我尝试了几种不同的方法来计算点是否在外接圆之内，但有时，我会不断得到不与原始超三角形不共享点的三角形。
我的代码：
#include <iostream>
#include <vector>
#include <random>
#include <algorithm>
#include <chrono>
bool is_point_inside_circumcircle(const int32_t pX, const int32_t pY,
int32_t aX, int32_t aY, int32_t bX, int32_t bY, int32_t cX, int32_t cY)
{
const uint32_t ab = aX * aX + aY * aY;
const uint32_t bc = bX * bX + bY * bY;
const uint32_t ca = cX * cX + cY * cY;
uint32_t circumX = (ab * (cY - bY) + bc * (aY - cY) + ca * (bY - aY)) / (aX * (cY - bY) + bX * (aY - cY) + cX * (bY - aY));
uint32_t circumY = (ab * (cX - bX) + bc * (aX - cX) + ca * (bX - aX)) / (aY * (cX - bX) + bY * (aX - cX) + cY * (bX - aX));
circumX /= 2;
circumY /= 2;
uint32_t dX = aX - circumX;
uint32_t dY = aY - circumY;
uint32_t circumR = dX * dX + dY * dY;
dX = pX - circumX;
dY = pY - circumY;
uint32_t d = dX * dX + dY * dY;
return d <= circumR;
}
struct Edge
{
int32_t x1;
int32_t y1;
int32_t x2;
int32_t y2;
};
bool operator==(const Edge& lhs, const Edge& rhs)
{
return (lhs.x1 == rhs.x1 && lhs.y1 == rhs.y1 && lhs.x2 == rhs.x2 && lhs.y2 == rhs.y2)
|| (lhs.x1 == rhs.x2 && lhs.y1 == rhs.y2 && lhs.x2 == rhs.x1 && lhs.y2 == rhs.y1);
}
struct Triangle
{
Edge ab;
Edge bc;
Edge ca;
};
bool triangle_contains(const Triangle& tri, int32_t x, int32_t y)
{
return (tri.ab.x1 == x && tri.ab.y1 == y) ||
(tri.bc.x1 == x && tri.bc.y1 == y) ||
(tri.ca.x1 == x && tri.ca.y1 == y);
}
bool operator==(const Triangle& lhs, const Triangle& rhs)
{
return
((lhs.ab.x1 == rhs.ab.x1 && lhs.ab.y1 == rhs.ab.y1) ||
(lhs.ab.x1 == rhs.bc.x1 && lhs.ab.y1 == rhs.bc.y1) ||
(lhs.ab.x1 == rhs.ca.x1 && lhs.ca.y1 == rhs.ca.y1)) &&
((lhs.bc.x1 == rhs.ab.x1 && lhs.bc.y1 == rhs.ab.y1) ||
(lhs.bc.x1 == rhs.bc.x1 && lhs.bc.y1 == rhs.bc.y1) ||
(lhs.bc.x1 == rhs.ca.x1 && lhs.bc.y1 == rhs.ca.y1)) &&
((lhs.ca.x1 == rhs.ab.x1 && lhs.ca.y1 == rhs.ab.y1) ||
(lhs.ca.x1 == rhs.bc.x1 && lhs.ca.y1 == rhs.bc.y1) ||
(lhs.ca.x1 == rhs.ca.x1 && lhs.ca.y1 == rhs.ca.y1));
}
bool operator!=(const Triangle& lhs, const Triangle& rhs)
{
return !(lhs==rhs);
}
std::vector<Triangle> generate_galaxy(const std::size_t numSystems)
{
std::vector<std::pair<int32_t,int32_t>> galaxy;
std::mt19937 gen(std::chrono::system_clock::now().time_since_epoch().count());
std::uniform_int_distribution<> dimension(0, numSystems*3-1);
for(std::size_t i = 0; i < numSystems; ++i)
{
galaxy.emplace_back(dimension(gen),dimension(gen));
}
std::vector<Triangle> triangulation;
Triangle super;
int32_t minX = std::numeric_limits<int32_t>::max();
int32_t minY = std::numeric_limits<int32_t>::max();
int32_t maxX = 0;
int32_t maxY = 0;
for(const std::pair<int32_t,int32_t>& n : galaxy)
{
if(n.first < minX) minX = n.first;
if(n.second < minY) minY = n.second;
if(n.first > maxX) maxX = n.first;
if(n.second > maxY) maxY = n.second;
}
int32_t dX = maxX - minX;
int32_t dY = maxY - minY;
int32_t dM = std::max(dX,dY);
int32_t midX = (minX + maxX) / 2;
int32_t midY = (minY + maxY) / 2;
super.ab.x1 = super.ca.x2 = midX - 20 * dM;
super.ab.y1 = super.ca.y2 = midY - dM;
super.ab.x2 = super.bc.x1 = midX;
super.ab.y2 = super.bc.y1 = midY + 20 * dM;
