UT_Vector3.h

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/*
 * PROPRIETARY INFORMATION.  This software is proprietary to
 * Side Effects Software Inc., and is not to be reproduced,
 * transmitted, or disclosed in any way without written permission.
 *
 * NAME:        Utility Library (C++)
 *
 * COMMENTS:
 *      This class handles fpreal vectors of dimension 3.
 *
 * WARNING:
 *      This class should NOT contain any virtual methods, nor should it
 *      define more member data. The size of UT_Vector3 must always be
 *      12 bytes (3 floats).
 */

#pragma once

#ifndef __UT_Vector3_h__
#define __UT_Vector3_h__

#include "UT_API.h"
#include "UT_Assert.h"
#include "UT_FixedVectorTraits.h"
#include "UT_Storage.h"
#include "UT_FixedArrayMath.h"
#include "UT_VectorTypes.h"
#include "UT_Vector2.h"
#include <SYS/SYS_Deprecated.h>
#include <SYS/SYS_Inline.h>
#include <SYS/SYS_Math.h>
#include <SYS/SYS_TypeTraits.h>
#include <iosfwd>
#include <limits>

class UT_IStream;
class UT_JSONWriter;
class UT_JSONValue;
class UT_JSONParser;

// Free floating functions:

// Right-multiply operators (M*v) have been removed. They had previously
// been defined to return v*M, which was too counterintuitive. Once HDK
// etc. users have a chance to update their code (post 7.0) we could
// reintroduce a right-multiply operator that does a colVecMult.

template <typename T, typename S>
inline UT_Vector3T<T> operator*(const UT_Vector3T<T> &v, const UT_Matrix3T<S>  &m);
template <typename T, typename S>
inline UT_Vector3T<T> operator*(const UT_Vector3T<T> &v, const UT_Matrix4T<S>  &m);
template <typename T>
constexpr UT_Vector3T<T> operator+(const UT_Vector3T<T> &a, const UT_Vector3T<T> &b) noexcept;
template <typename T>
constexpr UT_Vector3T<T> operator-(const UT_Vector3T<T> &a, const UT_Vector3T<T> &b) noexcept;
template <typename T, typename S>
constexpr UT_Vector3T<T> operator+(const UT_Vector3T<T> &v, S scalar) noexcept;
template <typename T, typename S>
constexpr UT_Vector3T<T> operator-(const UT_Vector3T<T> &v, S scalar) noexcept;
template <typename T, typename S>
constexpr UT_Vector3T<T> operator*(const UT_Vector3T<T> &v, S scalar) noexcept;
template <typename T, typename S>
constexpr UT_Vector3T<T> operator/(const UT_Vector3T<T> &v, S scalar) noexcept;
template <typename T, typename S>
constexpr UT_Vector3T<T> operator+(S scalar, const UT_Vector3T<T> &v) noexcept;
template <typename T, typename S>
constexpr UT_Vector3T<T> operator-(S scalar, const UT_Vector3T<T> &v) noexcept;
template <typename T, typename S>
constexpr UT_Vector3T<T> operator*(S scalar, const UT_Vector3T<T> &v) noexcept;
template <typename T, typename S>
constexpr UT_Vector3T<T> operator/(S scalar, const UT_Vector3T<T> &v) noexcept;

/// The dot and cross products between two vectors (see operator*() too)
// @{
template <typename T>
constexpr auto dot( const UT_Vector3T< T >& a, const UT_Vector3T< T >& b ) noexcept;

template <typename T>
constexpr UT_Vector3T<T> cross(const UT_Vector3T<T> &a, const UT_Vector3T<T> &b) noexcept;
// @}

/// The angle between two vectors in radians
// @{
template <typename T>
inline fpreal64 UTangleBetween(const UT_Vector3T<T> &v1, const UT_Vector3T<T> &v2);
// @}

/// Componentwise min and maximum
template <typename T>
inline UT_Vector3T<T> SYSmin(const UT_Vector3T<T> &v1, const UT_Vector3T<T> &v2);
template <typename T>
inline UT_Vector3T<T> SYSmax(const UT_Vector3T<T> &v1, const UT_Vector3T<T> &v2);
/// Componentwise equality.
template <typename T, typename S>
inline bool             SYSisEqual(const UT_Vector3T<T> &a, const UT_Vector3T<T> &b, S tol = SYS_FTOLERANCE);

/// Componentwise integer test
template <typename T>
inline bool SYSisInteger(const UT_Vector3T<T> &v1) 
{ return SYSisInteger(v1.x()) && SYSisInteger(v1.y()) && SYSisInteger(v1.z()); }

/// Componentwise linear interpolation
template <typename T,typename S>
inline UT_Vector3T<T> SYSlerp(const UT_Vector3T<T> &v1, const UT_Vector3T<T> &v2, S t);

/// Componentwise inverse linear interpolation
template <typename T>
inline UT_Vector3T<T> SYSinvlerp(const UT_Vector3T<T> &a, const UT_Vector3T<T> &v1, const UT_Vector3T<T> &v2);

/// Bilinear interpolation
template <typename T,typename S>
inline UT_Vector3T<T> SYSbilerp(const UT_Vector3T<T> &u0v0, const UT_Vector3T<T> &u1v0,
                                  const UT_Vector3T<T> &u0v1, const UT_Vector3T<T> &u1v1,
                                  S u, S v)
{ return SYSlerp(SYSlerp(u0v0, u0v1, v), SYSlerp(u1v0, u1v1, v), u); }

/// Barycentric interpolation
template <typename T, typename S>
inline UT_Vector3T<T> SYSbarycentric(const UT_Vector3T<T> &v0,
        const UT_Vector3T<T> &v1, const UT_Vector3T<T> &v2, S u, S v)
{ return v0 * (1 - u - v) + v1 * u + v2 *v; }


/// Trilinear hat function over kernel widths in s.
template <typename T>
inline T SYStrihat(const UT_Vector3T<T> &v, const UT_Vector3T<T> &s);

/// Gradient of trilinear hat function over kernel widths in s.
template <typename T>
inline UT_Vector3T<T> SYStrihatgrad(const UT_Vector3T<T> &v, const UT_Vector3T<T> &s);

/// The orthogonal projection of a vector u onto a vector v
template <typename T>
inline UT_Vector3T<T> project(const UT_Vector3T<T> &u, const UT_Vector3T<T> &v);

// TODO: make UT_Vector4 versions of these:

