实用计算几何学
前言GeometryPointLineSegmentPolyline
Algorithms基本运算Projection-投影Distance-求距离Side-求相对位置关系Intersection-相交Curvature-曲率Find closest segment-求polyline上距离给定点最近的线段
前言
前段时间在b站发布了关于二维平面下一些计算几何学知识的讲解，有许多小伙伴私戳我说能不能出个代码实现，所以这段时间就抽个时间用c++实现下视频里面讲的内容。
注： 本篇博客不再具体讲解理论内容，而是实现相关算法。想要进一步深入了解理论内容的小伙伴可以去回顾之前的视频讲解：bilibili。
代码仓库：https://github.com/CHH3213/Math_Geometry/
Geometry
Point
point的struct定义如下：
// Define point struct.
struct Point {
Point()=default;
Point(double x_in, double y_in) : x(x_in), y(y_in) {}
Point(const Point& p) : x(p.x), y(p.y) {}
Point& operator=(const Point& p) {
x = p.x;
y = p.y;
return *this;
}
Point operator+(const Point& p) const{
return {x + p.x, y + p.y};
}
Point operator-(const Point& p) const{
return {x - p.x, y - p.y};
}
double operator*(const Point& p) const{
return x * p.x+ y * p.y;
}
Point operator*(double k)const {
return {x *k, y * k};
}
friend Point operator*(double k, const Point& p) {
return {p.x * k, p.y * k};
}
bool operator==(const Point& p)const{
return p.x==x&&p.y==y;
}
bool operator!=(const Point& p)const{
return !(p.x==x&&p.y==y);
}
double modulus()const {
return sqrt(x * x + y * y);
}
double DistanceTo(const Point& other)const {
double dx = x - other.x;
double dy = y - other.y;
return sqrt(dx * dx + dy * dy);
}
friend std::ostream& operator<<(std::ostream& out, const Point& p) {
out << "(" << p.x << ", " << p.y << ")";
return out;
}
double x;
double y;
};
Line
line就是直线，直线的struct我们可以定义成：
//Define line segment.
struct Line {
Line()=default;
Line(Point p1_in, Point p2_in) : p1(p1_in), p2(p2_in),direction(p2_in-p1_in) {
}
friend std::ostream& operator<<(std::ostream& out, const Line& line) {
out << "Line: " << line.p1 << " ---> " << line.p2;
return out;
}
// A point in Line.
Point p1;
// Another point in line.
Point p2;
Point direction;
};
Segment
线段的struct如下：
// Define segment struct.
struct Segment {
Segment()=default;
Segment(Point start_in, Point end_in) : start(start_in), end(end_in),direction(end-start) {
}
Segment(const Segment& s) : start(s.start), end(s.end),direction(end-start) {}
Segment& operator=(const Segment& s) {
start = s.start;
end = s.end;
return *this;
}
Segment operator+(const Segment& rhs)const {
return {start + rhs.start, end + rhs.end};
}
Segment operator-(const Segment& rhs)const {
return {start - rhs.start, end - rhs.end};
}
double length() const{
return direction.modulus();
}
Point unit_direction() const{
double len = length();
if (len != 0) {
return {direction.x / len, direction.y / len};
} else {
// Handle the case where the length is zero (avoid division by zero).
throw std::runtime_error("Cannot calculate unit direction for a segment with zero length.");
}
}
friend std::ostream& operator<<(std::ostream& out, const Segment& s) {
out << "Segment: " << s.start << " ---> " << s.end;
return out;
}
Point start;
Point end;
Point direction;
};
Polyline
线段彼此相连便组成了折线段，也就是polyline，我们可以这样定义struct
// Define polyline struct.
// Note that a polyline we can use points vector or segments vector to represent.
