math3d.h

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#ifndef MATH3D_H
#define MATH3D_H

#include <string>
#include <iostream>
#include <cmath>
#include <vector>
#include <stdexcept>
#include <cstdlib>

#ifndef M_PI
#define M_PI (3.14159265358979323846)
#endif

typedef unsigned char uint8_t;
typedef unsigned int uint32_t;

namespace math3d
{

static const double pi = M_PI;
static const double rad_on_deg = pi / 180.;
static const double deg_on_rad = 180. / pi;

template <typename T>
inline bool almost_zero(T a, double e)
{
  return (a == T(0)) || (a > 0 && a < e) || (a < 0 && a > -e);
}

struct color_rgb24
{
  uint8_t r, g, b;
  color_rgb24(uint8_t R, uint8_t G, uint8_t B) : r(R), g(G), b(B) {}
};

template <typename T>
struct vec3d
{
  T x, y, z;

  explicit vec3d() : x(0), y(0), z(0) {}
  vec3d(T x_, T y_, T z_) : x(x_), y(y_), z(z_) {}

  template <typename S>
  vec3d(const vec3d<S>& s) : x(T(s.x)), y(T(s.y)), z(T(s.z)) {}

  template <typename S>
  vec3d(const S* s) : x(T(s[0])), y(T(s[1])), z(T(s[2])) {}

  vec3d<T> operator+(const vec3d<T>& p) const
  {
    return vec3d<T>(x + p.x, y + p.y, z + p.z);
  }

  vec3d<T> operator-(const vec3d<T>& p) const
  {
    return vec3d<T>(x - p.x, y - p.y, z - p.z);
  }

  vec3d<T> operator-() const
  {
    return vec3d<T>(-x, -y, -z);
  }

  template <typename S>
  vec3d<T>& operator+=(const vec3d<S>& p)
  {
    x += T(p.x);
    y += T(p.y);
    z += T(p.z);
    return *this;
  }

  // rules for partial ordering of function templates say that the new overload
  // is a better match, when it matches

  vec3d<T>& operator+=(const vec3d<T>& p)
  {
    x += p.x;
    y += p.y;
    z += p.z;
    return *this;
  }

  template <typename S>
  vec3d<T>& operator-=(const vec3d<S>& p)
  {
    x -= T(p.x);
    y -= T(p.y);
    z -= T(p.z);
    return *this;
  }

  vec3d<T>& operator-=(const vec3d<T>& p)
  {
    x -= p.x;
    y -= p.y;
    z -= p.z;
    return *this;
  }

  template <typename Scalar>
  vec3d<T>& operator/=(const Scalar& s)
  {
    const T i = T(1) / T(s);
    x *= i;
    y *= i;
    z *= i;
    return *this;
  }

  template <typename Scalar>
  vec3d<T>& operator*=(const Scalar& s)
  {
    x *= T(s);
    y *= T(s);
    z *= T(s);
    return *this;
  }

  bool operator==(const vec3d& o) const
  {
    return (x == o.x) && (y == o.y) && (z == o.z);
  }

  template <typename S>
  bool operator==(const vec3d<S>& o) const
  {
    return (x == T(o.x) && y == T(o.y) && z == T(o.z));
  }

  bool operator!=(const vec3d& o) const
  {
    return !(*this == o);
  }

  template <typename S>
  bool operator!=(const vec3d<S>& o) const
  {
    return !(*this == o);
  }

  template <typename Scalar>
  friend vec3d<T> operator*(const vec3d<T>& p, const Scalar& s)
  {
    return vec3d<T>(s * p.x, s * p.y, s * p.z);
  }

  template <typename Scalar>
  friend vec3d<T> operator*(const Scalar& s, const vec3d<T>& p)
  {
    return p * s;
  }

  template <typename Scalar>
  friend vec3d<T> operator/(const vec3d<T>& p, const Scalar& s)
  {
    return vec3d<T>(p.x / T(s), p.y / T(s), p.z / T(s));
  }

  friend std::ostream& operator<<(std::ostream& os, const vec3d<T>& p)
  {
    return (os << p.x << " " << p.y << " " << p.z);
  }

  friend std::istream& operator>>(std::istream& is, vec3d<T>& p)
  {
    return (is >> p.x >> p.y >> p.z);
  }