/// Compute the distance between two points
template <typename T>
inline T distance3d(const UT_Vector3T<T> &p1, const UT_Vector3T<T> &p2);
/// Compute the distance squared
template <typename T>
inline T distance2(const UT_Vector3T<T> &p1, const UT_Vector3T<T> &p2);
template <typename T>
inline T segmentPointDist2(const UT_Vector3T<T> &pos,
                                     const UT_Vector3T<T> &pt1,
                                     const UT_Vector3T<T> &pt2 );
/// Compute the squared minimum distance between a point and a semi-infinite
/// ray. Direction vector need not be normalized.
template <typename T>
inline T rayPointDist2(const UT_Vector3T<T> &pos,
                       const UT_Vector3T<T> &orig,
                       const UT_Vector3T<T> &dir);
/// Computes the squared minimum distance between a point and an infinite line.
/// Direction vector need not be normalized.
template <typename T>
inline T linePointDist2(const UT_Vector3T<T> &pos,
                        const UT_Vector3T<T> &orig,
                        const UT_Vector3T<T> &dir);

/// Intersect the lines p1 + v1 * t1 and p2 + v2 * t2.
/// t1 and t2 are set so that the lines intersect when
/// projected to the plane defined by the two lines.
/// This function returns a value which indicates how close
/// to being parallel the lines are. Closer to zero means
/// more parallel. This is done so that the user of this
/// function can decide what epsilon they want to use.
template <typename T>
UT_API double           intersectLines(const UT_Vector3T<T> &p1,
                                       const UT_Vector3T<T> &v1,
                                       const UT_Vector3T<T> &p2,
                                       const UT_Vector3T<T> &v2,
                                       T &t1, T &t2);

/// Returns true if the segments from p0 to p1 and from a to b intersect, and
/// t will contain the parametric value of the intersection on the segment a-b.
/// Otherwise returns false.  Parallel segments will return false.  T is
/// close to being between 0.0 and 1.0 if this function returns true.
/// NOTE: Does not test the actual distance of the projected points
/// on each segment, merely that they are parmetrically within the
/// segment!
template <typename T>
UT_API bool             intersectSegments(const UT_Vector3T<T> &p0,
                                          const UT_Vector3T<T> &p1,
                                          const UT_Vector3T<T> &a,
                                          const UT_Vector3T<T> &b, T &t);

/// Returns the U coordinates of the closest points on each of the two 
/// parallel line segments
template <typename T>
UT_API UT_Vector2T<T>   segmentClosestParallel(const UT_Vector3T<T> &p0,
                                               const UT_Vector3T<T> &p1,
                                               const UT_Vector3T<T> &a,
                                               const UT_Vector3T<T> &b);
/// Returns the U coordinates of the closest points on each of the two 
/// line segments
template <typename T>
UT_API UT_Vector2T<T>   segmentClosest(const UT_Vector3T<T> &p0,
                                       const UT_Vector3T<T> &p1,
                                       const UT_Vector3T<T> &a,
                                       const UT_Vector3T<T> &b);
/// Returns the U coordinate of the point on line segment p0->p1
/// that is closest to a.
template <typename T>
UT_API T segmentClosest(
    const UT_Vector3T<T> &p0,
    const UT_Vector3T<T> &p1,
    const UT_Vector3T<T> &a);

/// Returns the squared distance between two line segments: p0-p1 and a-b
template <typename T>
inline T                segmentDistance2(const UT_Vector3T<T> &p0,
                                         const UT_Vector3T<T> &p1,
                                         const UT_Vector3T<T> &a,
                                         const UT_Vector3T<T> &b);
/// Returns the distance between two line segments: p0-p1 and a-b
template <typename T>
inline T                segmentDistance(const UT_Vector3T<T> &p0,
                                        const UT_Vector3T<T> &p1,
                                        const UT_Vector3T<T> &a,
                                        const UT_Vector3T<T> &b);

/// 3D Vector class.
template <typename T>
class UT_API UT_Vector3T
{
public:
    typedef T value_type;
    static constexpr int tuple_size = 3;

    /// Default constructor.
    /// No data is initialized! Use it for extra speed.
    SYS_FORCE_INLINE UT_Vector3T() = default;

    constexpr SYS_FORCE_INLINE UT_Vector3T(const UT_Vector3T<T> &that) = default;
    constexpr SYS_FORCE_INLINE UT_Vector3T(UT_Vector3T<T> &&that) = default;

    constexpr SYS_FORCE_INLINE UT_Vector3T(const T vx, const T vy, const T vz) noexcept :
        vec{ vx, vy, vz }
    {}

    constexpr explicit SYS_FORCE_INLINE UT_Vector3T(const T v) noexcept :
        UT_Vector3T( v, v, v )
    {}

    constexpr SYS_FORCE_INLINE UT_Vector3T(const fpreal16 v[tuple_size]) noexcept :
        UT_Vector3T( v[0], v[1], v[2] )
    {}
    constexpr SYS_FORCE_INLINE UT_Vector3T(const fpreal32 v[tuple_size]) noexcept :
        UT_Vector3T( v[0], v[1], v[2] )
    {}
    constexpr SYS_FORCE_INLINE UT_Vector3T(const fpreal64 v[tuple_size]) noexcept :
        UT_Vector3T( v[0], v[1], v[2] )
    {}
    constexpr SYS_FORCE_INLINE UT_Vector3T(const int32 v[tuple_size]) noexcept :
        UT_Vector3T( v[0], v[1], v[2] )
    {}
    constexpr SYS_FORCE_INLINE UT_Vector3T(const int64 v[tuple_size]) noexcept :
        UT_Vector3T( v[0], v[1], v[2] )
    {}

    SYS_FORCE_INLINE explicit UT_Vector3T(const UT_Vector2T<T> &v);
    SYS_FORCE_INLINE explicit UT_Vector3T(const UT_Vector4T<T> &v);

    /// Our own type of any given value_type.
    template <typename S>
    constexpr SYS_FORCE_INLINE UT_Vector3T(const UT_Vector3T<S>& v) noexcept :
        UT_Vector3T( v[0], v[1], v[2] )
    {}

    constexpr SYS_FORCE_INLINE UT_Vector3T<T> &operator=(const UT_Vector3T<T> &that) = default;
    constexpr SYS_FORCE_INLINE UT_Vector3T<T> &operator=(UT_Vector3T<T> &&that) = default;

    template <typename S>
    SYS_FORCE_INLINE UT_Vector3T<T> &operator=(const UT_Vector3T<S> &v)
    { vec[0] = v[0]; vec[1] = v[1]; vec[2] = v[2]; return *this; }
    
    constexpr SYS_FORCE_INLINE const T& operator[]( exint i ) const noexcept
    {
        UT_ASSERT_P( ( 0 <= i ) && ( i < tuple_size ) );

        return vec[ i ];
    }
    
    constexpr SYS_FORCE_INLINE T& operator[]( exint i ) noexcept
    {
        UT_ASSERT_P( ( 0 <= i ) && ( i < tuple_size ) );

        return vec[ i ];
    }

    constexpr SYS_FORCE_INLINE const T* data() const noexcept
    {
        return vec;
    }

    constexpr SYS_FORCE_INLINE T* data() noexcept
    {
        return vec;
    }

    constexpr SYS_FORCE_INLINE UT_Vector3T& operator+=( const UT_Vector3T& a ) noexcept
    {
        UT::FA::Add< T, tuple_size >{}( vec, a.vec );
        return *this;
    }

