
*/
struct Line{
Point s, e;
Line(){}
Line(Point _s, Point _e){
s = _s;
e = _e;
}
bool operator ==(Line v){
return (s == v.s) && (e == v.e);
}
//根据一个点和倾斜角 angle 确定直线,0<=angle<pi
Line(Point p, double angle){
s = p;
if (sgn(angle - pi / 2) == 0){
e = (s + Point(0, 1));
}
else{
e = (s + Point(1, tan(angle)));
}
}
//ax+by+c=0
Line(double a, double b, double c){
if (sgn(a) == 0){
s = Point(0, -c / b);
e = Point(1, -c / b);
}
else if (sgn(b) == 0){
s = Point(-c / a, 0);
e = Point(-c / a, 1);
}
else{
s = Point(0, -c / b);
e = Point(1, (-c - a) / b);
}
}
void input(){
s.input();
e.input();
}
void adjust(){
if (e < s)swap(s, e);
}
//求线段长度
double length(){
return s.distance(e);
}
//返回直线倾斜角 0<=angle<pi
double angle(){
double k = atan2(e.y - s.y, e.x - s.x);
if (sgn(k) < 0)k += pi;
if (sgn(k - pi) == 0)k -= pi;
return k;
}
//点和直线关系
//1 在左侧
//2 在右侧
//3 在直线上
int relation(Point p){
int c = sgn((p - s) ^ (e - s));
if (c < 0)return 1;
else if (c > 0)return 2;
else return 3;
}
// 点在线段上的判断
bool pointonseg(Point p){
return sgn((p - s) ^ (e - s)) == 0 && sgn((p - s)*(p - e)) <= 0;
}
//两向量平行 (对应直线平行或重合)
bool parallel(Line v){
return sgn((e - s) ^ (v.e - v.s)) == 0;
}
//两线段相交判断
//2 规范相交
//1 非规范相交
//0 不相交
int segcrossseg(Line v){
int d1 = sgn((e - s) ^ (v.s - s));
int d2 = sgn((e - s) ^ (v.e - s));
int d3 = sgn((v.e - v.s) ^ (s - v.s));
int d4 = sgn((v.e - v.s) ^ (e - v.s));
if ((d1^d2) == -2 && (d3^d4) == -2)return 2;
return (d1 == 0 && sgn((v.s - s)*(v.s - e)) <= 0) ||
(d2 == 0 && sgn((v.e - s)*(v.e - e)) <= 0) ||
(d3 == 0 && sgn((s - v.s)*(s - v.e)) <= 0) ||
(d4 == 0 && sgn((e - v.s)*(e - v.e)) <= 0);
}
//直线和线段相交判断
//-*this line -v seg
//2 规范相交
//1 非规范相交
//0 不相交
int linecrossseg(Line v){
int d1 = sgn((e - s) ^ (v.s - s));
int d2 = sgn((e - s) ^ (v.e - s));
if ((d1^d2) == -2) return 2;
return (d1 == 0 || d2 == 0);
}
//两直线关系
//0 平行
//1 重合
//2 相交
int linecrossline(Line v){
if ((*this).parallel(v))
return v.relation(s) == 3;
return 2;
}
//求两直线的交点
//要保证两直线不平行或重合
Point crosspoint(Line v){
double a1 = (v.e - v.s) ^ (s - v.s);
double a2 = (v.e - v.s) ^ (e - v.s);
return Point((s.x*a2 - e.x*a1) / (a2 - a1), (s.y*a2 - e.y*a1) / (a2 - a1
));
}
//点到直线的距离
double dispointtoline(Point p){
return fabs((p - s) ^ (e - s)) / length();
}
//点到线段的距离
double dispointtoseg(Point p){
if (sgn((p - s)*(e - s)) < 0 || sgn((p - e)*(s - e)) < 0)
return min(p.distance(s), p.distance(e));
return dispointtoline(p);
}
//返回线段到线段的距离
//前提是两线段不相交,相交距离就是 0 了
double dissegtoseg(Line v){
return min(min(dispointtoseg(v.s), dispointtoseg(v.e)), min(v
.dispointtoseg(s), v.dispointtoseg(e)));
}
//返回点 p 在直线上的投影
Point lineprog(Point p){
return s + (((e - s)*((e - s)*(p - s))) / ((e - s).len2()));
}
//返回点 p 关于直线的对称点
Point symmetrypoint(Point p){
Point q = lineprog(p);
return Point(2 * q.x - p.x, 2 * q.y - p.y);
}
};
//圆
struct circle{
Point p;//圆心
double r;//半径
circle(){}
circle(Point _p, double _r){
p = _p;
r = _r;
}
circle(double x, double y, double _r){
p = Point(x, y);
r = _r;
}
//三角形的外接圆
//需要 Point 的 + / rotate() 以及 Line 的 crosspoint()
//利用两条边的中垂线得到圆心
//测试:UVA12304
circle(Point a, Point b, Point c){
Line u = Line((a + b) / 2, ((a + b) / 2) + ((b - a).rotleft()));
Line v = Line((b + c) / 2, ((b + c) / 2) + ((c - b).rotleft()));
p = u.crosspoint(v);
r = p.distance(a);
}
//三角形的内切圆
//参数 bool t 没有作用,只是为了和上面外接圆函数区别
//测试:UVA12304
circle(Point a, Point b, Point c, bool t){
Line u, v;
double m = atan2(b.y - a.y, b.x - a.x), n = atan2(c.y - a.y, c.x - a.
