gsCurveCurveIntersection.h

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#pragma once
 
#include <gsCore/gsLinearAlgebra.h>
 
namespace gismo
{
 
namespace internal
{
 
#define EPSILON_CCI ( 100*std::numeric_limits<T>::epsilon() )
 
template<class T=real_t>
struct gsPoint2d
{
    gsPoint2d(const T xCoord, const T yCoord)
    {
        pt.x() = xCoord;
        pt.y() = yCoord;
    }
 
    inline T x() const { return pt.x();}
    inline T y() const { return pt.y();}
 
    gsPoint<2,T> pt;
};
 
// To find orientation of ordered triplet (p, q, r).
// The function returns following values
// 0 --> p, q and r are collinear
// 1 --> Clockwise
// 2 --> Counterclockwise
template<class T>
short_t orientationxx(const gsPoint2d<T> &p, const gsPoint2d<T> &q, const gsPoint2d<T> &r)
{
    // See https://www.geeksforgeeks.org/orientation-3-ordered-points/
    // for details of below formula.
    T val = (q.y() - p.y()) * (r.x() - q.x()) - (q.x() - p.x()) * (r.y() - q.y());
 
    // Caution: numerically val is almost never exactly zero
    if (val == 0) return 0;  // collinear
 
    return (val > 0) ? 1 : 2; // clock or counterclock wise
}
 
// Given three collinear points p, q, r, the function checks if
// point q lies on the line segment 'pr'
template<class T>
bool onSegment(const gsPoint2d<T> &p, const gsPoint2d<T> &q, const gsPoint2d<T> &r)
{
    if (q.x() <= math::max(p.x(), r.x()) && q.x() >= math::min(p.x(), r.x()) &&
        q.y() <= math::max(p.y(), r.y()) && q.y() >= math::min(p.y(), r.y()))
        return true;
 
    return false;
}
 
// The main function that returns true if line segment 'p1q1'
// and 'p2q2' intersect.
template<class T>
bool doIntersect(const gsPoint2d<T> &p1, const gsPoint2d<T> &q1,
                 const gsPoint2d<T> &p2, const gsPoint2d<T> &q2)
{
    // Find the four orientations needed for general and special cases
    short_t o1 = orientationxx(p1, q1, p2);
    short_t o2 = orientationxx(p1, q1, q2);
    short_t o3 = orientationxx(p2, q2, p1);
    short_t o4 = orientationxx(p2, q2, q1);
 
    // General case
    if (o1 != o2 && o3 != o4)
        return true;
 
    // Special Cases
    // p1, q1 and p2 are collinear and p2 lies on segment p1q1
    if (o1 == 0 && onSegment(p1, p2, q1)) return true;
 
    // p1, q1 and q2 are collinear and q2 lies on segment p1q1
    if (o2 == 0 && onSegment(p1, q2, q1)) return true;
 
    // p2, q2 and p1 are collinear and p1 lies on segment p2q2
    if (o3 == 0 && onSegment(p2, p1, q2)) return true;
 
    // p2, q2 and q1 are collinear and q1 lies on segment p2q2
    if (o4 == 0 && onSegment(p2, q1, q2)) return true;
 
    return false; // Doesn't fall in any of the above cases
}
 
template<class T=real_t>
struct gsCurveIntersectionResult
{
  // Explicit constructor for direct initialization
    explicit gsCurveIntersectionResult( const T paramOnCurve1,
                                        const T paramOnCurve2,
                                        const gsVector<T> & pt)
    :
    m_paramOnCurve1(paramOnCurve1),
    m_paramOnCurve2(paramOnCurve2),
    m_point(pt)
    {}
 
    // Equality operators
    bool operator==(const gsCurveIntersectionResult& other) const
    {
    return  m_paramOnCurve1 == other.m_paramOnCurve1 &&
            m_paramOnCurve2 == other.m_paramOnCurve2 &&
            m_point == other.m_point;
    }
 
    bool operator!=(const gsCurveIntersectionResult& other) const
    {
        return !(*this == other);
    }
 
    T getParamOnCurve1() const { return m_paramOnCurve1; }
    T getParamOnCurve2() const { return m_paramOnCurve2; }
    gsVector<T> getPoint() const { return m_point; }
 
private:
    T m_paramOnCurve1;
    T m_paramOnCurve2;
    gsVector<T> m_point;
};
 
template<class T=real_t>
class gsInterval
{
public:
    gsInterval(const T min, const T max)
    :
    m_min(min),
    m_max(max)
    {}
 
    bool almostEqual(const T a, const T b, const T tolerance = EPSILON_CCI) const
    {
        return gsClose(a,b,tolerance);
    }
 
    bool operator==(const gsInterval<T> &other) const
    {
        return almostEqual(m_min, other.getMin()) && almostEqual(m_max, other.getMax());
    }
 