    constexpr SYS_FORCE_INLINE UT_Vector3T& operator-=( const UT_Vector3T& a ) noexcept
    {
        UT::FA::Subtract< T, tuple_size >{}( vec, a.vec );
        return *this;
    }

    constexpr SYS_FORCE_INLINE UT_Vector3T& operator+=( const T& a ) noexcept
    {
        UT::FA::AddUniform< T, tuple_size >{}( vec, a );
        return *this;
    }

    constexpr SYS_FORCE_INLINE UT_Vector3T& operator-=( const T& a ) noexcept
    {
        UT::FA::SubtractUniform< T, tuple_size >{}( vec, a );
        return *this;
    }

    constexpr SYS_FORCE_INLINE UT_Vector3T& operator*=( const T& a ) noexcept
    {
        UT::FA::Scale< T, tuple_size >{}( vec, a );
        return *this;
    }
    
    constexpr SYS_FORCE_INLINE UT_Vector3T& operator/=( const T& a ) noexcept
    {
        using MF = UT_StorageMathFloat_t< T >;
        UT::FA::Scale< T, tuple_size, MF >{}( vec, MF{1} / a );
        return *this;
    }

    constexpr SYS_FORCE_INLINE UT_Vector3T& operator*=( const UT_Vector3T& a ) noexcept
    {
        UT::FA::ScaleComponentwise< T, tuple_size >{}( vec, a.vec );
        return *this;
    }

    constexpr SYS_FORCE_INLINE UT_Vector3T& operator/=( const UT_Vector3T& a ) noexcept
    {
        UT::FA::DivideComponentwise< T, tuple_size >{}( vec, a.vec );
        return *this;
    }

    constexpr SYS_FORCE_INLINE void negate() noexcept
    {
        UT::FA::Negate< T, tuple_size >{}( vec );
    }
    
    constexpr SYS_FORCE_INLINE T length2() const noexcept
    {
        return UT::FA::Length2< T, tuple_size >{}( vec );
    }

    constexpr SYS_FORCE_INLINE T length() const noexcept
    {
        return SYSsqrt( length2() );
    }

    constexpr SYS_FORCE_INLINE T distance2( const UT_Vector3T& b ) const noexcept
    {
        return UT::FA::Distance2< T, tuple_size >{}( vec, b.vec );
    }

    constexpr SYS_FORCE_INLINE T distance( const UT_Vector3T& b ) const noexcept
    {
        return SYSsqrt( distance2( b ) );
    }

    SYS_FORCE_INLINE UT_StorageMathFloat_t< T > normalize() noexcept
    {
        using MF = UT_StorageMathFloat_t< T >;
        using U = UT_StorageAtLeast32Bit_t<T,T>;
        return UT::FA::Normalize< T, tuple_size, MF, U >{}( vec, std::numeric_limits< MF >::min() );
    }

    constexpr SYS_FORCE_INLINE bool isNan() const noexcept
    {
        return UT::FA::AnyOf< T, tuple_size >{}( vec, [ & ]( const T& a ) { return SYSisNan( a ); } );
    }

    constexpr SYS_FORCE_INLINE bool isFinite() const noexcept
    {
        return UT::FA::AllOf< T, tuple_size >{}( vec, [ & ]( const T& a ) { return SYSisFinite( a ); } );
    }

    constexpr SYS_FORCE_INLINE bool isZero() const noexcept
    {
        return UT::FA::IsUniformZero< T, tuple_size >{}( vec );
    }
    
    constexpr SYS_FORCE_INLINE bool equalZero( const T tolerance = SYS_FTOLERANCE ) const noexcept
    {
        return UT::FA::MaxNormIsLEQ< T, tuple_size >{}( vec, tolerance );
    }
    
    constexpr SYS_FORCE_INLINE bool isEqual( const UT_Vector3T& b, const T tolerance = SYS_FTOLERANCE ) const noexcept
    {
        return UT::FA::MaxMetricIsLEQ< T, tuple_size >{}( vec, b.vec, tolerance );
    }
    
    constexpr SYS_FORCE_INLINE T maxComponent() const noexcept
    {
        return UT::FA::Max< T, tuple_size >{}( vec );
    }
    
    constexpr SYS_FORCE_INLINE T minComponent() const noexcept
    {
        return UT::FA::Min< T, tuple_size >{}( vec );
    }
    
    constexpr SYS_FORCE_INLINE T avgComponent() const noexcept
    {
        return UT::FA::Sum< T, tuple_size >{}( vec ) / T{ tuple_size };
    }

    /// Assignment operator that truncates a V4 to a V3.
    /// TODO: remove this. This should require an explicit UT_Vector3()
    /// construction, since it's unsafe.
    SYS_DEPRECATED_HDK_REPLACE(16.0,explicit UT_Vector3 constructor to avoid implicit conversion from UT_Vector4)
    UT_Vector3T<T>      &operator=(const UT_Vector4T<T> &v);

    constexpr SYS_FORCE_INLINE UT_Vector3T& operator=( const T a ) noexcept;

    constexpr SYS_FORCE_INLINE UT_Vector3T<T> operator-() const noexcept
    {
        return UT_Vector3T<T>(-vec[0], -vec[1], -vec[2]);
    }

    void        clampZero(T tol = T(0.00001f))
                {
                    if (vec[0] >= -tol && vec[0] <= tol) vec[0] = 0;
                    if (vec[1] >= -tol && vec[1] <= tol) vec[1] = 0;
                    if (vec[2] >= -tol && vec[2] <= tol) vec[2] = 0;
                }

    SYS_FORCE_INLINE
    void         multiplyComponents(const UT_Vector3T<T> &v)
                 {
                     vec[0] *= v.vec[0];
                     vec[1] *= v.vec[1];
                     vec[2] *= v.vec[2];
                 }

    /// If you need a multiplication operator that left multiplies the vector
    /// by a matrix (M * v), use the following colVecMult() functions. If
    /// you'd rather not use operator*=() for right-multiplications (v * M),
    /// use the following rowVecMult() functions. The methods that take a 4x4
    /// matrix first extend this vector to 4D by adding an element equal to 1.0.
    /// @internal These are implemented in UT_Matrix3.h and UT_Matrix4.h
    // @{
    SYS_FORCE_INLINE void rowVecMult(const UT_Matrix3F &m);
    SYS_FORCE_INLINE void rowVecMult(const UT_Matrix4F &m);
    SYS_FORCE_INLINE void rowVecMult(const UT_Matrix3D &m);
    SYS_FORCE_INLINE void rowVecMult(const UT_Matrix4D &m);
    SYS_FORCE_INLINE void colVecMult(const UT_Matrix3F &m);
    SYS_FORCE_INLINE void colVecMult(const UT_Matrix4F &m);
    SYS_FORCE_INLINE void colVecMult(const UT_Matrix3D &m);
    SYS_FORCE_INLINE void colVecMult(const UT_Matrix4D &m);
    // @}


