Coord.h

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// Copyright Contributors to the OpenVDB Project
// SPDX-License-Identifier: Apache-2.0

#ifndef OPENVDB_MATH_COORD_HAS_BEEN_INCLUDED
#define OPENVDB_MATH_COORD_HAS_BEEN_INCLUDED

#include <functional>// for std::hash
#include <algorithm> // for std::min(), std::max()
#include <array> // for std::array
#include <iostream>
#include <limits>
#include <openvdb/Platform.h>
#include <openvdb/util/Assert.h>
#include "Math.h"
#include "Vec3.h"

#include <tbb/blocked_range.h> // for tbb::split

namespace openvdb {
OPENVDB_USE_VERSION_NAMESPACE
namespace OPENVDB_VERSION_NAME {
namespace math {

/// @brief Signed (x, y, z) 32-bit integer coordinates
class Coord
{
public:
    using Int32 = int32_t;
    using Index32 = uint32_t;
    using Vec3i = Vec3<Int32>;
    using Vec3I = Vec3<Index32>;

    using ValueType = Int32;
    using Limits = std::numeric_limits<ValueType>;

    Coord(): mVec{{0, 0, 0}} {}
    constexpr explicit Coord(Int32 xyz): mVec{{xyz, xyz, xyz}} {}
    constexpr Coord(Int32 x, Int32 y, Int32 z): mVec{{x, y, z}} {}
    explicit Coord(const Vec3i& v): mVec{{v[0], v[1], v[2]}} {}
    explicit Coord(const Vec3I& v): mVec{{Int32(v[0]), Int32(v[1]), Int32(v[2])}} {}
    explicit Coord(const Int32* v): mVec{{v[0], v[1], v[2]}} {}

    /// @brief Return the smallest possible coordinate
    static Coord min() { return Coord(Limits::min()); }

    /// @brief Return the largest possible coordinate
    static Coord max() { return Coord(Limits::max()); }

    /// @brief Return @a xyz rounded to the closest integer coordinates
    /// (cell centered conversion).
    template<typename T> static Coord round(const Vec3<T>& xyz)
    {
        return Coord(Int32(Round(xyz[0])), Int32(Round(xyz[1])), Int32(Round(xyz[2])));
    }
    /// @brief Return the largest integer coordinates that are not greater
    /// than @a xyz (node centered conversion).
    template<typename T> static Coord floor(const Vec3<T>& xyz)
    {
        return Coord(Int32(Floor(xyz[0])), Int32(Floor(xyz[1])), Int32(Floor(xyz[2])));
    }

    /// @brief Return the largest integer coordinates that are not greater
    /// than @a xyz+1 (node centered conversion).
    template<typename T> static Coord ceil(const Vec3<T>& xyz)
    {
        return Coord(Int32(Ceil(xyz[0])), Int32(Ceil(xyz[1])), Int32(Ceil(xyz[2])));
    }

    /// @brief Reset all three coordinates with the specified arguments
    Coord& reset(Int32 x, Int32 y, Int32 z)
    {
        mVec[0] = x;
        mVec[1] = y;
        mVec[2] = z;
        return *this;
    }
    /// @brief Reset all three coordinates with the same specified argument
    Coord& reset(Int32 xyz) { return this->reset(xyz, xyz, xyz); }

    Coord& setX(Int32 x) { mVec[0] = x; return *this; }
    Coord& setY(Int32 y) { mVec[1] = y; return *this; }
    Coord& setZ(Int32 z) { mVec[2] = z; return *this; }

    Coord& offset(Int32 dx, Int32 dy, Int32 dz)
    {
        mVec[0] += dx;
        mVec[1] += dy;
        mVec[2] += dz;
        return *this;
    }
    Coord& offset(Int32 n) { return this->offset(n, n, n); }
    Coord offsetBy(Int32 dx, Int32 dy, Int32 dz) const
    {
        return Coord(mVec[0] + dx, mVec[1] + dy, mVec[2] + dz);
    }
    Coord offsetBy(Int32 n) const { return offsetBy(n, n, n); }

    Coord& operator+=(const Coord& rhs)
    {
        mVec[0] += rhs[0];
        mVec[1] += rhs[1];
        mVec[2] += rhs[2];
        return *this;
    }
    Coord& operator-=(const Coord& rhs)
    {
        mVec[0] -= rhs[0];
        mVec[1] -= rhs[1];
        mVec[2] -= rhs[2];
        return *this;
    }
    Coord operator+(const Coord& rhs) const
    {
        return Coord(mVec[0] + rhs[0], mVec[1] + rhs[1], mVec[2] + rhs[2]);
    }
    Coord operator-(const Coord& rhs) const
    {
        return Coord(mVec[0] - rhs[0], mVec[1] - rhs[1], mVec[2] - rhs[2]);
    }
    Coord operator-() const { return Coord(-mVec[0], -mVec[1], -mVec[2]); }