x);
u.s = a;
u.e = u.s + Point(cos((n + m) / 2), sin((n + m) / 2));
v.s = b;
m = atan2(a.y - b.y, a.x - b.x), n = atan2(c.y - b.y, c.x - b.x);
v.e = v.s + Point(cos((n + m) / 2), sin((n + m) / 2));
p = u.crosspoint(v);
r = Line(a, b).dispointtoseg(p);
}
//输入
void input(){
p.input();
scanf("%lf", &r);
}
//输出
void output(){
printf("%.2lf?%.2lf?%.2lf\n", p.x, p.y, r);
}
bool operator == (circle v){
return (p == v.p) && sgn(r - v.r) == 0;
}
bool operator < (circle v)const{
return ((p < v.p) || ((p == v.p) && sgn(r - v.r) < 0));
}
//面积
double area(){
return pi*r*r;
}
//周长
double circumference(){
return 2 * pi*r;
}
//点和圆的关系
//0 圆外
//1 圆上
//2 圆内
int relation(Point b){
double dst = b.distance(p);
if (sgn(dst - r) < 0)return 2;
else if (sgn(dst - r) == 0)return 1;
return 0;
}
//线段和圆的关系
//比较的是圆心到线段的距离和半径的关系
int relationseg(Line v){
double dst = v.dispointtoseg(p);
if (sgn(dst - r) < 0)return 2;
else if (sgn(dst - r) == 0)return 1;
return 0;
}
//直线和圆的关系
//比较的是圆心到直线的距离和半径的关系
int relationline(Line v){
double dst = v.dispointtoline(p);
if (sgn(dst - r) < 0)return 2;
else if (sgn(dst - r) == 0)return 1;
return 0;
}
//两圆的关系
//5 相离
//4 外切
//3 相交
//2 内切
//1 内含
//需要 Point 的 distance
//测试:UVA12304
int relationcircle(circle v){
double d = p.distance(v.p);
if (sgn(d - r - v.r) > 0)return 5;
if (sgn(d - r - v.r) == 0)return 4;
double l = fabs(r - v.r);
if (sgn(d - r - v.r) < 0 && sgn(d - l) > 0)return 3;
if (sgn(d - l) == 0)return 2;
if (sgn(d - l) < 0)return 1;
}
//求两个圆的交点,返回 0 表示没有交点,返回 1 是一个交点,2 是两个交点
//需要 relationcircle
//测试:UVA12304
int pointcrosscircle(circle v, Point &p1, Point &p2){
int rel = relationcircle(v);
if (rel == 1 || rel == 5)return 0;
double d = p.distance(v.p);
double l = (d*d + r*r - v.r*v.r) / (2 * d);
double h = sqrt(r*r - l*l);
Point tmp = p + (v.p - p).trunc(l);
p1 = tmp + ((v.p - p).rotleft().trunc(h));
p2 = tmp + ((v.p - p).rotright().trunc(h));
if (rel == 2 || rel == 4)
return 1;
return 2;
}
//求直线和圆的交点,返回交点个数
int pointcrossline(Line v, Point &p1, Point &p2){
if (!(*this).relationline(v))return 0;
Point a = v.lineprog(p);
double d = v.dispointtoline(p);
d = sqrt(r*r - d*d);
if (sgn(d) == 0){
p1 = a;
p2 = a;
return 1;
}
p1 = a + (v.e - v.s).trunc(d);
p2 = a - (v.e - v.s).trunc(d);
return 2;
}
//得到过 a,b 两点,半径为 r1 的两个圆
int gercircle(Point a, Point b, double r1, circle &c1, circle &c2){
circle x(a, r1), y(b, r1);
int t = x.pointcrosscircle(y, c1.p, c2.p);
if (!t)return 0;
c1.r = c2.r = r;
return t;
}
//得到与直线 u 相切,过点 q, 半径为 r1 的圆
//测试:UVA12304
int getcircle(Line u, Point q, double r1, circle &c1, circle &c2){
double dis = u.dispointtoline(q);
if (sgn(dis - r1 * 2) > 0)return 0;
if (sgn(dis) == 0){
c1.p = q + ((u.e - u.s).rotleft().trunc(r1));
c2.p = q + ((u.e - u.s).rotright().trunc(r1));
c1.r = c2.r = r1;
return 2;
}
Line u1 = Line((u.s + (u.e - u.s).rotleft().trunc(r1)), (u.e +