    /// This multiply will not extend the vector by adding a fourth element.
    /// Instead, it converts the Matrix4 to a Matrix3.  This means that
    /// the translate component of the matrix is not applied to the vector
    /// @internal These are implemented in UT_Matrix4.h
    // @{
    SYS_FORCE_INLINE void rowVecMult3(const UT_Matrix4F &m);
    SYS_FORCE_INLINE void rowVecMult3(const UT_Matrix4D &m);
    SYS_FORCE_INLINE void colVecMult3(const UT_Matrix4F &m);
    SYS_FORCE_INLINE void colVecMult3(const UT_Matrix4D &m);
    // @}



    /// The *=, multiply, multiply3 and multiplyT routines are provided for
    /// legacy reasons. They all assume that *this is a row vector. Generally,
    /// the rowVecMult and colVecMult methods are preferred, since they're
    /// more explicit about the row vector assumption.
    // @{
    template <typename S>
    SYS_FORCE_INLINE UT_Vector3T<T> &operator*=(const UT_Matrix3T<S> &m);
    template <typename S>
    SYS_FORCE_INLINE UT_Vector3T<T> &operator*=(const UT_Matrix4T<S> &m);

    template <typename S>
    SYS_FORCE_INLINE void multiply3(const UT_Matrix4T<S> &mat);
    // @}

    /// This multiply will multiply the (row) vector by the transpose of the
    /// matrix instead of the matrix itself.  This is faster than
    /// transposing the matrix, then multiplying (as well there's potentially
    /// less storage requirements).
    // @{
    template <typename S>
    SYS_FORCE_INLINE void multiplyT(const UT_Matrix3T<S> &mat);
    template <typename S>
    SYS_FORCE_INLINE void multiply3T(const UT_Matrix4T<S> &mat);
    // @}

    /// The following methods implement multiplies (row) vector by a matrix,
    /// however, the resulting vector is specified by the dest parameter
    /// These operations are safe even if "dest" is the same as "this".
    // @{
    template <typename S>
    SYS_FORCE_INLINE void multiply3(UT_Vector3T<T> &dest,
                                    const UT_Matrix4T<S> &mat) const;
    template <typename S>
    SYS_FORCE_INLINE void multiplyT(UT_Vector3T<T> &dest,
                                    const UT_Matrix3T<S> &mat) const;
    template <typename S>
    SYS_FORCE_INLINE void multiply3T(UT_Vector3T<T> &dest,
                                     const UT_Matrix4T<S> &mat) const;
    template <typename S>
    SYS_FORCE_INLINE void multiply(UT_Vector3T<T> &dest,
                                   const UT_Matrix4T<S> &mat) const;
    template <typename S>
    SYS_FORCE_INLINE void multiply(UT_Vector3T<T> &dest,
                                   const UT_Matrix3T<S> &mat) const;
    // @}

    constexpr SYS_FORCE_INLINE
    T dot(const UT_Vector3T& b) const noexcept
    {       
        return UT::FA::Dot< T, tuple_size >{}( vec, b.vec );
    }

    constexpr SYS_FORCE_INLINE void cross(const UT_Vector3T<T> &v) noexcept
                 {
                    operator=(::cross(*this, v));
                 }

    SYS_FORCE_INLINE void normal(const UT_Vector3T<T> &va, const UT_Vector3T<T> &vb)
                 {
                    vec[0] += (va.vec[2]+vb.vec[2])*(vb.vec[1]-va.vec[1]);
                    vec[1] += (va.vec[0]+vb.vec[0])*(vb.vec[2]-va.vec[2]);
                    vec[2] += (va.vec[1]+vb.vec[1])*(vb.vec[0]-va.vec[0]);
                 }

    /// Finds an arbitrary perpendicular to v, and sets this to it.
    void         arbitraryPerp(const UT_Vector3T<T> &v);
    /// Makes this orthogonal to the given vector.  If they are colinear,
    /// does an arbitrary perp
    void         makeOrthonormal(const UT_Vector3T<T> &v);

    /// These allow you to find out what indices to use for different axes
    // @{
    int          findMinAbsAxis() const
                 {
                    T   ax = SYSabs(x()), ay = SYSabs(y());
                    if (ax < ay)
                        return (SYSabs(z()) < ax) ? 2 : 0;
                    else
                        return (SYSabs(z()) < ay) ? 2 : 1;
                 }
    int          findMaxAbsAxis() const
                 {
                    T   ax = SYSabs(x()), ay = SYSabs(y());
                    if (ax >= ay)
                        return (SYSabs(z()) >= ax) ? 2 : 0;
                    else
                        return (SYSabs(z()) >= ay) ? 2 : 1;
                 }
    // @}

    /// Given this vector as the z-axis, get a frame of reference such that the
    /// X and Y vectors are orthonormal to the vector.  This vector should be
    /// normalized.
    void         getFrameOfReference(UT_Vector3T<T> &X, UT_Vector3T<T> &Y) const
                 {
                         if (SYSabs(x()) < 0.6F) Y = UT_Vector3T<T>(1, 0, 0);
                    else if (SYSabs(z()) < 0.6F) Y = UT_Vector3T<T>(0, 1, 0);
                    else                        Y = UT_Vector3T<T>(0, 0, 1);
                    X = ::cross(Y, *this);
                    X.normalize();
                    Y = ::cross(*this, X);
                 }

    /// Calculates the orthogonal projection of a vector u on the *this vector 
    UT_Vector3T<T>       project(const UT_Vector3T<T> &u) const;

    /// Create a matrix of projection onto this vector: the matrix transforms
    /// a vector v into its projection on the direction of (*this) vector,
    /// ie. dot(*this, v) * this->normalize();
    /// If we need to be normalized, set norm to non-false.
    template <typename S>
    UT_Matrix3T<S> project(bool norm=true);

    /// Vector p (representing a point in 3-space) and vector v define
    /// a line. This member returns the projection of "this" onto the
    /// line (the point on the line that is closest to this point).
    UT_Vector3T<T>       projection(const UT_Vector3T<T> &p,
                            const UT_Vector3T<T> &v) const;

    /// Projects this onto the line segement [a,b].  The returned point
    /// will lie between a and b.
    UT_Vector3T<T>       projectOnSegment(const UT_Vector3T<T> &va,
                                  const UT_Vector3T<T> &vb) const;
    /// Projects this onto the line segment [a, b].  The fpreal t is set
    /// to the parametric position of intersection, a being 0 and b being 1.
    UT_Vector3T<T>       projectOnSegment(const UT_Vector3T<T> &va, const UT_Vector3T<T> &vb,
                                  T &t) const;

