1 Vector Rotate(Vector A, double rad) {
2 return Vector(A.x * cos(rad) - A.y * sin(rad), A.x * sin(rad) + A.y * cos(rad));
3 }
4 double angle(Vector v) {return atan2(v.y, v.x);}
5 struct Circle {
6 Point c;
7 double r;
8 Circle(Point c, double r) {
9 this->c = c;
10 this->r = r;
11 }
12 Point point(double a) {
13 return Point(c.x + cos(a) * r, c.y + sin(a) * r);
14 }
15 };
16
17 Vector AngleBisector(Point p, Vector v1, Vector v2){//给定两个向量，求角平分线
18 double rad = Angle(v1, v2);
19 return Rotate(v1, dcmp(Cross(v1, v2)) * 0.5 * rad);
20 }
21
22 //求直线与圆的交点
23 int getLineCircleIntersection(Point p, Vector v, Circle c, vector<point> &sol) {
24 double a1 = v.x, b1 = p.x - c.c.x, c1 = v.y, d1 = p.y - c.c.y;
25 double e1 = a1 * a1 + c1 * c1, f1 = 2 * (a1 * b1 + c1 * d1), g1 = b1 * b1 + d1 * d1 - c.r * c.r;
26 double delta = f1 * f1 - 4 * e1 * g1, t;
27 if(dcmp(delta) < 0) return 0;
28 else if(dcmp(delta) == 0){
29 t = (-f1) / (2 * e1);
30 sol.push_back(p + v * t);
31 return 1;
32 } else{
33 t = (-f1 + sqrt(delta)) / (2 * e1); sol.push_back(p + v * t);
34 t = (-f1 - sqrt(delta)) / (2 * e1); sol.push_back(p + v * t);
35 return 2;
36 }
37 }
38
39 //点到圆的切线
40 int getTangents(Point p, Circle C, Vector *v) {
41 Vector u = C.c - p;
42 double dist = Length(u);
43 if (dist < C.r) return 0;
44 else if (dcmp(dist - C.r) == 0) {
45 v[0] = Rotate(u, PI / 2);
46 return 1;
47 } else {
48 double ang = asin(C.r / dist);
49 v[0] = Rotate(u, -ang);
50 v[1] = Rotate(u, +ang);
51 return 2;
52 }
53 }
54
55 //两圆公切线
56 //a[i], b[i]分别是第i条切线在圆A和圆B上的切点
57 int getCircleTangents(Circle A, Circle B, Point *a, Point *b) {
58 int cnt = 0;
59 if (A.r < B.r) { swap(A, B); swap(a, b); }
60 //圆心距的平方
61 double d2 = (A.c.x - B.c.x) * (A.c.x - B.c.x) + (A.c.y - B.c.y) * (A.c.y - B.c.y);
62 double rdiff = A.r - B.r;
63 double rsum = A.r + B.r;
64 double base = angle(B.c - A.c);
65 //重合有无限多条
66 if (d2 == 0 && dcmp(A.r - B.r) == 0) return -1;
67 //内切
68 if (dcmp(d2 - rdiff * rdiff) == 0) {
69 a[cnt] = A.point(base);
70 b[cnt] = B.point(base);
71 cnt++;
72 return 1;
73 }
74 //有外公切线
75 double ang = acos((A.r - B.r) / sqrt(d2));
76 a[cnt] = A.point(base + ang); b[cnt] = B.point(base + ang); cnt++;
77 a[cnt] = A.point(base - ang); b[cnt] = B.point(base - ang); cnt++;
78
79 //一条内切线，两条内切线
80 if (dcmp(d2 - rsum*rsum) == 0) {
81 a[cnt] = A.point(base); b[cnt] = B.point(PI + base); cnt++;
82 } else if (dcmp(d2 - rsum*rsum) > 0) {
83 double ang = acos((A.r + B.r) / sqrt(d2));
84 a[cnt] = A.point(base + ang); b[cnt] = B.point(base + ang); cnt++;
85 a[cnt] = A.point(base - ang); b[cnt] = B.point(base - ang); cnt++;
86 }
87 return cnt;
88 }
89
90 //求经过点p1,与直线(p2, w)相切,半径为r的一组圆
91 int CircleThroughAPointAndTangentToALineWithRadius(Point p1, Point p2, Vector w, double r, vector<point> &sol) {
92 Circle c1 = Circle(p1, r);
93 double t = r / Length(w);
94 Vector u = Vector(-w.y, w.x);
95 Point p4 = p2 + u * t;
96 int tot = getLineCircleIntersection(p4, w, c1, sol);
97 u = Vector(w.y, -w.x);
98 p4 = p2 + u * t;
99 tot += getLineCircleIntersection(p4, w, c1, sol);
100 return tot;
101 }
102
103 //给定两个向量，求两向量方向内夹着的圆的圆心。圆与两线均相切，圆的半径已给定
104 Point Centre_CircleTangentTwoNonParallelLineWithRadius(Point p1, Vector v1, Point p2, Vector v2, double r){
105 Point p0 = GetLineIntersection(p1, v1, p2, v2);
106 Vector u = AngleBisector(p0, v1, v2);
107 double rad = 0.5 * Angle(v1, v2);
108 double l = r / sin(rad);
109 double t = l / Length(u);
110 return p0 + u * t;
111 }
112
113 //求与两条不平行的直线都相切的4个圆，圆的半径已给定
114 int CircleThroughAPointAndTangentALineWithRadius(Point p1, Vector v1, Point p2, Vector v2, double r, Point *sol) {
115 int ans = 0;
116 sol[ans++] = Centre_CircleTangentTwoNonParallelLineWithRadius(p1, v1, p2, v2, r);
117 sol[ans++] = Centre_CircleTangentTwoNonParallelLineWithRadius(p1, v1 * -1, p2, v2, r);
118 sol[ans++] = Centre_CircleTangentTwoNonParallelLineWithRadius(p1, v1, p2, v2 * -1, r);
119 sol[ans++] = Centre_CircleTangentTwoNonParallelLineWithRadius(p1, v1 * -1, p2, v2 * -1, r);
120 return ans;
121 }
122
123 //求与两个相离的圆均外切的一组圆，三种情况
124 int CircleTangentToTwoDisjointCirclesWithRadius(Circle c1, Circle c2, double r, Point *sol){
125 double dis1 = c1.r + r + r + c2.r;
126 double dis2= Length(c1.c - c2.c);
127 if(dcmp(dis1 - dis2) < 0) return 0;
128 Vector u = c2.c - c1.c;
129 double t = (r + c1.r) / Length(u);
130 if(dcmp(dis1 - dis2)==0){
131 Point p0 = c1.c + u * t;
132 sol[0] = p0;
133 return 1;
134 }
135 double aa = Length(c1.c - c2.c);
136 double bb = r + c1.r, cc = r + c2.r;
137 double rad = acos((aa * aa + bb * bb - cc * cc) / (2 * aa * bb));
138 Vector w = Rotate(u, rad);
139 Point p0 = c1.c + w * t;
140 sol[0] = p0;
141 w = Rotate(u, -rad);
142 p0 = c1.c + w * t;
143 sol[1] = p0;
144 return 2;
145 }