    Coord  operator>> (size_t n) const { return Coord(mVec[0]>>n, mVec[1]>>n, mVec[2]>>n); }
    Coord  operator<< (size_t n) const { return Coord(mVec[0]<<n, mVec[1]<<n, mVec[2]<<n); }
    Coord& operator<<=(size_t n) { mVec[0]<<=n; mVec[1]<<=n; mVec[2]<<=n; return *this; }
    Coord& operator>>=(size_t n) { mVec[0]>>=n; mVec[1]>>=n; mVec[2]>>=n; return *this; }
    Coord  operator&  (Int32 n) const { return Coord(mVec[0] & n, mVec[1] & n, mVec[2] & n); }
    Coord  operator|  (Int32 n) const { return Coord(mVec[0] | n, mVec[1] | n, mVec[2] | n); }
    Coord& operator&= (Int32 n) { mVec[0]&=n; mVec[1]&=n; mVec[2]&=n; return *this; }
    Coord& operator|= (Int32 n) { mVec[0]|=n; mVec[1]|=n; mVec[2]|=n; return *this; }

    Int32 x() const { return mVec[0]; }
    Int32 y() const { return mVec[1]; }
    Int32 z() const { return mVec[2]; }
    Int32 operator[](size_t i) const { OPENVDB_ASSERT(i < 3); return mVec[i]; }
    Int32& x() { return mVec[0]; }
    Int32& y() { return mVec[1]; }
    Int32& z() { return mVec[2]; }
    Int32& operator[](size_t i) { OPENVDB_ASSERT(i < 3); return mVec[i]; }

    const Int32* data() const { return mVec.data(); }
    Int32* data() { return mVec.data(); }
    const Int32* asPointer() const { return mVec.data(); }
    Int32* asPointer() { return mVec.data(); }
    Vec3d asVec3d() const { return Vec3d(double(mVec[0]), double(mVec[1]), double(mVec[2])); }
    Vec3s asVec3s() const { return Vec3s(float(mVec[0]), float(mVec[1]), float(mVec[2])); }
    Vec3i asVec3i() const { return Vec3i(mVec.data()); }
    Vec3I asVec3I() const { return Vec3I(Index32(mVec[0]), Index32(mVec[1]), Index32(mVec[2])); }
    void asXYZ(Int32& x, Int32& y, Int32& z) const { x = mVec[0]; y = mVec[1]; z = mVec[2]; }

    bool operator==(const Coord& rhs) const
    {
        return (mVec[0] == rhs.mVec[0] && mVec[1] == rhs.mVec[1] && mVec[2] == rhs.mVec[2]);
    }
    bool operator!=(const Coord& rhs) const { return !(*this == rhs); }

    /// Lexicographic less than
    bool operator<(const Coord& rhs) const
    {
        return this->x() < rhs.x() ? true : this->x() > rhs.x() ? false
             : this->y() < rhs.y() ? true : this->y() > rhs.y() ? false
             : this->z() < rhs.z() ? true : false;
    }
    /// Lexicographic less than or equal to
    bool operator<=(const Coord& rhs) const
    {
        return this->x() < rhs.x() ? true : this->x() > rhs.x() ? false
             : this->y() < rhs.y() ? true : this->y() > rhs.y() ? false
             : this->z() <=rhs.z() ? true : false;
    }
    /// Lexicographic greater than
    bool operator>(const Coord& rhs) const { return !(*this <= rhs); }
    /// Lexicographic greater than or equal to
    bool operator>=(const Coord& rhs) const { return !(*this < rhs); }

    /// Perform a component-wise minimum with the other Coord.
    void minComponent(const Coord& other)
    {
        mVec[0] = std::min(mVec[0], other.mVec[0]);
        mVec[1] = std::min(mVec[1], other.mVec[1]);
        mVec[2] = std::min(mVec[2], other.mVec[2]);
    }

    /// Perform a component-wise maximum with the other Coord.
    void maxComponent(const Coord& other)
    {
        mVec[0] = std::max(mVec[0], other.mVec[0]);
        mVec[1] = std::max(mVec[1], other.mVec[1]);
        mVec[2] = std::max(mVec[2], other.mVec[2]);
    }