super.bc.x2 = super.ca.x1 = midX + 20 * dM;
super.bc.y2 = super.ca.y1 = midY - dM;
triangulation.push_back(super);
for(const std::pair<int32_t,int32_t>& n : galaxy)
{
std::vector<Triangle> badTriangles;
for(const Triangle& triangle : triangulation)
{
if(is_point_inside_circumcircle(n.first, n.second, triangle.ab.x1, triangle.ab.x2, triangle.bc.x1, triangle.bc.y1,
triangle.ca.x1, triangle.ca.y1))
{
badTriangles.push_back(triangle);
}
}
std::cout << "\n--------------------------------------------" << std::endl;
std::cout << "\n\nAFTER NEW NODE CREATING BAD TRIANGLES" << std::endl;
std::cout << "NODE X,Y: " << n.first << "," << n.second
<< "\nTriangulation (" << triangulation.size() << ")";
for(const Triangle& tri : triangulation)
std::cout << "\n\t" << tri.ab.x1 << "," << tri.ab.y1 << " "
<< tri.bc.x1 << "," << tri.bc.y1 << " "
<< tri.ca.x1 << "," << tri.ca.y1;
std::cout << std::endl;
std::cout << "NODE X,Y: " << n.first << "," << n.second
<< "\nBad (" << badTriangles.size() << ")";
for(const Triangle& tri : badTriangles)
std::cout << "\n\t" << tri.ab.x1 << "," << tri.ab.y1 << " "
<< tri.bc.x1 << "," << tri.bc.y1 << " "
<< tri.ca.x1 << "," << tri.ca.y1;
std::cout << std::endl;
std::vector<Edge> polygon;
std::vector<Triangle>::iterator triangleItr = badTriangles.begin();
while(triangleItr != badTriangles.end())
{
if(std::find_if(badTriangles.begin(),badTriangles.end(),[&](const Triangle& tri)
{
return
*triangleItr != tri && (
triangleItr->ab == tri.ab ||
triangleItr->ab == tri.bc ||
triangleItr->ab == tri.ca);
}) == badTriangles.end())
{
polygon.push_back(triangleItr->ab);
}
if(std::find_if(badTriangles.begin(),badTriangles.end(),[&](const Triangle& tri)
{
return
*triangleItr != tri && (
triangleItr->bc == tri.ab ||
triangleItr->bc == tri.bc ||
triangleItr->bc == tri.ca);
}) == badTriangles.end())
{
polygon.push_back(triangleItr->bc);
}
if(std::find_if(badTriangles.begin(),badTriangles.end(),[&](const Triangle& tri)
{
return
*triangleItr != tri && (
triangleItr->ca == tri.ab ||
triangleItr->ca == tri.bc ||
triangleItr->ca == tri.ca);
}) == badTriangles.end())
{
polygon.push_back(triangleItr->ca);
}
++triangleItr;
}
triangulation.erase(std::remove_if(triangulation.begin(),triangulation.end(),[&](const Triangle& tri)
{
for(const Triangle& triangle : badTriangles)
{
if(tri == triangle)
return true;
}
return false;
}),triangulation.end());
std::cout << "\n\nAFTER ADDING EDGES TO POLYGON BECAUSE NOT SHARED IN BAD TRIANGLES AND REMOVE BAD TRIANGLES FROM TRIANGULATION" << std::endl;
std::cout << "NODE X,Y: " << n.first << "," << n.second
<< "\nTriangulation (" << triangulation.size() << ")";
for(const Triangle& tri : triangulation)
std::cout << "\n\t" << tri.ab.x1 << "," << tri.ab.y1 << " "
<< tri.bc.x1 << "," << tri.bc.y1 << " "
<< tri.ca.x1 << "," << tri.ca.y1;
std::cout << std::endl;
std::cout << "NODE X,Y: " << n.first << "," << n.second
<< "\nBad (" << badTriangles.size() << ")";
for(const Triangle& tri : badTriangles)
std::cout << "\n\t" << tri.ab.x1 << "," << tri.ab.y1 << " "
<< tri.bc.x1 << "," << tri.bc.y1 << " "