    T getMin() const { return m_min; }
    T getMax() const { return m_max; }
 
    void setMin(T mmin) { m_min = mmin; }
    void setMax(T mmax) { m_min = mmax; }
 
private:
    T m_min{0}, m_max{0};
};
 
template<class T=real_t>
 
class gsCurveBoundingBox
{
public:
    explicit gsCurveBoundingBox(const gsBSpline<T> &curve)
    :
    range(curve.domainStart(),curve.domainEnd())
    {
        low.resize( curve.geoDim() );
        high.resize(curve.geoDim() );
        for (short_t i = 0; i != curve.geoDim(); ++i)
        {
          low[i] = std::numeric_limits<T>::max();
          high[i] = std::numeric_limits<T>::lowest();
        }
 
        // compute min / max from control points
        for (index_t ipt = 0; ipt != curve.coefsSize(); ++ipt)
        {
            typename gsMatrix<T>::ConstRowXpr p = curve.coef(ipt); // Access once per iteration
            for (short_t i = 0; i != curve.geoDim(); ++i)
            {
                high[i] = math::max(high[i], p(i));
                low[i] = math::min(low[i], p(i));
            }
        }
    }
 
 
  bool intersect(const gsCurveBoundingBox<T> &other,
                 T eps = EPSILON_CCI) const
  {
    GISMO_ASSERT( this->getLow().size() == other.getLow().size(), "Size mismatch between this and other size" );
    for (index_t i = 0; i != other.getLow().size(); ++i)
    {
        if (math::max(low[i], other.low[i]) > math::min(high[i], other.high[i]) + eps)
            return false;
    }
    return true;
  }
 
    gsVector<T> getLow() const { return low; }
    gsVector<T> getHigh() const { return high; }
    gsInterval<T> getRange() const { return range; }
 
private:
    gsVector<T> low, high;
    gsInterval<T> range;
};
 
template<class T=real_t>
struct gsBoundingBoxPair
{
    gsBoundingBoxPair(const gsCurveBoundingBox<T> &i1, const gsCurveBoundingBox<T> &i2)
    :
    b1(i1),
    b2(i2)
    {}
 
    gsCurveBoundingBox<T> b1;
    gsCurveBoundingBox<T> b2;
};
 
template<class T=real_t>
std::vector<gsBoundingBoxPair<T>> getPotentialIntersectionRanges(const gsBSpline<T> &curve1,
                                                                 const gsBSpline<T> &curve2,
                                                                 const T MAX_CURVATURE)
{
    gsCurveBoundingBox<T> h1(curve1);
    gsCurveBoundingBox<T> h2(curve2);
 
    std::vector<gsBoundingBoxPair<T>> result;
 
    // Bounding boxes do not intersect. No intersection possible
    if (!h1.intersect(h2))
        return result;
 
    T crv1Curvature = curve1.pseudoCurvature();
    T crv2Curvature = curve2.pseudoCurvature();
    //  static const T MAX_CURVATURE = 1.0 + 5e-6;
 
    // Check for intersection between endpoint line segments if curves are linear enough
    if (crv1Curvature <= MAX_CURVATURE && crv2Curvature <= MAX_CURVATURE)
    {
        if ( curve1.geoDim() == 2 )
        {
            // line segment intersection check for the planar case
            gsPoint2d<T> pt1(curve1.coef(0).x(), curve1.coef(0).y());
            gsPoint2d<T> pt2(curve1.coef(curve1.coefsSize() - 1).x(), curve1.coef(curve1.coefsSize() - 1).y());
            gsPoint2d<T> pt3(curve2.coef(0).x(), curve2.coef(0).y());
            gsPoint2d<T> pt4(curve2.coef(curve2.coefsSize() - 1).x(), curve2.coef(curve2.coefsSize() - 1).y());
            if (doIntersect(pt1, pt2, pt3, pt4))
            {
                result.push_back(gsBoundingBoxPair<T>(h1, h2));
                return result;
            }
        }
        else
        {
            result.push_back(gsBoundingBoxPair<T>(h1, h2));
            return result;
        }
    }
 
    // Recursive subdividing for curves with high curvature
    if (crv1Curvature > MAX_CURVATURE && crv2Curvature > MAX_CURVATURE)
    {
        // Subdivide both curves by splitting them in the parametric center
        T curve1MidParm = 0.5 * (curve1.domainStart() + curve1.domainEnd());
        T curve2MidParm = 0.5 * (curve2.domainStart() + curve2.domainEnd());
 
        gsBSpline<T> c11, c12, c21, c22;
        curve1.splitAt(curve1MidParm, c11, c12);
        curve2.splitAt(curve2MidParm, c21, c22);
 
        std::vector<gsBoundingBoxPair<T>> result1 = getPotentialIntersectionRanges<T>(c11, c21, MAX_CURVATURE);
        std::vector<gsBoundingBoxPair<T>> result2 = getPotentialIntersectionRanges<T>(c11, c22, MAX_CURVATURE);
        std::vector<gsBoundingBoxPair<T>> result3 = getPotentialIntersectionRanges<T>(c12, c21, MAX_CURVATURE);
        std::vector<gsBoundingBoxPair<T>> result4 = getPotentialIntersectionRanges<T>(c12, c22, MAX_CURVATURE);
 
