Wrong inexact intersection between 3D triangles (CGAL)

Question

I'm having a really weird bug when trying to intersect two triangles inside a 3D space while using the CGAL::Exact_predicates_inexact_constructions_kernel kernel. Essentially, I have two triangles that should not intersect. The function CGAL::do_intersect returns always false when testing them, but the function CGAL::intersection builds an intersection, depending on the order of the vertices of the triangles.

The bug disappears when I use the CGAL::Exact_predicates_exact_constructions_kernel kernel, but I can't afford to use it in the real case scenario.

Below is a minimal code with the bug. Triangles B and C are equal (up to a permutation of the vertices), and should return the same intersection with Triangle A.

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Intersections.h>

#include <iostream>
#include <vector>

typedef CGAL::Exact_predicates_inexact_constructions_kernel Kernel;

typedef Kernel::Point_3     Point_3;
typedef Kernel::Triangle_3  Triangle_3;


int main(int argc, char *argv[])
{
    std::vector<Point_3> APoints(3);
    std::vector<Point_3> BPoints(3);

    APoints[0] = Point_3(2, 2, 0.9423616295572568);
    APoints[1] = Point_3(0.9685134704003172, 2, 0.9678422992674797);
    APoints[2] = Point_3(2, 1.124710354419025, 1.068692504586136);

    BPoints[0] = Point_3(2.5, 2.5, 1.442361629557257);
    BPoints[1] = Point_3(1.588259113885977, 2.5, 0.5);
    BPoints[2] = Point_3(2.5, 1.624710354419025, 1.568692504586136);

    Triangle_3 TriangleA(APoints[0],APoints[1],APoints[2]);
    Triangle_3 TriangleB(BPoints[0],BPoints[1],BPoints[2]);
    Triangle_3 TriangleC(BPoints[2],BPoints[1],BPoints[0]);

    std::cout.precision(16);
    std::cout   << "   - Tried to intersect: " << std::endl;
    std::cout   << "   - Triangle (A) " << " : "
                << "(" << TriangleA.vertex(0) << ") "
                << "(" << TriangleA.vertex(1) << ") "
                << "(" << TriangleA.vertex(2) << ") " << std::endl;
    std::cout   << "   - Triangle (B) " << " : "
                << "(" << TriangleB.vertex(0) << ") "
                << "(" << TriangleB.vertex(1) << ") "
                << "(" << TriangleB.vertex(2) << ") " << std::endl;
    std::cout   << "   - Triangle (C) " << " : "
                << "(" << TriangleC.vertex(0) << ") "
                << "(" << TriangleC.vertex(1) << ") "
                << "(" << TriangleC.vertex(2) << ") " << std::endl;

    if( TriangleB.vertex(0)==TriangleC.vertex(2) && 
        TriangleB.vertex(1)==TriangleC.vertex(1) && 
        TriangleB.vertex(2)==TriangleC.vertex(0))
    {
        std::cout << "   - Triangles (B) and (C) have the same vertices " << std::endl;
    }

    bool bIntersectAB = CGAL::do_intersect(TriangleA,TriangleB);
    bool bIntersectAC = CGAL::do_intersect(TriangleA,TriangleC);

    bool bIntersectInexactAB = CGAL::intersection(TriangleA,TriangleB);
    bool bIntersectInexactAC = CGAL::intersection(TriangleA,TriangleC);

    if(bIntersectAB)
    {
        std::cout << " --> A and B are intersecting (exact) ..." << std::endl;
    }

    if(bIntersectAC)
    {
        std::cout << " --> A and C are intersecting (exact) ..." << std::endl;
    }

    if(bIntersectInexactAB)
    {
        std::cout << " --> A and B are intersecting (inexact) ..." << std::endl;
    }

    if(bIntersectInexactAC)
    {
        std::cout << " --> A and C are intersecting (inexact) ..." << std::endl;
    }

    return 0;
}

Here's the output ...

   - Tried to intersect: 
   - Triangle (A)  : (2 2 0.9423616295572568) (0.9685134704003172 2 0.9678422992674797) (2 1.124710354419025 1.068692504586136) 
   - Triangle (B)  : (2.5 2.5 1.442361629557257) (1.588259113885977 2.5 0.5) (2.5 1.624710354419025 1.568692504586136) 
   - Triangle (C)  : (2.5 1.624710354419025 1.568692504586136) (1.588259113885977 2.5 0.5) (2.5 2.5 1.442361629557257) 
   - Triangles (B) and (C) have the same vertices 
 --> A and C are intersecting (inexact) ...

... and a figure with the two triangles (A: vertices 1, 2, 3 ; B: vertices 11,12,13) and the "intersection" (segment 21 - 22), found using a similar version of this program.

What could be wrong? I'm using CGAL 4.6.1 on OS X 10.10.5 (Yosemite). Thanks in advance!

Answer 1

I've also sent this question to CGAL's mailing list, and the developers answered that this behaviour is not a bug, although it is unfortunate. intersection is a generic function, implemented the same way for all CGAL kernels, and it uses one step that is not always handled correctly by inexact kernels - hence the intersection error. According to this thread at CGAL's GitHub page,

In order to keep using a kernel with inexact constructions, I usually advice to first call the do_intersect predicate and then call the intersection function using EPECK on primitives converted on the fly using CGAL::Cartesian_converter. You'll have to convert the output using another CGAL::Cartesian_converter. The call to do_intersect is not mandatory, it usually depends on your setting.