    /// Return the component-wise minimum of the two Coords.
    static inline Coord minComponent(const Coord& lhs, const Coord& rhs)
    {
        return Coord(std::min(lhs.x(), rhs.x()),
                     std::min(lhs.y(), rhs.y()),
                     std::min(lhs.z(), rhs.z()));
    }

    /// Return the component-wise maximum of the two Coords.
    static inline Coord maxComponent(const Coord& lhs, const Coord& rhs)
    {
        return Coord(std::max(lhs.x(), rhs.x()),
                     std::max(lhs.y(), rhs.y()),
                     std::max(lhs.z(), rhs.z()));
    }

    /// Return true if any of the components of @a a are smaller than the
    /// corresponding components of @a b.
    static inline bool lessThan(const Coord& a, const Coord& b)
    {
        return (a[0] < b[0] || a[1] < b[1] || a[2] < b[2]);
    }

    /// @brief Return the index (0, 1 or 2) with the smallest value.
    size_t minIndex() const { return MinIndex(mVec); }

    /// @brief Return the index (0, 1 or 2) with the largest value.
    size_t maxIndex() const { return MaxIndex(mVec); }

    void read(std::istream& is) { is.read(reinterpret_cast<char*>(mVec.data()), sizeof(mVec)); }
    void write(std::ostream& os) const
    {
        os.write(reinterpret_cast<const char*>(mVec.data()), sizeof(mVec));
    }

    /// @brief Return a hash value for this coordinate
    /// @note Log2N is the binary logarithm of the hash table size.
    /// @details The hash function is originally taken from the SIGGRAPH paper:
    /// "VDB: High-resolution sparse volumes with dynamic topology"
    /// and the prime numbers are modified based on the ACM Transactions on Graphics paper:
    /// "Real-time 3D reconstruction at scale using voxel hashing"
    template<int Log2N = 20>
    size_t hash() const
    {
        const uint32_t* vec = reinterpret_cast<const uint32_t*>(mVec.data());
        return ((1<<Log2N)-1) & (vec[0]*73856093 ^ vec[1]*19349669 ^ vec[2]*83492791);
    }

private:
    std::array<Int32, 3> mVec;
}; // class Coord


////////////////////////////////////////


/// @brief Axis-aligned bounding box of signed integer coordinates
/// @note The range of the integer coordinates, [min, max], is inclusive.
/// Thus, a bounding box with min = max is not empty but rather encloses
/// a single coordinate.
class CoordBBox
{
public:
    using Index64 = uint64_t;
    using ValueType = Coord::ValueType;

    /// @brief Iterator over the Coord domain covered by a CoordBBox
    /// @note If ZYXOrder is @c true, @e z is the fastest-moving coordinate,
    /// otherwise the traversal is in XYZ order (i.e., @e x is fastest-moving).
    template<bool ZYXOrder>
    class Iterator
    {
    public:
        /// @brief C-tor from a bounding box
        Iterator(const CoordBBox& b): mPos(b.min()), mMin(b.min()), mMax(b.max()) {}
        /// @brief Increment the iterator to point to the next coordinate.
        /// @details Iteration stops one past the maximum coordinate
        /// along the axis determined by the template parameter.
        Iterator& operator++() { ZYXOrder ? next<2,1,0>() : next<0,1,2>(); return *this; }
        /// @brief Return @c true if the iterator still points to a valid coordinate.
        operator bool() const { return ZYXOrder ? (mPos[0] <= mMax[0]) : (mPos[2] <= mMax[2]); }
        /// @brief Return a const reference to the coordinate currently pointed to.
        const Coord& operator*() const { return mPos; }
        /// Return @c true if this iterator and the given iterator point to the same coordinate.
        bool operator==(const Iterator& other) const
        {
            return ((mPos == other.mPos) && (mMin == other.mMin) && (mMax == other.mMax));
        }
        /// Return @c true if this iterator and the given iterator point to different coordinates.
        bool operator!=(const Iterator& other) const { return !(*this == other); }
    private:
        template<size_t a, size_t b, size_t c>
        void next()
        {
            if (mPos[a] < mMax[a]) { ++mPos[a]; } // this is the most common case
            else if (mPos[b] < mMax[b]) { mPos[a] = mMin[a]; ++mPos[b]; }
            else if (mPos[c] <= mMax[c]) { mPos[a] = mMin[a]; mPos[b] = mMin[b]; ++mPos[c]; }
        }
        Coord mPos, mMin, mMax;
        friend class CoordBBox; // for CoordBBox::end()
    };// CoordBBox::Iterator

    using ZYXIterator = Iterator</*ZYX=*/true>;
    using XYZIterator = Iterator</*ZYX=*/false>;