<< tri.ca.x1 << "," << tri.ca.y1;
std::cout << std::endl;
std::cout << "NODE X,Y: " << n.first << "," << n.second
<< "\nPoly (" << polygon.size() << ")";
for(const Edge& e : polygon)
std::cout << "\n\t" << e.x1 << "," << e.y1 << " > " << e.x2 << "," << e.y2;
std::cout << std::endl;
for(const Edge& edge : polygon)
{
Triangle newTri;
newTri.ab = edge;
Edge bc;
bc.x1 = edge.x2;
bc.y1 = edge.y2;
bc.x2 = n.first;
bc.y2 = n.second;
Edge ca;
ca.x1 = n.first;
ca.y1 = n.second;
ca.x2 = edge.x1;
ca.y2 = edge.y1;
newTri.bc = bc;
newTri.ca = ca;
triangulation.push_back(newTri);
}
std::cout << "\n\nADDING NEW TRIANGLES FROM EDGES TO TRIANGULATION" << std::endl;
std::cout << "NODE X,Y: " << n.first << "," << n.second
<< "\nTriangulation (" << triangulation.size() << ")";
for(const Triangle& tri : triangulation)
std::cout << "\n\t" << tri.ab.x1 << "," << tri.ab.y1 << " "
<< tri.bc.x1 << "," << tri.bc.y1 << " "
<< tri.ca.x1 << "," << tri.ca.y1;
std::cout << std::endl;
std::cout << "NODE X,Y: " << n.first << "," << n.second
<< "\nBad (" << badTriangles.size() << ")";
for(const Triangle& tri : badTriangles)
std::cout << "\n\t" << tri.ab.x1 << "," << tri.ab.y1 << " "
<< tri.bc.x1 << "," << tri.bc.y1 << " "
<< tri.ca.x1 << "," << tri.ca.y1;
std::cout << std::endl;
std::cout << "NODE X,Y: " << n.first << "," << n.second
<< "\nPoly (" << polygon.size() << ")";
for(const Edge& e : polygon)
std::cout << "\n\t" << e.x1 << "," << e.y1 << " > " << e.x2 << "," << e.y2;
std::cout << std::endl;
}
triangulation.erase(std::remove_if(triangulation.begin(),triangulation.end(),[&](const Triangle& tri)
{
return triangle_contains(tri,super.ab.x1,super.ab.y1) ||
triangle_contains(tri,super.bc.x1,super.bc.y1) ||
triangle_contains(tri,super.ca.x1,super.ca.y1);
}),triangulation.end());
// every edge in triangulation is now the connection of vertices
return triangulation;
}
int main()
{
std::vector<Triangle> galaxy = generate_galaxy(4);
std::cout << "\n\nOUTCOMES:\n" << std::endl;
for(const Triangle& tri : galaxy)
{
std::cout << tri.ab.x1 << "," << tri.ab.y1 << " > " << tri.ab.x2 << "," << tri.ab.y2 << std::endl;
std::cout << tri.bc.x1 << "," << tri.bc.y1 << " > " << tri.bc.x2 << "," << tri.bc.y2 << std::endl;
std::cout << tri.ca.x1 << "," << tri.ca.y1 << " > " << tri.ca.x2 << "," << tri.ca.y2 << std::endl;
}
return 0;
}
我也尝试采用https://github.com/Bl4ckb0ne/delaunay-triangulation/blob/master/dt/delaunay.cpp中的一些方法，希望能有所帮助，但似乎没有太大的不同。
在没有结果三角形的情况下，上面的输出如下所示：
--------------------------------------------
AFTER NEW NODE CREATING BAD TRIANGLES
NODE X,Y: 2,10
Triangulation (1)
-154,-1 6,167 166,-1
NODE X,Y: 2,10
Bad (1)
-154,-1 6,167 166,-1
AFTER ADDING EDGES TO POLYGON BECAUSE NOT SHARED IN BAD TRIANGLES AND REMOVE BAD TRIANGLES FROM TRIANGULATION
NODE X,Y: 2,10
Triangulation (0)
NODE X,Y: 2,10
Bad (1)
-154,-1 6,167 166,-1
NODE X,Y: 2,10
Poly (3)
-154,-1 > 6,167
6,167 > 166,-1
166,-1 > -154,-1
ADDING NEW TRIANGLES FROM EDGES TO TRIANGULATION
NODE X,Y: 2,10
Triangulation (3)
-154,-1 6,167 2,10
6,167 166,-1 2,10
166,-1 -154,-1 2,10
NODE X,Y: 2,10
Bad (1)
-154,-1 6,167 166,-1
NODE X,Y: 2,10
