This document discusses algorithms and data structures for computational geometry problems involving points and line segments in a plane. It defines a Point class to represent points as complex numbers, and a Line class to represent line segments as pairs of Points. It provides functions for calculating distances and angles between Points, determining if a point lies on a line segment, finding the closest distance between two line segments, checking if two line segments intersect, and finding their intersection point. The document also describes an algorithm to solve the closest pair of line segments problem from a given set of line segments.

1. ICPC ~ ~

2. ● !
● !
● !
● !
● !
● !
●

3. ● x,y
!
● 2

4. ● C++ STL complex !
◦ !
● typedef complex<double> Point;!
◦ !
● x real() y imag() !
◦ a(x,y) !
● Point a;!
● a.real() = x; a.imag() = y;!
◦ !
● x:
y:
a(x,y)
O

5. ●
!
◦ a !
●
x
y
a(x,y)
O

6. complex
● !
◦ complex
!
●
!
struct Point{!
double x,y;!
Point(double a, double b){ x = a; y = b; }!
};

7. ● a,b a-b
!
◦ double dis(Point a, Point b){ return abs(a-b); }
b(xb,yb)
a(xa,ya)

8. θ b(xb,yb)
a(xa,ya)

9. θ b(xb,yb)
a(xa,ya)

10. ● = |a| * |b| * sinθ
= |b| * |b xa |
= × !
● = × / 2
= / 2
10
θ

11. ● 2 a,b c !
● ccw
11
a
b

12. ccw
int ccw(Point a, Point b, Point c){
if(cross(b-a,c-a)>EPS)return 1;
if(cross(b-a,c-a)<-EPS)return -1;
if(dot(b-a,c-a)<-EPS)return 2;
if(abs(b-a)+EPS<abs(c-a)return -2;
return 0;
}
12
a
b

13. : AOJ0035 Is It Convex?
● !
●
13

14. ● 2
!
● 2
!
◦ typedef pair<Point, Point> Line;!
●
a
b
a
b
ab ab

15.
a
b
c
θ

16. ● ab c
a,b c !
●
a
b
c
a
b
c

17. ● a : ∠BAC > 90°!
● b : ∠ABC > 90°!
● : !
∠BAC <= 90° ABC <= 90°!
● cosθ(= )
a
b
c
a
b
c

18. double seg_to_point_dis(Line l, Point p){!
Point a = l.first, b = l.second, c = p;!
if(dot(b-a,c-a)<EPS)return abs(c-a);!
if(dot(a-b,c-b)<EPS)return abs(c-b);!
return line_to_point_dis(l,p);!
}
a
b
c
a
b
c

19. ●
!
double seg_to_seg_dis(Line a, Line b){!
double res = seg_to_point_dis(a, b.first);!
res = min(res, set_to_point_dis(a, b.second) );!
res = min(res, set_to_point_dis(b, a.first) );!
res = min(res, set_to_point_dis(b, a.second) );!
return res;!
}

20. ● a b !
◦ a 2 b !
◦ b 2 a !
●
a
b
a
b

21. ● a,b !
● b 2 a !
● a 2 b !
● ccw
α
-β
b1
b2

22. bool is_cross(Line a, Line b){!
if(ccw(a.first,a.second,b.first)*!
ccw(a.first,a.second,b.second) <= 0 &&
ccw(b.first,b.second,a.first)*!
ccw(b.first,b.second,a.second) <= 0){!
return true;!
}!
return false;!
}

23. : AOJ2351 Closest Segment Pair
● N !
● !
● 2 ≦ N ≦ 105
!
● 0 ≦ xi,yi ≦ 100
23

24.
b1
b2
cp
● a , b !
● a !
●

25. Point cross_point(Line a, Line b){
double d1 = cross(b.second - b.first,
b.first - a.first);
double d2 = cross(b.second - b.first,
a.second - a.first);
return a.first + (a.second - a.first) * d1/d2;
}
25
b1
b2
cp
d1 = |b2-b1| * |b1-a1| * sinx!
= |b2-b1| * |o1-a1|!
d2 = |b2-b1| * |a1-a1| * siny!
= |b2-b1| * |o2-a1|!
!
d1/d2 = |o1-a1| / |o2-a1|!
= |cp-a1| / |a1-a1|
o1
o2

26. : AOJ0187 Stoning Fortune
● 3 3
!
●
3 !
● -1000 ≦ xi,yi ≦ 1000
26