    /// @brief The default constructor produces an empty bounding box.
    CoordBBox(): mMin(Coord::max()), mMax(Coord::min()) {}
    /// @brief Construct a bounding box with the given @a min and @a max bounds.
    CoordBBox(const Coord& min, const Coord& max): mMin(min), mMax(max) {}
    /// @brief Construct from individual components of the min and max bounds.
    CoordBBox(ValueType xMin, ValueType yMin, ValueType zMin,
              ValueType xMax, ValueType yMax, ValueType zMax)
        : mMin(xMin, yMin, zMin), mMax(xMax, yMax, zMax)
    {
    }
    /// @brief Splitting constructor for use in TBB ranges
    /// @note The other bounding box is assumed to be divisible.
    CoordBBox(CoordBBox& other, const tbb::split&): mMin(other.mMin), mMax(other.mMax)
    {
        OPENVDB_ASSERT(this->is_divisible());
        const size_t n = this->maxExtent();
        mMax[n] = (mMin[n] + mMax[n]) >> 1;
        other.mMin[n] = mMax[n] + 1;
    }

    static CoordBBox createCube(const Coord& min, ValueType dim)
    {
        return CoordBBox(min, min.offsetBy(dim - 1));
    }

    /// Return an "infinite" bounding box, as defined by the Coord value range.
    static CoordBBox inf() { return CoordBBox(Coord::min(), Coord::max()); }

    const Coord& min() const { return mMin; }
    const Coord& max() const { return mMax; }

    Coord& min() { return mMin; }
    Coord& max() { return mMax; }

    void reset() { mMin = Coord::max(); mMax = Coord::min(); }
    void reset(const Coord& min, const Coord& max) { mMin = min; mMax = max; }
    void resetToCube(const Coord& min, ValueType dim) { mMin = min; mMax = min.offsetBy(dim - 1); }

    /// @brief Return the minimum coordinate.
    /// @note The start coordinate is inclusive.
    Coord getStart() const { return mMin; }
    /// @brief Return the maximum coordinate plus one.
    /// @note This end coordinate is exclusive.
    Coord getEnd() const { return mMax.offsetBy(1); }

    /// @brief Return a ZYX-order iterator that points to the minimum coordinate.
    ZYXIterator begin() const { return ZYXIterator{*this}; }
    /// @brief Return a ZYX-order iterator that points to the minimum coordinate.
    ZYXIterator beginZYX() const { return ZYXIterator{*this}; }
    /// @brief Return an XYZ-order iterator that points to the minimum coordinate.
    XYZIterator beginXYZ() const { return XYZIterator{*this}; }

    /// @brief Return a ZYX-order iterator that points past the maximum coordinate.
    ZYXIterator end() const { ZYXIterator it{*this}; it.mPos[0] = mMax[0] + 1; return it; }
    /// @brief Return a ZYX-order iterator that points past the maximum coordinate.
    ZYXIterator endZYX() const { return end(); }
    /// @brief Return an XYZ-order iterator that points past the maximum coordinate.
    XYZIterator endXYZ() const { XYZIterator it{*this}; it.mPos[2] = mMax[2] + 1; return it; }

    bool operator==(const CoordBBox& rhs) const { return mMin == rhs.mMin && mMax == rhs.mMax; }
    bool operator!=(const CoordBBox& rhs) const { return !(*this == rhs); }

    /// @brief Return @c true if this bounding box is empty (i.e., encloses no coordinates).
    bool empty() const
    {
#if defined(__GNUC__) && !defined(__INTEL_COMPILER)
  #pragma GCC diagnostic push
  #pragma GCC diagnostic ignored "-Wstrict-overflow"
#endif
        return (mMin[0] > mMax[0] || mMin[1] > mMax[1] || mMin[2] > mMax[2]);
#if defined(__GNUC__) && !defined(__INTEL_COMPILER)
  #pragma GCC diagnostic pop
#endif
    }
    /// @brief Return @c true if this bounding box is nonempty
    /// (i.e., encloses at least one coordinate).
    operator bool() const { return !this->empty(); }
    /// @brief Return @c true if this bounding box is nonempty
    /// (i.e., encloses at least one coordinate).
    bool hasVolume() const { return !this->empty(); }