Poly (3)
-154,-1 > 6,167
6,167 > 166,-1
166,-1 > -154,-1
--------------------------------------------
AFTER NEW NODE CREATING BAD TRIANGLES
NODE X,Y: 3,6
Triangulation (3)
-154,-1 6,167 2,10
6,167 166,-1 2,10
166,-1 -154,-1 2,10
NODE X,Y: 3,6
Bad (2)
-154,-1 6,167 2,10
6,167 166,-1 2,10
AFTER ADDING EDGES TO POLYGON BECAUSE NOT SHARED IN BAD TRIANGLES AND REMOVE BAD TRIANGLES FROM TRIANGULATION
NODE X,Y: 3,6
Triangulation (1)
166,-1 -154,-1 2,10
NODE X,Y: 3,6
Bad (2)
-154,-1 6,167 2,10
6,167 166,-1 2,10
NODE X,Y: 3,6
Poly (4)
-154,-1 > 6,167
2,10 > -154,-1
6,167 > 166,-1
166,-1 > 2,10
ADDING NEW TRIANGLES FROM EDGES TO TRIANGULATION
NODE X,Y: 3,6
Triangulation (5)
166,-1 -154,-1 2,10
-154,-1 6,167 3,6
2,10 -154,-1 3,6
6,167 166,-1 3,6
166,-1 2,10 3,6
NODE X,Y: 3,6
Bad (2)
-154,-1 6,167 2,10
6,167 166,-1 2,10
NODE X,Y: 3,6
Poly (4)
-154,-1 > 6,167
2,10 > -154,-1
6,167 > 166,-1
166,-1 > 2,10
--------------------------------------------
AFTER NEW NODE CREATING BAD TRIANGLES
NODE X,Y: 4,5
Triangulation (5)
166,-1 -154,-1 2,10
-154,-1 6,167 3,6
2,10 -154,-1 3,6
6,167 166,-1 3,6
166,-1 2,10 3,6
NODE X,Y: 4,5
Bad (4)
-154,-1 6,167 3,6
2,10 -154,-1 3,6
6,167 166,-1 3,6
166,-1 2,10 3,6
AFTER ADDING EDGES TO POLYGON BECAUSE NOT SHARED IN BAD TRIANGLES AND REMOVE BAD TRIANGLES FROM TRIANGULATION
NODE X,Y: 4,5
Triangulation (1)
166,-1 -154,-1 2,10
NODE X,Y: 4,5
Bad (4)
-154,-1 6,167 3,6
2,10 -154,-1 3,6
6,167 166,-1 3,6
166,-1 2,10 3,6
NODE X,Y: 4,5
Poly (4)
-154,-1 > 6,167
2,10 > -154,-1
6,167 > 166,-1
166,-1 > 2,10
ADDING NEW TRIANGLES FROM EDGES TO TRIANGULATION
NODE X,Y: 4,5
Triangulation (5)
166,-1 -154,-1 2,10
-154,-1 6,167 4,5
2,10 -154,-1 4,5
6,167 166,-1 4,5
166,-1 2,10 4,5
NODE X,Y: 4,5
Bad (4)
-154,-1 6,167 3,6
2,10 -154,-1 3,6
6,167 166,-1 3,6
166,-1 2,10 3,6
NODE X,Y: 4,5
Poly (4)
-154,-1 > 6,167
2,10 > -154,-1
6,167 > 166,-1
166,-1 > 2,10
--------------------------------------------
AFTER NEW NODE CREATING BAD TRIANGLES
NODE X,Y: 10,8
Triangulation (5)
166,-1 -154,-1 2,10
-154,-1 6,167 4,5
2,10 -154,-1 4,5
6,167 166,-1 4,5
166,-1 2,10 4,5
NODE X,Y: 10,8
Bad (4)
-154,-1 6,167 4,5
2,10 -154,-1 4,5
6,167 166,-1 4,5
166,-1 2,10 4,5
AFTER ADDING EDGES TO POLYGON BECAUSE NOT SHARED IN BAD TRIANGLES AND REMOVE BAD TRIANGLES FROM TRIANGULATION
NODE X,Y: 10,8
Triangulation (1)
166,-1 -154,-1 2,10
NODE X,Y: 10,8
Bad (4)
-154,-1 6,167 4,5
2,10 -154,-1 4,5
6,167 166,-1 4,5
166,-1 2,10 4,5
NODE X,Y: 10,8
Poly (4)
-154,-1 > 6,167
2,10 > -154,-1
6,167 > 166,-1
166,-1 > 2,10
ADDING NEW TRIANGLES FROM EDGES TO TRIANGULATION
NODE X,Y: 10,8
Triangulation (5)
166,-1 -154,-1 2,10
-154,-1 6,167 10,8
2,10 -154,-1 10,8
6,167 166,-1 10,8
166,-1 2,10 10,8
NODE X,Y: 10,8
Bad (4)
-154,-1 6,167 4,5
2,10 -154,-1 4,5
6,167 166,-1 4,5
166,-1 2,10 4,5
NODE X,Y: 10,8
Poly (4)
-154,-1 > 6,167
2,10 > -154,-1
6,167 > 166,-1
166,-1 > 2,10
OUTCOMES:
Process returned 0 (0x0) execution time : 0.620 s
Press any key to continue.
我可以看到，三角剖分只包含与计算出的原始超级三角形共享一个顶点的三角形。因此，我知道为什么将它们全部删除了，但是我无法在网上找到有关为什么会发生这种情况的任何信息。我逐步完成了该过程，并将其与Bowyer–Watson algorithm上的描述和图像进行了匹配，但看不到任何问题。