};

typedef vec3d<double> normal3d;
typedef vec3d<double> point3d;

class oriented_point3d : public point3d
{
public:
  normal3d n;

  explicit oriented_point3d() : point3d() {}
  oriented_point3d(const point3d& p) : point3d(p) {}
  oriented_point3d(double xx, double yy, double zz) : point3d(xx, yy, zz) {}
  oriented_point3d(const oriented_point3d& p) : point3d(p), n(p.n) {}
  oriented_point3d(const point3d& p, const normal3d& nn) : point3d(p), n(nn) {}
};


struct triangle
{
  oriented_point3d p0, p1, p2;
  int id0, id1, id2; // indices to vertices p0, p1, p2
  normal3d n;
  triangle() : id0(-1), id1(-1), id2(-1) {}
  triangle(int id0, int id1, int id2) : id0(id0), id1(id1), id2(id2) {}
  triangle(const oriented_point3d& p0_, const oriented_point3d& p1_, const oriented_point3d& p2_, const normal3d& n_)
    : p0(p0_), p1(p1_), p2(p2_), n(n_) {}
  triangle(const oriented_point3d& p0_, const oriented_point3d& p1_, const oriented_point3d& p2_, const normal3d& n_, int id0_, int id1_, int id2_)
    : p0(p0_), p1(p1_), p2(p2_), id0(id0_), id1(id1_), id2(id2_), n(n_) {}
  triangle(const point3d& p0_, const point3d& p1_, const point3d& p2_, const normal3d& n_)
    : p0(p0_), p1(p1_), p2(p2_), n(n_) {}
  friend std::ostream& operator<<(std::ostream& o, const triangle& t)
  {
    o << t.p0 << " " << t.p1 << " " << t.p2;
    return o;
  }
};


class invalid_vector : public std::logic_error
{
public:
  explicit invalid_vector() : std::logic_error("Exception invalid_vector caught.") {}
  invalid_vector(const std::string& msg) : std::logic_error("Exception invalid_vector caught: " + msg) {}
};


// ==============================================================
//                         Rotations
// ==============================================================

// NOTE: this is a std::vector derivation, thus a matrix<bool> will
// take 1 bit per element.

template<class T>
class matrix : private std::vector<T> // row-major order
{
private:
  typedef std::vector<T> super;
  int width_;
  int height_;

public:

  const int& width;
  const int& height;

  typedef typename super::iterator iterator;
  typedef typename super::const_iterator const_iterator;

  explicit matrix() : super(), width_(0), height_(0), width(width_), height(height_) {}

  matrix(int w, int h) : super(w * h), width_(w), height_(h), width(width_), height(height_) {}

  matrix(int w, int h, const T& v) : super(w * h, v), width_(w), height_(h), width(width_), height(height_) {}

  matrix(const matrix<T>& m) : super(), width(width_), height(height_)
  {
    resize(m.width_, m.height_);
    super::assign(m.begin(), m.end());
  }

  typename super::reference operator()(size_t r, size_t c)
  {
    return super::operator[](r * width_ + c);
  }
  typename super::const_reference operator()(size_t r, size_t c) const
  {
    return super::operator[](r * width_ + c);
  }

  typename super::reference at(const size_t r, const size_t c)
  {
    return super::at(r * width_ + c);
  }
  typename super::const_reference at(size_t r, size_t c) const
  {
    return super::at(r * width_ + c);
  }

  const T* to_ptr() const
  {
    return &(super::operator[](0)); // ok since std::vector is guaranteed to be contiguous
  }

  T* to_ptr()
  {
    return &(super::operator[](0));
  }

  void resize(int w, int h)
  {
    super::resize(w * h);
    width_ = w;
    height_ = h;
  }

  size_t size() const
  {
    return super::size();
  }

  iterator begin()
  {
    return super::begin();
  }
  const_iterator begin() const
  {
    return super::begin();
  }
  iterator end()
  {
    return super::end();
  }
  const_iterator end() const
  {
    return super::end();
  }

  bool operator==(const matrix<T>& m) const
  {
    if ((width_ != m.width_) || (height != m.height_))
      return false;

    const_iterator it1(begin()), it2(m.begin()), it1_end(end());

    for (; it1 != it1_end; ++it1, ++it2)
      if (*it1 != *it2) return false;

    return true;
  }

  bool operator!=(const matrix<T>& m) const
  {
    return !(*this == m);
  }

  matrix& operator=(const matrix<T>& m)
  {
    if (&m == this)
      return *this;

    if (width != m.width || height != m.height)
      throw invalid_vector("Cannot assign matrices with different sizes.");

    super::assign(m.begin(), m.end());
    return *this;
  }

  template <typename S>
  matrix<T>& operator*=(const S& s)
  {
    for (size_t i = 0; i < size(); ++i)
      super::operator[](i) *= T(s);
    return *this;
  }

  template <typename S>
  matrix<T>& operator/=(const S& s)
  {
    for (size_t i = 0; i < size(); ++i)
      super::operator[](i) /= T(s);
    return *this;
  }