    /// @brief Return the floating-point position of the center of this bounding box.
    Vec3d getCenter() const { return 0.5 * Vec3d((mMin + mMax).asPointer()); }

    /// @brief Return the dimensions of the coordinates spanned by this bounding box.
    /// @note Since coordinates are inclusive, a bounding box with min = max
    /// has dimensions of (1, 1, 1).
    Coord dim() const { return empty() ? Coord(0) : (mMax.offsetBy(1) - mMin); }
    /// @todo deprecate - use dim instead
    Coord extents() const { return this->dim(); }
    /// @brief Return the integer volume of coordinates spanned by this bounding box.
    /// @note Since coordinates are inclusive, a bounding box with min = max has volume one.
    Index64 volume() const
    {
        const Coord d = this->dim();
        return Index64(d[0]) * Index64(d[1]) * Index64(d[2]);
    }
    /// @brief Return @c true if this bounding box can be subdivided [mainly for use by TBB].
    bool is_divisible() const { return mMin[0]<mMax[0] && mMin[1]<mMax[1] && mMin[2]<mMax[2]; }

    /// @brief Return the index (0, 1 or 2) of the shortest axis.
    size_t minExtent() const { return this->dim().minIndex(); }

    /// @brief Return the index (0, 1 or 2) of the longest axis.
    size_t maxExtent() const { return this->dim().maxIndex(); }

    /// @brief Return @c true if point (x, y, z) is inside this bounding box.
    bool isInside(const Coord& xyz) const
    {
        return !(Coord::lessThan(xyz,mMin) || Coord::lessThan(mMax,xyz));
    }

    /// @brief Return @c true if the given bounding box is inside this bounding box.
    bool isInside(const CoordBBox& b) const
    {
        return !(Coord::lessThan(b.mMin,mMin) || Coord::lessThan(mMax,b.mMax));
    }

    /// @brief Return @c true if the given bounding box overlaps with this bounding box.
    bool hasOverlap(const CoordBBox& b) const
    {
        return !(Coord::lessThan(mMax,b.mMin) || Coord::lessThan(b.mMax,mMin));
    }

    /// @brief Pad this bounding box with the specified padding.
    void expand(ValueType padding)
    {
        mMin.offset(-padding);
        mMax.offset( padding);
    }

    /// @brief Return a new instance that is expanded by the specified padding.
    CoordBBox expandBy(ValueType padding) const
    {
        return CoordBBox(mMin.offsetBy(-padding),mMax.offsetBy(padding));
    }

    /// @brief Expand this bounding box to enclose point (x, y, z).
    void expand(const Coord& xyz)
    {
        mMin.minComponent(xyz);
        mMax.maxComponent(xyz);
    }

    /// @brief Union this bounding box with the given bounding box.
    void expand(const CoordBBox& bbox)
    {
          mMin.minComponent(bbox.min());
          mMax.maxComponent(bbox.max());
    }
    /// @brief Intersect this bounding box with the given bounding box.
    void intersect(const CoordBBox& bbox)
    {
        mMin.maxComponent(bbox.min());
        mMax.minComponent(bbox.max());
    }
    /// @brief Union this bounding box with the cubical bounding box
    /// of the given size and with the given minimum coordinates.
    void expand(const Coord& min, Coord::ValueType dim)
    {
        mMin.minComponent(min);
        mMax.maxComponent(min.offsetBy(dim-1));
    }
    /// @brief Translate this bounding box by
    /// (<i>t<sub>x</sub></i>, <i>t<sub>y</sub></i>, <i>t<sub>z</sub></i>).
    void translate(const Coord& t) { mMin += t; mMax += t; }

    /// @brief Move this bounding box to the specified min
    void moveMin(const Coord& min) { mMax += min - mMin; mMin = min; }

    /// @brief Move this bounding box to the specified max
    void moveMax(const Coord& max) { mMin += max - mMax; mMax = max; }

    /// @brief Populates an array with the eight corner points of this bounding box.
    /// @details The ordering of the corner points is lexicographic.
    /// @warning It is assumed that the pointer can be incremented at
    /// least seven times, i.e. has storage for eight Coord elements!
    void getCornerPoints(Coord *p) const
    {
        OPENVDB_ASSERT(p != nullptr);
        p->reset(mMin.x(), mMin.y(), mMin.z()); ++p;
        p->reset(mMin.x(), mMin.y(), mMax.z()); ++p;
        p->reset(mMin.x(), mMax.y(), mMin.z()); ++p;
        p->reset(mMin.x(), mMax.y(), mMax.z()); ++p;
        p->reset(mMax.x(), mMin.y(), mMin.z()); ++p;
        p->reset(mMax.x(), mMin.y(), mMax.z()); ++p;
        p->reset(mMax.x(), mMax.y(), mMin.z()); ++p;
        p->reset(mMax.x(), mMax.y(), mMax.z());
    }