    /// Create a matrix of symmetry around this vector: the matrix transforms 
    /// a vector v into its symmetry around (*this), ie. two times the 
    /// projection of v onto (*this) minus v.
    /// If we need to be normalized, set norm to non-false.
    UT_Matrix3   symmetry(bool norm=true);

    /// This method stores in (*this) the intersection between two 3D lines,
    /// p1+t*v1 and p2+u*v2. If the two lines do not actually intersect, we
    /// shift the 2nd line along the perpendicular on both lines (along the
    /// line of min distance) and return the shifted intersection point; this
    /// point thus lies on the 1st line.
    /// If we find an intersection point (shifted or not) we return 0; if
    /// the two lines are parallel we return -1; and if they intersect 
    /// behind our back we return -2. When we return -2 there still is a
    /// valid intersection point in (*this).
    int         lineIntersect(const UT_Vector3T<T> &p1, const UT_Vector3T<T> &v1,
                              const UT_Vector3T<T> &p2, const UT_Vector3T<T> &v2);

    /// Compute the intersection of vector p2+t*v2 and the line segment between
    /// points pa and pb. If the two lines do not intersect we shift the 
    /// (p2, v2) line along the line of min distance and return the point 
    /// where it intersects the segment. If we find an intersection point 
    /// along the stretch between pa and pb, we return 0. If the lines are
    /// parallel we return -1. If they intersect before pa we return -2, and 
    /// if after pb, we return -3. The intersection point is valid with 
    /// return codes 0,-2,-3.
    int         segLineIntersect(const UT_Vector3T<T> &pa,  const UT_Vector3T<T> &pb,
                                 const UT_Vector3T<T> &p2, const UT_Vector3T<T> &v2);

    /// Determines whether or not the points p0, p1 and "this" are collinear.
    /// If they are t contains the parametric value of where "this" is found
    /// on the segment from p0 to p1 and returns true.  Otherwise returns
    /// false.  If p0 and p1 are equal, t is set to
    /// std::numeric_limits<T>::max() and true is returned.
    bool        areCollinear(const UT_Vector3T<T> &p0, const UT_Vector3T<T> &p1,
                             T *t = 0, T tol = 1e-5) const;

    /// Compute (homogeneous) barycentric co-ordinates of this point
    /// relative to the triangle defined by t0, t1 and t2. (The point is
    /// projected into the triangle's plane.)
    UT_Vector3T<T>      getBary(const UT_Vector3T<T> &t0, const UT_Vector3T<T> &t1,
                        const UT_Vector3T<T> &t2, bool *degen = NULL) const;


    /// Compute the signed distance from us to a line.
    T   distance(const UT_Vector3T<T> &p1, const UT_Vector3T<T> &v1) const;
    /// Compute the signed distance between two lines.
    T   distance(const UT_Vector3T<T> &p1, const UT_Vector3T<T> &v1,
                         const UT_Vector3T<T> &p2, const UT_Vector3T<T> &v2) const;

    /// Return the components of the vector. The () operator does NOT check
    /// for the boundary condition.
    /// @{
    constexpr SYS_FORCE_INLINE T &x() noexcept          { return vec[0]; }
    constexpr SYS_FORCE_INLINE T  x() const noexcept    { return vec[0]; }
    constexpr SYS_FORCE_INLINE T &y() noexcept          { return vec[1]; }
    constexpr SYS_FORCE_INLINE T  y() const noexcept    { return vec[1]; }
    constexpr SYS_FORCE_INLINE T &z() noexcept          { return vec[2]; }
    constexpr SYS_FORCE_INLINE T  z() const noexcept    { return vec[2]; }
    constexpr SYS_FORCE_INLINE T &r() noexcept          { return vec[0]; }
    constexpr SYS_FORCE_INLINE T  r() const noexcept    { return vec[0]; }
    constexpr SYS_FORCE_INLINE T &g() noexcept          { return vec[1]; }
    constexpr SYS_FORCE_INLINE T  g() const noexcept    { return vec[1]; }
    constexpr SYS_FORCE_INLINE T &b() noexcept          { return vec[2]; }
    constexpr SYS_FORCE_INLINE T  b() const noexcept    { return vec[2]; }
    
    constexpr SYS_FORCE_INLINE T &operator()(unsigned i) noexcept
                 {
                     UT_ASSERT_P(i < tuple_size);
                     return vec[i];
                 }
    constexpr SYS_FORCE_INLINE T operator()(unsigned i) const noexcept
                 {
                     UT_ASSERT_P( i < tuple_size );
                     return vec[i];
                 }
    /// @}

    /// Compute a hash
    unsigned    hash() const    { return SYSvector_hash(data(), tuple_size); }

    // TODO: eliminate these methods. They're redundant, given good inline
    // constructors.
    /// Set the values of the vector components
    void         assign(T xx = 0.0f, T yy = 0.0f, T zz = 0.0f)
                 {
                     vec[0] = xx; vec[1] = yy; vec[2] = zz;
                 }
    /// Set the values of the vector components
    void         assign(const T *v)
                 {
                     vec[0]=v[0]; vec[1]=v[1]; vec[2]=v[2];
                 }

    /// Express the point in homogeneous coordinates or vice-versa
    // @{
    void         homogenize()
                 {
                     vec[0] *= vec[2];
                     vec[1] *= vec[2];
                 }
    void         dehomogenize()
                 {
                     if (vec[2] != 0)
                     {
                         T denom = 1.0f / vec[2];
                         vec[0] *= denom;
                         vec[1] *= denom;
                     }
                 }
    // @}

    /// assuming that "this" is a rotation (in radians, of course), the
    /// equivalent set of rotations which are closest to the "base" rotation
    /// are found. The equivalent rotations are the same as the original
    /// rotations +2*n*PI
    void         roundAngles(const UT_Vector3T<T> &base);

    /// conversion between degrees and radians
    // @{
    void         degToRad();
    void         radToDeg();
    // @}

    /// It seems that given any rotation matrix and transform order, 
    /// there are two distinct triples of rotations that will result in
    /// the same overall rotation. This method will find the closest of
    /// the two after finding the closest using the above method.
    void         roundAngles(const UT_Vector3T<T> &b, const UT_XformOrder &o);

    /// Return the dual of the vector
    /// The dual is a matrix which acts like the cross product when
    /// multiplied by other vectors.
    /// The following are equivalent:
    /// a.getDual(A);  c = colVecMult(A, b)
    /// c = cross(a, b)
    template <typename S>
    void         getDual(UT_Matrix3T<S> &dual) const;

    /// Protected I/O methods
    // @{
    void         save(std::ostream &os, bool binary = false) const;
    bool         load(UT_IStream &is);
    // @}

    /// @{
    /// Methods to serialize to a JSON stream.  The vector is stored as an
    /// array of 3 reals.
    bool                save(UT_JSONWriter &w) const;
    bool                save(UT_JSONValue &v) const;
    bool                load(UT_JSONParser &p);
    /// @}