  friend std::ostream& operator<<(std::ostream& s, const matrix<T>& m)
  {
    for (int y = 0; y < m.height_; ++y)
    {
      for (int x = 0; x < m.width_; ++x)
      {
        s << m[y * m.width_ + x] << " ";
      }
      s << std::endl;
    }
    return s;
  }
};


template<typename T>
struct matrix3x3
{
  T r00, r01, r02,
  r10, r11, r12,
  r20, r21, r22;

  int width, height;

  explicit matrix3x3()
    : r00(0), r01(0), r02(0)
    , r10(0), r11(0), r12(0)
    , r20(0), r21(0), r22(0)
    , width(3), height(3)
  {}

  template <typename S>
  explicit matrix3x3(const S* v)
    : r00(v[0]), r01(v[1]), r02(v[2])
    , r10(v[3]), r11(v[4]), r12(v[5])
    , r20(v[6]), r21(v[7]), r22(v[8])
    , width(3), height(3)
  {}

  void set_column(size_t c, const vec3d<T>& v)
  {
    T x = v.x;
    T y = v.y;
    T z = v.z;

    if (c == 0)
    {
      r00 = x;
      r10 = y;
      r20 = z;
    }
    else if (c == 1)
    {
      r01 = x;
      r11 = y;
      r21 = z;
    }
    else if (c == 2)
    {
      r02 = x;
      r12 = y;
      r22 = z;
    }
    else
      throw std::logic_error("Cannot set column for 3x3 matrix.");
  }

  T& operator()(size_t row, size_t col)
  {
    switch (row)
    {
    case 0:
      if (col == 0) return r00;
      if (col == 1) return r01;
      if (col == 2) return r02;
      else throw std::out_of_range("Cannot access element in 3x3 matrix");
    case 1:
      if (col == 0) return r10;
      if (col == 1) return r11;
      if (col == 2) return r12;
      else throw std::out_of_range("Cannot access element in 3x3 matrix");
    case 2:
      if (col == 0) return r20;
      if (col == 1) return r21;
      if (col == 2) return r22;
      else throw std::out_of_range("Cannot access element in 3x3 matrix");
    default:
      throw std::out_of_range("Cannot access element in 3x3 matrix");
    }
  }

  const T& operator()(size_t row, size_t col) const
  {
    switch (row)
    {
    case 0:
      if (col == 0) return r00;
      if (col == 1) return r01;
      if (col == 2) return r02;
      else throw std::out_of_range("Cannot access element in 3x3 matrix");
    case 1:
      if (col == 0) return r10;
      if (col == 1) return r11;
      if (col == 2) return r12;
      else throw std::out_of_range("Cannot access element in 3x3 matrix");
    case 2:
      if (col == 0) return r20;
      if (col == 1) return r21;
      if (col == 2) return r22;
      else throw std::out_of_range("Cannot access element in 3x3 matrix");
    default:
      throw std::out_of_range("Cannot access element in 3x3 matrix");
    }
  }

  friend std::ostream& operator<<(std::ostream& s, const matrix3x3<T>& m)
  {
    s << m.r00 << " " << m.r01 << " " << m.r02 << std::endl;
    s << m.r10 << " " << m.r11 << " " << m.r12 << std::endl;
    s << m.r20 << " " << m.r21 << " " << m.r22 << std::endl;
    return s;
  }
};


template <typename T>
inline matrix3x3<T> identity3x3()
{
  matrix3x3<T> m;
  set_identity(m);
  return m;
}


template <typename T>
inline void mult_matrix_inplace(const matrix3x3<T>& m1, const matrix3x3<T>& m2, matrix3x3<T>& r)
{
  const double r00 = m1.r00 * m2.r00 + m1.r01 * m2.r10 + m1.r02 * m2.r20;
  const double r01 = m1.r00 * m2.r01 + m1.r01 * m2.r11 + m1.r02 * m2.r21;
  const double r02 = m1.r00 * m2.r02 + m1.r01 * m2.r12 + m1.r02 * m2.r22;

  const double r10 = m1.r10 * m2.r00 + m1.r11 * m2.r10 + m1.r12 * m2.r20;
  const double r11 = m1.r10 * m2.r01 + m1.r11 * m2.r11 + m1.r12 * m2.r21;
  const double r12 = m1.r10 * m2.r02 + m1.r11 * m2.r12 + m1.r12 * m2.r22;

  const double r20 = m1.r20 * m2.r00 + m1.r21 * m2.r10 + m1.r22 * m2.r20;
  const double r21 = m1.r20 * m2.r01 + m1.r21 * m2.r11 + m1.r22 * m2.r21;
  const double r22 = m1.r20 * m2.r02 + m1.r21 * m2.r12 + m1.r22 * m2.r22;

  r.r00 = r00;
  r.r01 = r01;
  r.r02 = r02;
  r.r10 = r10;
  r.r11 = r11;
  r.r12 = r12;
  r.r20 = r20;
  r.r21 = r21;
  r.r22 = r22;
}

template <typename T>
inline void mult_matrix(const matrix3x3<T>& m1, const matrix3x3<T>& m2, matrix3x3<T>& r)
{
  if (&r == &m1 || &r == &m2) throw std::logic_error("math3d::mult_matrix() argument alias");
  return mult_matrix_inplace<T>(m1, m2, r);
}