struct Polyline {
Polyline()=default;
Polyline(const std::vector
for(int i = 1;i segs.push_back(Segment(points[i-1],points[i])); } } Polyline(const std::vector for(int i = 0;i points.push_back(segs[i].start); } points.push_back(segs[segs.size()-1].end); } void append(const Segment& seg) { if(!segs.empty() && segs.back().end != seg.start) { throw std::invalid_argument("Disconnected Segment"); } segs.push_back(seg); points.push_back(seg.end); } void append(const Point& p) { const auto seg = Segment(points.back(),p); points.push_back(p); segs.push_back(seg); } Polyline operator+(const Polyline& other) const { Polyline result; result.segs = this->segs; result.points = this->points; for(auto& seg : other.segs) { result.append(seg); } return result; } Segment GetSegmentByIndex(int index) { if(index < 0 || index >= segs.size()) { throw std::out_of_range("Index out of range"); } return segs[index]; } std::vector return points; } std::vector return segs; } private: std::vector std::vector }; Algorithms 基本运算 点积 // Calculates dot product. double DotProduct(const Vec& v1,const Vec& v2){ return v1.x*v2.x+v1.y*v2.y; } 叉积 // Calculates cross product. double CrossProduct(const Vec& v1,const Vec& v2) { return v1.x*v2.y-v2.x*v1.y; } Projection-投影 投影这里指的是求点到线段的投影点、投影长度。 点到线段的投影长度 // Compute projection length of point p. double ComputeProjectionLength(const Point& p,const Segment& segment){ const auto& p1p = p-segment.start; return DotProduct(p1p,segment.unit_direction()); } 点到线段的投影点 // Compute projection point of point p. Point ComputeProjection(const Point& p,const Segment& segment){ double projection_length = ComputeProjectionLength(p,segment); return segment.start + segment.unit_direction()*projection_length; } Distance-求距离 距离包括点-点，点-直线，点-线段，线段-线段等之间的距离。 point to point // Get distance between point p1 and point p2. double GetDistance(const Point& p1, const Point& p2){ return p1.DistanceTo(p2); } point to line // Get distance between point p and a straight line. double GetDistance(const Point& p, const Line& line){ Segment p1p2(line.p1,line.p2); Segment p1p(line.p1,p); return std::abs(CrossProduct(p1p2.direction,p1p.direction))/p1p2.length(); } point to segment // Get distance between point p and segment(p1,p2). double GetDistance(const Point& p, const Segment& segment){ Segment p1p(segment.start,p); Segment p2p(segment.end,p); const auto c1 = DotProduct(p1p.direction,segment.direction); const auto c2 = DotProduct(p2p.direction,segment.direction); if(c1<=0){ //distance(p,segment)=distacne(p1,p). return GetDistance(segment.start,p); } if(c2>=0){ //distance(p,segment)=distacne(p2,p). return GetDistance(segment.end,p); } return std::abs(CrossProduct(segment.direction,p1p.direction))/segment.length(); } segment to segment // Get distance between segment and segment (method 1), method 2 is to be determined. double GetDistance(const Segment& s1,const Segment& s2){ const double d1 = GetDistance(s1.start, s2); const double d2 = GetDistance(s1.end, s2); const double d3 = GetDistance(s2.start, s1); const double d4 = GetDistance(s2.end, s1); return std::min(std::min(d1, d2), std::min(d3, d4)); } 注：视频中讲解的另外一种方法暂时未实现，留个todo。 Side-求相对位置关系 对于一个点与线段之间的位置关系，我们可以定义一个enum来表示 enum class Side{ // When the segment's length is zero, it's useless to determine the side, so we use DEFAULT_NO_SIDE to show. DEFAULT_NO_SIDE=0, LEFT, RIGHT, // The three below states mean that the point is in line. ON_AFTER, ON_BEFORE, WITHIN }; 也就是说点与线段的相对位置关系包括以下几种： 点在线段的左边点在线段的右边点在线段所在的直线上 点在线段前面点在线段后面点在线段内部 判断点跟一条线段的相对位置关系 // Determine which side of the segment the point is. Side GetSide(const Point& p,const Segment& s){ const auto& p0 = s.start; const auto& p2 = s.end; const auto& a = p-p0; const auto& b = p2-p0; const auto cross_product = CrossProduct(a,b); if(cross_product!=0){ // Returns LEFT if p0,p,p2 are clockwise position, returns RIGHT means p0,p,p2 are counter-clockwise position. return cross_product<0?Side::LEFT:Side::RIGHT; } const auto dot_product = DotProduct(a,b); if(dot_product<0){ return Side::ON_BEFORE; }else if(dot_product>0){ if(b.modulus() return Side::ON_AFTER; }else{ return Side::WITHIN; } }else{ if(a.modulus()==0){ return Side::WITHIN; } return Side::DEFAULT_NO_SIDE; } } 判断点与两条相连的线段的相对位置关系——法一 // Determine which side of two segments the point is. //Method 1: directly use cross product to determine and only have left/right options. Side GetSide(const Point& p, const Segment& s1,const Segment& s2) { //DCHECK(s1.end==s2.start)<<"please ensure the two segments are connected."; if (s1.end != s2.start) { throw std::runtime_error("Error: The two segments are not connected."); } const auto& p0p = p-s1.start; const auto& p1p = p-s2.start; const auto c1 = CrossProduct(s1.direction,p0p); const auto c2 = CrossProduct(s2.direction,p1p); if(c1>0&&c2>0){ return Side::LEFT; } if(c1<0&&c2<0){ return Side::RIGHT; } return std::abs(c1)>std::abs(c2)?Side::LEFT:Side::RIGHT; } 判断点与两条相连的线段的相对位置关系——法二 // Determine which side of two segments the point is. // Method 2: through the side of p to one segment to determine, and except left/right side, we also provide other options. Side GetSide(const Point& p, const Segment& s1,const Segment& s2) { //DCHECK(s1.end==s2.start)<<"please ensure the two segments are connected."; if (s1.end != s2.start) { throw std::runtime_error("Error: The two segments are not connected."); } const auto side_1 = GetSide(p,s1); const auto side_2 = GetSide(p,s2); if(side_1==side_2){ return side_1; } if(side_1==Side::WITHIN||side_2==Side::WITHIN){ return Side::WITHIN; } const auto& p0p = p-s1.start; const auto& p1p = p-s2.start; const auto c1 = CrossProduct(s1.direction,p0p); const auto c2 = CrossProduct(s2.direction,p1p); return std::abs(c2) > std::abs(c1) ? side_2 : side_1; } Intersection-相交 这里相交主要指的是线段与线段之间的相交。