    /// @{
    /// Method to return the angle (in radians) between this and another vector
    fpreal64 angleTo(const UT_Vector3T<T> &v) const
    {
        return ::UTangleBetween(*this, v);
    }
    /// @}

    /// Returns the vector size
    static int          entries()       { return tuple_size; }

    T vec[tuple_size];

private:
    
    friend constexpr bool isZero( const UT_Vector3T& a ) noexcept
    {
        return UT::FA::IsUniformZero< T, tuple_size >{}( a.vec );
    }

    friend constexpr auto length2( const UT_Vector3T& a ) noexcept
    {
        return UT::FA::Length2< T, tuple_size >{}( a.vec );
    }

    friend constexpr auto distance2( const UT_Vector3T& a, const UT_Vector3T& b ) noexcept
    {
        return UT::FA::Distance2< T, tuple_size >{}( a.vec, b.vec );
    }

    friend constexpr bool operator==( const UT_Vector3T& a, const UT_Vector3T& b ) noexcept
    {
        return UT::FA::AreEqual< T, tuple_size >{}( a.vec, b.vec );
    }

    friend constexpr bool operator!=( const UT_Vector3T& a, const UT_Vector3T& b ) noexcept
    {
        return ! UT::FA::AreEqual< T, tuple_size >{}( a.vec, b.vec );
    }

    /// Lexicographic order comparison operators
    /// @{
    friend constexpr bool operator<( const UT_Vector3T& a, const UT_Vector3T& b ) noexcept
    {
        return UT::FA::TernaryOrder< T, tuple_size >{}( a.vec, b.vec ) < 0;
    }

    friend constexpr bool operator<=( const UT_Vector3T& a, const UT_Vector3T& b ) noexcept
    {
        return UT::FA::TernaryOrder< T, tuple_size >{}( a.vec, b.vec ) <= 0;
    }

    friend constexpr bool operator>( const UT_Vector3T& a, const UT_Vector3T& b ) noexcept
    {
        return UT::FA::TernaryOrder< T, tuple_size >{}( a.vec, b.vec ) > 0;
    }

    friend constexpr bool operator>=( const UT_Vector3T& a, const UT_Vector3T& b ) noexcept
    {
        return UT::FA::TernaryOrder< T, tuple_size >{}( a.vec, b.vec ) >= 0;
    }
    /// @}
    
    /// I/O friends
    // @{
    friend std::ostream &operator<<(std::ostream &os, const UT_Vector3T<T> &v)
                        {
                            v.save(os);
                            return os;
                        }
    // @}
};

// Required for constructor
#include "UT_Vector2.h"
#include "UT_Vector4.h"

template <typename T>
SYS_FORCE_INLINE UT_Vector3T<T>::UT_Vector3T(const UT_Vector2T<T> &v)
{
    vec[0] = v.x();
    vec[1] = v.y();
    vec[2] = T(0);
}
template <typename T>
SYS_FORCE_INLINE UT_Vector3T<T>::UT_Vector3T(const UT_Vector4T<T> &v)
{
    vec[0] = v.x();
    vec[1] = v.y();
    vec[2] = v.z();
}

template <typename T>
constexpr SYS_FORCE_INLINE UT_Vector3T< T >& UT_Vector3T< T >::operator=( const T a ) noexcept
{
    for ( int i = 0; i != tuple_size; ++i )
    {
        vec[i] = a;
    }

    return *this;
}

// Free floating functions:
template <typename T>
constexpr
UT_Vector3T<T>  operator+(const UT_Vector3T<T> &a, const UT_Vector3T<T> &b) noexcept
{
    return UT_Vector3T<T>(a.vec[0]+b.vec[0], a.vec[1]+b.vec[1], a.vec[2]+b.vec[2]);
}

template <typename T>
constexpr
UT_Vector3T<T>  operator-(const UT_Vector3T<T> &a, const UT_Vector3T<T> &b) noexcept
{
    return UT_Vector3T<T>(a.vec[0]-b.vec[0], a.vec[1]-b.vec[1], a.vec[2]-b.vec[2]);
}

template <typename T, typename S>
constexpr
UT_Vector3T<T>  operator+(const UT_Vector3T<T> &v, S scalar) noexcept
{
    return UT_Vector3T<T>(v.vec[0]+scalar, v.vec[1]+scalar, v.vec[2]+scalar);
}

template <typename T>
constexpr
UT_Vector3T<T>  operator*(const UT_Vector3T<T> &v1, const UT_Vector3T<T> &v2) noexcept
{
    return UT_Vector3T<T>(v1.vec[0]*v2.vec[0], v1.vec[1]*v2.vec[1], v1.vec[2]*v2.vec[2]);
}

template <typename T>
constexpr
UT_Vector3T<T>  operator/(const UT_Vector3T<T> &v1, const UT_Vector3T<T> &v2) noexcept
{
    return UT_Vector3T<T>(v1.vec[0]/v2.vec[0], v1.vec[1]/v2.vec[1], v1.vec[2]/v2.vec[2]);
}

template <typename T, typename S>
constexpr
UT_Vector3T<T>  operator+(S scalar, const UT_Vector3T<T> &v) noexcept
{
    return UT_Vector3T<T>(v.vec[0]+scalar, v.vec[1]+scalar, v.vec[2]+scalar);
}

template <typename T, typename S>
constexpr
UT_Vector3T<T>  operator-(const UT_Vector3T<T> &v, S scalar) noexcept
{
    return UT_Vector3T<T>(v.vec[0]-scalar, v.vec[1]-scalar, v.vec[2]-scalar);
}

template <typename T, typename S>
constexpr
UT_Vector3T<T>  operator-(S scalar, const UT_Vector3T<T> &v) noexcept
{
    return UT_Vector3T<T>(scalar-v.vec[0], scalar-v.vec[1], scalar-v.vec[2]);
}

template <typename T, typename S>
constexpr
UT_Vector3T<T>  operator*(const UT_Vector3T<T> &v, S scalar) noexcept
{
    return UT_Vector3T<T>(v.vec[0]*scalar, v.vec[1]*scalar, v.vec[2]*scalar);
}

template <typename T, typename S>
constexpr
UT_Vector3T<T>  operator*(S scalar, const UT_Vector3T<T> &v) noexcept
{
    return UT_Vector3T<T>(v.vec[0]*scalar, v.vec[1]*scalar, v.vec[2]*scalar);
}

template <typename T, typename S>
constexpr
UT_Vector3T<T> operator/(const UT_Vector3T<T> &v, S scalar) noexcept
{
    //TODO: in C++17, this can be "if constexpr"
    if ( SYS_IsFloatingPoint_v< T > )
    {
        // This has to be T because S may be int for "v = v/2" code
        // For the same reason we must cast the 1
        T inv = T(1) / scalar;
        return UT_Vector3T<T>(v.vec[0]*inv, v.vec[1]*inv, v.vec[2]*inv);
    }
    return UT_Vector3T<T>(v.vec[0]/scalar, v.vec[1]/scalar, v.vec[2]/scalar);
}