    //@{
    /// @brief Bit-wise operations performed on both the min and max members
    CoordBBox  operator>> (size_t n) const { return CoordBBox(mMin>>n, mMax>>n); }
    CoordBBox  operator<< (size_t n) const { return CoordBBox(mMin<<n, mMax<<n); }
    CoordBBox& operator<<=(size_t n) { mMin <<= n; mMax <<= n; return *this; }
    CoordBBox& operator>>=(size_t n) { mMin >>= n; mMax >>= n; return *this; }
    CoordBBox  operator&  (Coord::Int32 n) const { return CoordBBox(mMin & n, mMax & n); }
    CoordBBox  operator|  (Coord::Int32 n) const { return CoordBBox(mMin | n, mMax | n); }
    CoordBBox& operator&= (Coord::Int32 n) { mMin &= n; mMax &= n; return *this; }
    CoordBBox& operator|= (Coord::Int32 n) { mMin |= n; mMax |= n; return *this; }
    //@}

    /// @brief Unserialize this bounding box from the given stream.
    void read(std::istream& is) { mMin.read(is); mMax.read(is); }
    /// @brief Serialize this bounding box to the given stream.
    void write(std::ostream& os) const { mMin.write(os); mMax.write(os); }

private:
    Coord mMin, mMax;
}; // class CoordBBox


////////////////////////////////////////


inline std::ostream& operator<<(std::ostream& os, const Coord& xyz)
{
    os << xyz.asVec3i(); return os;
}


inline Coord
Abs(const Coord& xyz)
{
    return Coord(Abs(xyz[0]), Abs(xyz[1]), Abs(xyz[2]));
}


//@{
/// Allow a Coord to be added to or subtracted from a Vec3.
template<typename T>
inline Vec3<typename promote<T, typename Coord::ValueType>::type>
operator+(const Vec3<T>& v0, const Coord& v1)
{
    Vec3<typename promote<T, typename Coord::ValueType>::type> result(v0);
    result[0] += v1[0];
    result[1] += v1[1];
    result[2] += v1[2];
    return result;
}

template<typename T>
inline Vec3<typename promote<T, typename Coord::ValueType>::type>
operator+(const Coord& v1, const Vec3<T>& v0)
{
    Vec3<typename promote<T, typename Coord::ValueType>::type> result(v0);
    result[0] += v1[0];
    result[1] += v1[1];
    result[2] += v1[2];
    return result;
}
//@}


//@{
/// Allow a Coord to be subtracted from a Vec3.
template <typename T>
inline Vec3<typename promote<T, Coord::ValueType>::type>
operator-(const Vec3<T>& v0, const Coord& v1)
{
    Vec3<typename promote<T, Coord::ValueType>::type> result(v0);
    result[0] -= v1[0];
    result[1] -= v1[1];
    result[2] -= v1[2];
    return result;
}

template <typename T>
inline Vec3<typename promote<T, Coord::ValueType>::type>
operator-(const Coord& v1, const Vec3<T>& v0)
{
    Vec3<typename promote<T, Coord::ValueType>::type> result(v0);
    result[0] -= v1[0];
    result[1] -= v1[1];
    result[2] -= v1[2];
    return -result;
}
//@}

inline std::ostream&
operator<<(std::ostream& os, const CoordBBox& b)
{
    os << b.min() << " -> " << b.max();
    return os;
}

} // namespace math
} // namespace OPENVDB_VERSION_NAME
} // namespace openvdb

////////////////////////////////////////

// template specialization of std::hash with Coord, which
// allows for Coord to be used as the key in std::unordered_map
namespace std {// injected in namespace std

template<>
struct hash<openvdb::math::Coord>
{
    using Coord = openvdb::math::Coord;
    using argument_type = Coord;
    using result_type = std::size_t;
    std::size_t operator()(const Coord& ijk) const noexcept { return ijk.Coord::hash<>(); }
};// std::hash<openvdb::math::Coord>

}// namespace std

#endif // OPENVDB_MATH_COORD_HAS_BEEN_INCLUDED