很显然相交分为两步： 判断是否相交 // Determine whether segment 1 intersects segment 2. bool IsIntersection(const Segment& s1, const Segment& s2){ const double o1 = CrossProduct(s2.start - s1.start, s1.direction); const double o2 = CrossProduct(s2.end - s1.start, s1.direction); const double o3 = CrossProduct(s1.start - s2.start, s2.direction); const double o4 = CrossProduct(s1.end - s2.start, s2.direction); // Segments are considered intersecting if they have different orientations. if(o1 * o2 < 0 && o3 * o4 < 0){ return true; } auto on_segment = [](const Point &p, const Point &q, const Point &r){ return (q.x <= std::max(p.x, r.x) && q.x >= std::min(p.x, r.x) && q.y <= std::max(p.y, r.y) && q.y >= std::min(p.y, r.y)); }; // Additional checks for collinear points. if(o1 == 0 && on_segment(s1.start, s2.start, s1.end)){ return true; } if(o2 == 0 && on_segment(s1.start, s2.end, s1.end)){ return true; } if(o3 == 0 && on_segment(s2.start, s1.start, s2.end)){ return true; } if(o4 == 0 && on_segment(s2.start, s1.end, s2.end)){ return true; } return false; } 若相交则求出相交点 //Calculate the intersection between segment 푝0푝1 and segment 푝2푝3. Point GetIntersectionPoint(const Segment& s1, const Segment& s2){ if(!IsIntersection(s1, s2)){ std::cout << "No intersection, return a deafult point value:(0,0)!"; return Point(0, 0); } const auto& u = s1.direction; const auto& v = s2.direction; const auto& w = s1.start - s2.end; const auto c1 = CrossProduct(w, v); const auto c2 = CrossProduct(v, u); if(c2 != 0){ const double t = std::abs(c1 / c2); return s1.start + t * u; } // Address collinear and overlapping situation. If so, we return overlaping start or end. const auto side_1 = GetSide(s1.start, s2); const auto side_2 = GetSide(s1.end, s2); const auto side_3 = GetSide(s2.start, s1); const auto side_4 = GetSide(s2.end, s1); if(side_1 == Side::WITHIN){ return s1.start; } if(side_2 == Side::WITHIN){ return s1.end; } if(side_3 == Side::WITHIN){ return s2.start; } if(side_4 == Side::WITHIN){ return s2.end; } throw std::runtime_error("Something is wrong, please check."); } Curvature-曲率 通过三点求曲率： // Obtain curvature according to p1,p2,p3. // NOTE : curvature has a sign, not just an unsigned value. double GetCurvature(const Point& p1, const Point& p2, const Point& p3){ const auto& p1p2 = p2 - p1; const auto& p2p3 = p3 - p2; const auto& p1p3 = p3 - p1; const auto& a = p1p2.modulus(); const auto& b = p2p3.modulus(); const auto& c = p1p3.modulus(); const auto cross_product = CrossProduct(p1p2, p2p3); return 2 * cross_product / (a * b * c); } Find closest segment-求polyline上距离给定点最近的线段 求距离给定点最近的线段。 实现方式1 // Find the given point's closest segment in polyline using linear search. // Option 1. Segment FindClosestSegmentByLinearSearch(const Point& point, const Polyline& polyline){ const auto points = polyline.Points(); const auto segments = polyline.Segments(); //Compute the square distance between given point and first point in polyline. double min_dist_sq = GetDistance(point,points[0]); int closest_segment_index = 0; for(int i=1;i const auto& p1 = points[i-1]; const auto& p2 = points[i]; const auto& seg = segments[i-1]; const auto& v = seg.unit_direction(); const auto& w = point to p1; const double c1 = DotProduct(w,v); if(c1<=0){ continue; } double dist_sq= 0.0; const double c2 = seg.length(); if(c2<=c1){ dist_sq = GetDistance(point,p2); }else{ dist_sq = GetDistance(point,seg); } if(dist_sq min_dist_sq = dist_sq; closest_segment_index=i-1; if(min_dist_sq break; } } } return segments[closest_segment_index]; } 实现方式2 // Find the given point's closest segment in polyline using linear search. // Option 2: since we have implemented distance function, we can directly use it. Segment FindClosestSegmentByLinearSearch(const Point& point, const Polyline& polyline){ const auto& points = polyline.Points(); const auto& segments = polyline.Segments(); //Compute the square distance between given point and first point in polyline. double min_dist_sq = GetDistance(point,points[0]); int closest_segment_index = 0; for(int i=0;i const auto& seg = segments[i]; const double dist_sq = GetDistance(point,seg); if(dist_sq min_dist_sq = dist_sq; closest_segment_index=i; if(min_dist_sq break; } } } return segments[closest_segment_index]; } 视频中主要涉及的内容实现基本完成了，还有一些额外的没有实现，后续会把它完善。 以上所有代码均存放于github仓库中，欢迎访问：https://github.com/CHH3213/Math_Geometry/ 精彩链接
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