template <typename T, typename S>
constexpr
UT_Vector3T<T>  operator/(S scalar, const UT_Vector3T<T> &v) noexcept
{
    return UT_Vector3T<T>(scalar/v.vec[0], scalar/v.vec[1], scalar/v.vec[2]);
}

template <typename T>
constexpr auto dot( const UT_Vector3T< T >& a, const UT_Vector3T< T >& b ) noexcept
{
    return UT::FA::Dot< T, 3 >{}( a.vec, b.vec );
}

template <typename T>
constexpr
UT_Vector3T<T>  cross(const UT_Vector3T<T> &a, const UT_Vector3T<T> &b) noexcept
{
    return UT_Vector3T<T>(
        a.vec[1]*b.vec[2] - a.vec[2]*b.vec[1],
        a.vec[2]*b.vec[0] - a.vec[0]*b.vec[2],
        a.vec[0]*b.vec[1] - a.vec[1]*b.vec[0]
    );
}

template <typename T>
inline
fpreal64        UTangleBetween(const UT_Vector3T<T> &v1, const UT_Vector3T<T> &v2)
{
    UT_Vector3T<fpreal64> v1crossv2 = cross(v1, v2);
    fpreal v1dotv2 = dot(v1, v2);
    return SYSatan2(v1crossv2.length(), v1dotv2);
}

template <typename T>
inline
UT_Vector3T<T>  SYSabs(const UT_Vector3T<T> &v)
{
    return UT_Vector3T<T>(SYSabs(v.x()), SYSabs(v.y()), SYSabs(v.z()));
}

template <typename T>
inline
UT_Vector3T<T>  SYSmin(const UT_Vector3T<T> &v1, const UT_Vector3T<T> &v2)
{
    return UT_Vector3T<T>(
            SYSmin(v1.x(), v2.x()),
            SYSmin(v1.y(), v2.y()),
            SYSmin(v1.z(), v2.z())
    );
}

template <typename T>
inline
UT_Vector3T<T>  SYSmax(const UT_Vector3T<T> &v1, const UT_Vector3T<T> &v2)
{
    return UT_Vector3T<T>(
            SYSmax(v1.x(), v2.x()),
            SYSmax(v1.y(), v2.y()),
            SYSmax(v1.z(), v2.z())
    );
}

template <typename T, typename S>
inline bool           
SYSisEqual(const UT_Vector3T<T> &a, const UT_Vector3T<T> &b, S tol)
{
    return a.isEqual(b, tol);
}

template <typename T,typename S>
inline
UT_Vector3T<T> SYSlerp(const UT_Vector3T<T> &v1, const UT_Vector3T<T> &v2, S t)
{
    return UT_Vector3T<T>(
            SYSlerp(v1.x(), v2.x(), t),
            SYSlerp(v1.y(), v2.y(), t),
            SYSlerp(v1.z(), v2.z(), t));
}

template <typename T>
inline
UT_Vector3T<T> SYSlerp(const UT_Vector3T<T> &v1,
                    const UT_Vector3T<T> &v2,
                    const UT_Vector3T<T> &t)
{
    return UT_Vector3T<T>(
            SYSlerp(v1.x(), v2.x(), t.x()),
            SYSlerp(v1.y(), v2.y(), t.y()),
            SYSlerp(v1.z(), v2.z(), t.z()));
}

template <typename T>
inline
UT_Vector3T<T> SYSinvlerp(const UT_Vector3T<T> &a,
             const UT_Vector3T<T> &v1,
             const UT_Vector3T<T> &v2)
{
    return UT_Vector3T<T>(
            SYSinvlerp(a.x(), v1.x(), v2.x()),
            SYSinvlerp(a.y(), v1.y(), v2.y()),
            SYSinvlerp(a.z(), v1.z(), v2.z()));
}

template <typename T>
inline
UT_Vector3T<T> SYSclamp(const UT_Vector3T<T> &v,
                    const UT_Vector3T<T> &min,
                    const UT_Vector3T<T> &max)
{
    return UT_Vector3T<T>(
            SYSclamp(v.x(), min.x(), max.x()),
            SYSclamp(v.y(), min.y(), max.y()),
            SYSclamp(v.z(), min.z(), max.z()));
}

template <typename T>
inline
bool SYSequalZero(const UT_Vector3T<T> &v)
{
    return SYSequalZero(v.x()) && SYSequalZero(v.y()) && SYSequalZero(v.z());
}

template <typename T>
inline
bool SYSisFinite(const UT_Vector3T<T> &v)
{
    return SYSisFinite(v.x()) && SYSisFinite(v.y()) && SYSisFinite(v.z());
}

template <typename T>
inline
UT_Vector3T<T> SYSrecip(const UT_Vector3T<T> &v)
{
    return UT_Vector3T<T>(SYSrecip(v[0]), SYSrecip(v[1]), SYSrecip(v[2]));
}

template <typename T>
inline
T SYStrihat(const UT_Vector3T<T> &v, const UT_Vector3T<T> &s)
{
    return SYShat(v.x(), s.x()) * SYShat(v.y(), s.y()) * SYShat(v.z(), s.z());
}

template <typename T>
inline
UT_Vector3T<T> SYStrihatgrad(const UT_Vector3T<T> &v, const UT_Vector3T<T> &s)
{
    const T xhat = SYShat(v.x(), s.x());
    const T yhat = SYShat(v.y(), s.y());
    const T zhat = SYShat(v.z(), s.z());
    return UT_Vector3T<T>(SYSdhat(v.x(), s.x()) * yhat * zhat,
                         xhat * SYSdhat(v.y(), s.y()) * zhat,
                         xhat * yhat * SYSdhat(v.z(), s.z()));
}

template <typename T>
inline
UT_Vector3T<T>  project(const UT_Vector3T<T> &u, const UT_Vector3T<T> &v)
{
    return dot(u, v) / v.length2() * v;
}

template <typename T>
inline
T       distance3d(const UT_Vector3T<T> &v1, const UT_Vector3T<T> &v2)
{
    return (v1 - v2).length();
}
template <typename T>
inline
T       distance2(const UT_Vector3T<T> &v1, const UT_Vector3T<T> &v2)
{
    return (v1 - v2).length2();
}

