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计算几何相关模板�Q�更��C��……）

LSM — Wed, 22 Aug 2007 10:39:00 GMT

计算几何相关模板�Q�更��C��……�Q?br>

最�q�在学计��几何，边学�Q�边整理模板�Q�有错的话请大家指出�Q?br>

1//计算几何模板 ~ alpc02
2const double PRECISION = 1e-8;
3struct Point {
4    double x, y;
5};
6int dblcmp(double d) {
7    return (fabs(d) < PRECISION) ? 0:(d>0 ? 1:-1);
8} //三叉口函敎ͼ�避免�_�ֺ�误差
9double length(double x, double y) {
10    return sqrt(x*x + y*y);
11} //向量长度
12double dotdet(double x1, double y1, double x2, double y2) {
13    return x1*x2 + y1*y2;
14} //点积
15double det(double x1, double y1, double x2, double y2) {
16    return x1*y2 - x2*y1;
17} //叉积
18int cross(const Point &a, const Point &c, const Point &d) {
19    return dblcmp( det(a.x-c.x, a.y-c.y, d.x-c.x, d.y-c.y) );
20} //��x��螺旋定则�Q?——a在cd右侧�Q?1——a在cd左侧�Q?——三点共�U?/span>
21bool between(const Point &a, const Point &c, const Point &d) {
22    return dblcmp( dotdet(c.x-a.x, c.y-a.y, d.x-a.x, d.y-a.y) ) != 1;
23} //在cross(a,c,d)==0的基��上，可判断点a是否在cd内部
24int segIntersect(const Point &a, const Point &b, const Point &c, const Point &d) {
25    int a_cd = cross(a,c,d);
26    if(a_cd == 0 && between(a,c,d))    return 2;
27    int b_cd = cross(b,c,d);
28    if(b_cd == 0 && between(b,c,d))    return 2;
29    int c_ab = cross(c,a,b);
30    if(c_ab == 0 && between(c,a,b))    return 2;
31    int d_ab = cross(d,a,b);
32    if(d_ab == 0 && between(d,a,b))    return 2;
33    if ((a_cd ^ b_cd) == -2 && (c_ab ^ d_ab) == -2)
34        return 1;
35    return 0;
36} //两线�D늛�交情况：0——不�怺��Q?——规范相交，2——不规范�怺��Q�交于端�Ҏ��重合�Q?/span>
37void intersectPoint(const Point &a, const Point &b, const Point &c, const Point &d, Point &e) {
38    double sc, sd;
39    sc = fabs( det(b.x-a.x, b.y-a.y, c.x-a.x, c.y-a.y) );
40    sd = fabs( det(b.x-a.x, b.y-a.y, d.x-a.x, d.y-a.y) );
41    e.x = (sc * d.x + sd * c.x) / (sc + sd);
42    e.y = (sc * d.y + sd * c.y) / (sc + sd);
43} //两线�D�规范相交时�Q�求交点坐标
44int linesegIntersect(const Point &a, const Point &b, const Point &c, const Point &d) {
45    int c_ab = cross(c,a,b);
46    if(c_ab == 0)    return 2;
47    int d_ab = cross(d,a,b);
48    if(d_ab == 0)    return 2;
49    if(c_ab ^ d_ab == -2)
50        return 1;
51    return 0;
52} //直线ab和线�D�cd�怺�情况�Q?——不�怺��Q?——规范相交，2——不规范�怺��Q�交于端�Ҏ��重合�Q?/span>
53int lineIntersect(const Point &a, const Point &b, const Point &c, const Point &d) {
54    if(dblcmp(det(b.x-a.x, b.y-a.y, d.x-c.x, d.y-c.y)) != 0)
55        return 1;
56    if(cross(a,c,d) == 0)
57        return 2;
58    return 0;
59} //两直�U�相交情况：0——��^行，1——规范相交，2——不规范�怺��Q�重合）
60

LSM 2007-08-22 18:39 发表评论

计算几何入门题（更新中……）

LSM — Wed, 22 Aug 2007 10:28:00 GMT

摘要: 计算几何入门题（更新�?#8230;…�Q?http://acm.pku.edu.cn/JudgeOnline/problem?id=1696Space Ant用到叉积�Q�点�U�。这个题的代码稍微改改，貌似可以认�ؓ是凸包的o(n^2)��法�Q�即卷包心菜�?Code highlighting produced by Actipro CodeHighlighter (freeware)http... 阅读全文

LSM 2007-08-22 18:28 发表评论

LSM — Tue, 21 Aug 2007 00:52:00 GMT

计算几何初步认识

—�?/span>Wjx昨晚�l�我速成的计��几�?/span>

一�?/span>炏V�?/span>

l 点的坐标A(x1, y1)�Q?/span>B(x2, y2)

二�?/span>向量�?/span>

l 向量AB = (x2-x1, y2-y1) = (x3,y3) �Q?/span>CD = (x4, y4)�?/span>

l 向量的模|AB| = sqrt(x3*x3 + y3*y3) 卛_��量的长度�?/span>

三�?/span>点积�?/span>

l 点积的结果�ؓ一个数倹{�?/span>

l 数��D��方�?/span>AB * CD �Q?/span> x3*x4 + y3*y4�?/span>

l 几何意义AB * CD = |AB| * |CD| * cos(a)�Q?/span>a为向�?/span>AB逆时针�{�?/span>CD的角度，0<=a<360�Q�也可以认�ؓ是两向量的夹角，0<=a<=180。一般用于求夹角�Q?/span>a = acos( (AB * CD) / (|AB| * |CD|) )。也可：|CD| * cos(a) = AB * CD / |AB|�Q�即向量CD�?/span>AB上的投媄�?/span>

四�?/span>叉积�?/span>

l 叉积的结果�ؓ一个向量�?/span>

l AB×CD数�?/span>= x3*y4 – x4*y3�?/span>

l 方向由右手螺旋定则判定�?/span>

l 几何意义�Q?/span>AB×CD = |AB| * |CD| * sin(a)�Q�取�l�对值即是以AB�?/span>CD��的��^行四边�Ş面积�?/span>

五�?/span>�U�段�怺�的判定（判定�U�段AB和线�D?/span>CD是否�怺��Q�属于哪�U�相交）�?/span>

l 规范�怺��Q�交点只有一个，且不是线�D늚�端点�?/span>

1�?/span>充要条�g�Q�点A和点B�?/span>CD的两侧�ƈ且点C和点D�?/span>AB的两侧�?/span>

2�?/span>如何判断�?/span>A在向�?/span>CD的左侧还是右侧：CA×CD的数值大�?/span>0则左侧，��于0则右侧。于�?/span>CA×CD的数值和CB×CD的数值异��P��?/span>A和点B�?/span>CD的两侧，同理可判断点C和点D是否�?/span>AB的两侧（注意�Q�必��M��格异��P��?/span>

l 不规范相交：交点为某个线�D늚�端点�Q�甚至两�U�段有一�D�重合�?/span>

1�?/span>充要条�g�Q�一条线�D늚�一个端点在另一条线�D�上。即A�?/span>CD上或B�?/span>CD上或C�?/span>AB上或D�?/span>AB上�?/span>

2�?/span>A在线�D?/span>CD上的充要条�g�Q?/span>

a) AC×AD = 0�Q�几何意义，AC�?/span>AD�l�成的��^行四边�Ş的面�U��ؓ0�Q�即A�?/span>C�?/span>D三点��q��?/span>

b) A点在CD之间�Q?/span>A.x处于C.x�?/span>D.x之间�q�且A.y处于C.y�?/span>D.y之间

LSM 2007-08-21 08:52 发表评论