(u.e - u.s).rotleft().trunc(r1)));
Line u2 = Line((u.s + (u.e - u.s).rotright().trunc(r1)), (u.e
+ (u.e - u.s).rotright().trunc(r1)));
circle cc = circle(q, r1);
Point p1, p2;
if (!cc.pointcrossline(u1, p1, p2))cc.pointcrossline(u2, p1, p2)
;
c1 = circle(p1, r1);
if (p1 == p2){
c2 = c1;
return 1;
}
c2 = circle(p2, r1);
return 2;
}
//同时与直线 u,v 相切,半径为 r1 的圆
//测试:UVA12304
int getcircle(Line u, Line v, double r1, circle &c1, circle &c2,
circle &c3, circle &c4){
if (u.parallel(v))return 0;//两直线平行
Line u1 = Line(u.s + (u.e - u.s).rotleft().trunc(r1), u.e + (u
.e - u.s).rotleft().trunc(r1));
Line u2 = Line(u.s + (u.e - u.s).rotright().trunc(r1), u.e + (
u.e - u.s).rotright().trunc(r1));
Line v1 = Line(v.s + (v.e - v.s).rotleft().trunc(r1), v.e + (v
.e - v.s).rotleft().trunc(r1));
Line v2 = Line(v.s + (v.e - v.s).rotright().trunc(r1), v.e + (
v.e - v.s).rotright().trunc(r1));
c1.r = c2.r = c3.r = c4.r = r1;
c1.p = u1.crosspoint(v1);
c2.p = u1.crosspoint(v2);
c3.p = u2.crosspoint(v1);
c4.p = u2.crosspoint(v2);
return 4;
}
//同时与不相交圆 cx,cy 相切,半径为 r1 的圆
//测试:UVA12304
int getcircle(circle cx, circle cy, double r1, circle &c1, circle &
c2){
circle x(cx.p, r1 + cx.r), y(cy.p, r1 + cy.r);
int t = x.pointcrosscircle(y, c1.p, c2.p);
if (!t)return 0;
c1.r = c2.r = r1;
return t;
}
//过一点作圆的切线 (先判断点和圆的关系)
//测试:UVA12304
int tangentline(Point q, Line &u, Line &v){
int x = relation(q);
if (x == 2)return 0;
if (x == 1){
u = Line(q, q + (q - p).rotleft());
v = u;
return 1;
}
double d = p.distance(q);
double l = r*r / d;
double h = sqrt(r*r - l*l);
u = Line(q, p + ((q - p).trunc(l) + (q - p).rotleft().trunc(h)))
;
v = Line(q, p + ((q - p).trunc(l) + (q - p).rotright().trunc(h))
);
return 2;
}
//求两圆相交的面积
double areacircle(circle v){
int rel = relationcircle(v);
if (rel >= 4)return 0.0;
if (rel <= 2)return min(area(), v.area());
double d = p.distance(v.p);
double hf = (r + v.r + d) / 2.0;
double ss = 2 * sqrt(hf*(hf - r)*(hf - v.r)*(hf - d));
double a1 = acos((r*r + d*d - v.r*v.r) / (2.0*r*d));
a1 = a1*r*r;
double a2 = acos((v.r*v.r + d*d - r*r) / (2.0*v.r*d));
a2 = a2*v.r*v.r;
return a1 + a2 - ss;
}
//求圆和三角形 pab 的相交面积
//测试:POJ3675 HDU3982 HDU2892
double areatriangle(Point a, Point b){
if (sgn((p - a) ^ (p - b)) == 0)return 0.0;
Point q[5];
int len = 0;
q[len++] = a;
Line l(a, b);
Point p1, p2;
if (pointcrossline(l, q[1], q[2]) == 2){
if (sgn((a - q[1])*(b - q[1])) < 0)q[len++] = q[1];
if (sgn((a - q[2])*(b - q[2])) < 0)q[len++] = q[2];
}
q[len++] = b;
if (len == 4 && sgn((q[0] - q[1])*(q[2] - q[1])) > 0)swap(q[1], q
[2]);
double res = 0;
for (int i = 0; i < len - 1; i++){
if (relation(q[i]) == 0 || relation(q[i + 1]) == 0){
double arg = p.rad(q[i], q[i + 1]);
res += r*r*arg / 2.0;
}
else{
res += fabs((q[i] - p) ^ (q[i + 1] - p)) / 2.0;
}
}
return res;
}
};
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