// calculate distance squared of pos to the line segment defined by pt1 to pt2
template <typename T>
inline
T       segmentPointDist2(const UT_Vector3T<T> &pos, 
                          const UT_Vector3T<T> &pt1, const UT_Vector3T<T> &pt2 )
{
    UT_Vector3T<T>      vec;
    T   proj_t;
    T   veclen2;

    vec = pt2 - pt1;
    proj_t = vec.dot( pos - pt1 );
    veclen2 = vec.length2();

    if( proj_t <= (T)0.0 )
    {
        // in bottom cap region, calculate distance from pt1
        vec = pos - pt1;
    }
    else if( proj_t >= veclen2 )
    {
        // in top cap region, calculate distance from pt2
        vec = pos - pt2;
    }
    else
    {
        // middle region, calculate distance from projected pt
        proj_t /= veclen2;
        vec = (pt1 + (proj_t * vec)) - pos;
    }

    return dot(vec, vec);
}

template <typename T>
inline
T       rayPointDist2(const UT_Vector3T<T> &pos,
                      const UT_Vector3T<T> &orig,
                      const UT_Vector3T<T> &dir)
{
    UT_Vector3T<T> dir_hat = dir;
    dir_hat.normalize();
    UT_Vector3T<T> vec = pos - orig;
    T proj = vec.dot(dir_hat);
    return distance2(SYSmax(proj, T(0)) * dir, vec);
}

template <typename T>
inline
T       linePointDist2(const UT_Vector3T<T> &pos,
                       const UT_Vector3T<T> &orig,
                       const UT_Vector3T<T> &dir)
{
    UT_Vector3T<T> dir_hat = dir;
    dir_hat.normalize();
    UT_Vector3T<T> vec = pos - orig;
    T proj = vec.dot(dir_hat);
    return distance2(proj * dir, vec);
}

// TODO: review the effiency of the following routine, there is a faster
// way to get just the distance.
template <typename T>
inline
T        segmentDistance2(const UT_Vector3T<T> &p0, const UT_Vector3T<T> &p1,
                          const UT_Vector3T<T> &q0, const UT_Vector3T<T> &q1)
{
    UT_Vector2 t = segmentClosest(p0, p1, q0, q1);
    UT_Vector3 a = p0 + (p1 - p0) * t[0];
    UT_Vector3 b = q0 + (q1 - q0) * t[1];
    return distance2(a, b);
}

template <typename T>
inline
T       segmentDistance(const UT_Vector3T<T> &p0, const UT_Vector3T<T> &p1,
                        const UT_Vector3T<T> &q0, const UT_Vector3T<T> &q1)
{
    return SYSsqrt(segmentDistance2(p0, p1, q0, q1));
}

/// Given a 3D position, input, and a 3D parallelpiped with corner p0 and
/// directions du, dv, and dw, finds the 0 or 1 locations in the parameter
/// space of that parallelpiped that correspond with the input position.
/// Only a parameter location approximately between 0 and 1
/// is accepted.  The return value is the number of accepted parameter locations,
/// i.e. 0 or 1.
template <typename T>
int UTinverseTrilerpFlat(const UT_Vector3T<T> &input,
    const UT_Vector3T<T> &p0,
    const UT_Vector3T<T> &du, const UT_Vector3T<T> &dv, const UT_Vector3T<T> &dw,
    UT_Vector3T<T> &output)
{
    const UT_Vector3T<T> orig = input - p0;

    const UT_Matrix3T<T> matrix(
        du.x(), dv.x(), dw.x(),
        du.y(), dv.y(), dw.y(),
        du.z(), dv.z(), dw.z()
    );

    bool failed = matrix.solve(orig.x(), orig.y(), orig.z(), output);
    return !failed &&
        SYSisGreaterOrEqual(output.x(), 0) && SYSisLessOrEqual(output.x(), 1) &&
        SYSisGreaterOrEqual(output.y(), 0) && SYSisLessOrEqual(output.y(), 1) &&
        SYSisGreaterOrEqual(output.z(), 0) && SYSisLessOrEqual(output.z(), 1);
}

template <typename T>
inline size_t hash_value(const UT_Vector3T<T> &val)
{
    return val.hash();
}

// Overload for custom formatting of UT_Vector3T<T> with UTformat.
template<typename T>
UT_API size_t format(char *buffer, size_t buffer_size, const UT_Vector3T<T> &v);

template< typename T, exint D >
class UT_FixedVector;

template<typename T>
struct UT_FixedVectorTraits<UT_Vector3T<T> >
{
    typedef UT_FixedVector<T,3> FixedVectorType;
    typedef T DataType;
    static const exint TupleSize = 3;
    static const bool isVectorType = true;
};

// UT_Vector3T in the role of a fixed array-like type.

template< typename T > 
struct SYS_IsFixedArrayNoCVRef< UT_Vector3T< T > > : SYS_TrueType {};

template< typename T > 
struct SYS_FixedArrayElementNoCVRef< UT_Vector3T< T > > : SYS_TypeIdentity< T > {};

template< typename T > 
struct SYS_FixedArraySizeNoCVRef< UT_Vector3T< T > > : std::integral_constant< std::size_t, 3 > {};


// UT_Vector3TFromUnbounded<T> is a function object that
// creates a UT_Vector2T<T> from an unbounded array-like type 'as'.
// 'as' must have at size at least 3.
template <typename T>
struct UT_Vector3TFromUnbounded
{    
    template< typename TS >
    constexpr SYS_FORCE_INLINE UT_Vector3T<T> operator()(const TS& as) const noexcept
    {
        return UT_Vector3T<T>( as[0], as[1], as[2] );
    }
};

// UT_FromUnbounded<V> creates a V from an unbounded array-like type

// Primary
template <typename V >
struct UT_FromUnbounded;

// Partial specialization for UT_Vector3T
template <typename T>
struct UT_FromUnbounded< UT_Vector3T<T> > : UT_Vector3TFromUnbounded<T> {};


// UT_Vector3TFromFixed<T> is a function object that
// creates a UT_Vector3T<T> from a fixed array-like type TS,
// examples of which include T[3], UT_FixedVector<T,3> and UT_FixedArray<T,3> (AKA std::array<T,3>)
template <typename T>
struct UT_Vector3TFromFixed
{    
    template< typename TS >
    constexpr SYS_FORCE_INLINE UT_Vector3T<T> operator()(const TS& as) const noexcept
    {        
        SYS_STATIC_ASSERT( SYS_IsFixedArrayOf_v< TS, T, 3 > );

        return UT_Vector3TFromUnbounded< T >{}( as );
    }
};

// Convert a fixed array-like type TS into a UT_Vector3T< T >.
// This allows conversion to UT_Vector3T without fixing T.
// Instead, the element type of TS determines the type T.
template< typename TS >
constexpr UT_Vector3T< SYS_FixedArrayElement_t< TS > >
UTmakeVector3T( const TS& as ) noexcept
{
    using T = SYS_FixedArrayElement_t< TS >;

    return UT_Vector3TFromFixed< T >{}( as );
} 

// UT_FromFixed<V> creates a V from a flat, fixed array-like representation

// Primary
template <typename V >
struct UT_FromFixed;

// Partial specialization for UT_Vector3T
template <typename T>
struct UT_FromFixed< UT_Vector3T<T> > : UT_Vector3TFromFixed<T> {};

// Relocation traits for UT_Vector3T are defined in UT_VectorTypes.h

#endif