        // append all results
        result.insert(result.end(), result1.begin(), result1.end());
        result.insert(result.end(), result2.begin(), result2.end());
        result.insert(result.end(), result3.begin(), result3.end());
        result.insert(result.end(), result4.begin(), result4.end());
 
        return result;
    }
    else if (crv1Curvature <= MAX_CURVATURE && MAX_CURVATURE < crv2Curvature)
    {
        // Subdivide only curve 2
        T curve2MidParm = 0.5 * (curve2.domainStart() + curve2.domainEnd());
        gsBSpline<T> c21, c22;
        curve2.splitAt(curve2MidParm, c21, c22);
 
        std::vector<gsBoundingBoxPair<T>> result1 = getPotentialIntersectionRanges<T>(curve1, c21, MAX_CURVATURE);
        std::vector<gsBoundingBoxPair<T>> result2 = getPotentialIntersectionRanges<T>(curve1, c22, MAX_CURVATURE);
 
        result.insert(result.end(), result1.begin(), result1.end());
        result.insert(result.end(), result2.begin(), result2.end());
        return result;
    }
    else if (crv2Curvature <= MAX_CURVATURE && MAX_CURVATURE < crv1Curvature)
    {
        // Subdivide only curve 1
        T curve1MidParm = 0.5 * (curve1.domainStart() + curve1.domainEnd());
        gsBSpline<T> c11, c12;
        curve1.splitAt(curve1MidParm, c11, c12);
 
        std::vector<gsBoundingBoxPair<T>> result1 = getPotentialIntersectionRanges<T>(c11, curve2, MAX_CURVATURE);
        std::vector<gsBoundingBoxPair<T>> result2 = getPotentialIntersectionRanges<T>(c12, curve2, MAX_CURVATURE);
 
        result.insert(result.end(), result1.begin(), result1.end());
        result.insert(result.end(), result2.begin(), result2.end());
        return result;
    }
 
    return result;
}
 
template<class T=real_t>
class gsCurveCurveDistanceSystem
{
public:
    gsCurveCurveDistanceSystem(const gsBSpline<T> &crv1, const gsBSpline<T> &crv2)
    :
    m_crv1(crv1),
    m_crv2(crv2)
    {
        assert(crv1.geoDim() == crv2.geoDim());
    }
 
    void Values(const gsVector<T,2> &uv,
                gsMatrix<T> &funcVal,
                gsMatrix<T> &invJac) const
    {
        std::vector<gsMatrix<T>> crv1Der = m_crv1.evalAllDers(uv.row(0), 1);
        std::vector<gsMatrix<T>> crv2Der = m_crv2.evalAllDers(uv.row(1), 1);
 
        funcVal = crv1Der[0] - crv2Der[0];
 
        gsMatrix<T> jac(m_crv1.geoDim(), 2);
        jac.col(0) =  crv1Der[1];
        jac.col(1) = -crv2Der[1];
        invJac = jac.completeOrthogonalDecomposition().pseudoInverse();
    }
 
    T compute(gsVector<T,2> &uv, T tolerance = 1e-5, size_t maxIter = 20) const
    {
        gsMatrix<T> funVal;
        gsMatrix<T> invJac;
        Values(uv, funVal, invJac); // Assumes Values correctly inverts the Jacobian
        T currentResidual = funVal.norm();
 
        gsVector<T,2> deltaUv = invJac * funVal;
        gsVector<T,2> newUv;
        for (size_t iter = 1; iter != maxIter; ++iter)
        {
            uv -= deltaUv; // update uv
 
            // project uv into its parameter range
            newUv.at(0) = math::min(math::max(uv.at(0), m_crv1.domainStart()), m_crv1.domainEnd());
            newUv.at(1) = math::min(math::max(uv.at(1), m_crv2.domainStart()), m_crv2.domainEnd());
            // using C++17 standard
            //      newUv(0, 0) = std::clamp(newUv(0, 0), m_crv1.domainStart(), m_crv1.domainEnd());
            //      newUv(1, 0) = std::clamp(newUv(1, 0), m_crv2.domainStart(), m_crv2.domainEnd());
 
            T delta = (uv - newUv).norm();
            uv = newUv;
 
            if (delta > EPSILON_CCI)
            {
                Values(uv, funVal, invJac); // Re-evaluate funVal and invJac
                currentResidual = funVal.norm();
            break;
            }
 
            Values(uv, funVal, invJac); // Re-evaluate funVal and invJac
            currentResidual = funVal.norm();
            if (currentResidual < tolerance)
                break;
 
            deltaUv = invJac * funVal;
            if (deltaUv.norm() < 1e-3*tolerance)
                break;
 
        //      gsInfo << "iter. = " << iter << ", residual = " << currentResidual << "\n";
        }
        return currentResidual;
  }
 
private:
    const gsBSpline<T> m_crv1{}, m_crv2{};
};
 
} // namespace internal
 
} // namespace gismo