// NO in-place!
template <typename Rot1, typename Rot2, typename Rot3>
void mult_matrix(const Rot1& m1, const Rot2& m2, Rot3& r)
{
  if ((char*)&r == (char*)&m1 || (char*)&r == (char*)&m2)
    throw std::logic_error("math3d::mult_matrix() argument alias");

  if (m1.width != m2.height || r.height != m1.height || r.width != m2.width)
    throw std::logic_error("Incompatible size matrices");

  double sum;

  for (int is = 0; is < m1.height; ++is)
  {
    for (int jd = 0; jd < m2.width; ++jd)
    {
      sum = 0.;
      for (int js = 0; js < m1.width; ++js)
      {
        sum += m1(is, js) * m2(js, jd);
      }
      r(is, jd) = sum;
    }
  }
}

// ==============================================================
//                         Quaternions
// ==============================================================

template <typename T>
struct quaternion
{
  T w, i, j, k;

  explicit quaternion(T v = 0) : w(v), i(0), j(0), k(0) {}
  //explicit quaternion(const T* p) : w(p[0]), i(p[1]), j(p[2]), k(p[3]) {}
  quaternion(T ww, T ii, T jj, T kk) : w(ww), i(ii), j(jj), k(kk) {}

  static quaternion<T> convert(const vec3d<T>& p)
  {
    return quaternion<T>(0, p.x, p.y, p.z);
  }

  static quaternion<T> convert(const T* p)
  {
    return quaternion<T>(p[0], p[1], p[2], p[3]);
  }

  quaternion<T>& operator+= (const quaternion<T>& a)
  {
    w += a.w;
    i += a.i;
    j += a.j;
    k += a.k;
    return *this;
  }

  quaternion<T>& operator*= (T a)
  {
    w *= a;
    i *= a;
    j *= a;
    k *= a;
    return *this;
  }

  void to_vector(T* p) const
  {
    p[0] = w;
    p[1] = i;
    p[2] = j;
    p[3] = k;
  }

  friend std::ostream& operator<<(std::ostream& os, const quaternion<T>& q)
  {
    return os << "[ " << q.w << " " << q.i << " " << q.j << " " << q.k << " ]";
  }

  friend std::istream& operator>>(std::istream& is, quaternion<T>& q)
  {
    std::string dump;
    return (is >> dump >> q.w >> q.i >> q.j >> q.k >> dump);
  }
};

template <typename T>
quaternion<T> operator+ (const quaternion<T>& a, const quaternion<T>& b)
{
  quaternion<T> result(a);
  result += b;
  return result;
}

template <typename T>
T dot(const quaternion<T>& a, const quaternion<T>& b)
{
  return a.w * b.w + a.i * b.i + a.j * b.j + a.k * b.k;
}

template <typename T>
T norm(const quaternion<T>& a)
{
  return std::sqrt(dot(a, a));
}

template <typename T>
quaternion<T> operator* (const quaternion<T>& a, const quaternion<T>& b)
{
  quaternion<T> result;

  result.w = a.w * b.w - a.i * b.i - a.j * b.j - a.k * b.k;
  result.i = a.i * b.w + a.w * b.i - a.k * b.j + a.j * b.k;
  result.j = a.j * b.w + a.k * b.i + a.w * b.j - a.i * b.k;
  result.k = a.k * b.w - a.j * b.i + a.i * b.j + a.w * b.k;

  return result;
}

template <typename T>
quaternion<T> operator~(const quaternion<T>& a)
{
  return quaternion<T>(a.w, -a.i, -a.j, -a.k);
}

template <typename T>
inline void conjugate(quaternion<T>& q)
{
  q.i = -q.i;
  q.j = -q.j;
  q.k = -q.k;
}

template <typename T>
inline void normalize(quaternion<T>& q)
{
  T mag = q.w * q.w + q.i * q.i + q.j * q.j + q.k * q.k;
  if (!almost_zero(mag - T(1), 1e-10))
  {
    mag = std::sqrt(mag);
    q.w /= mag;
    q.i /= mag;
    q.j /= mag;
    q.k /= mag;
  }
}

template <typename T>
inline void set_identity(quaternion<T>& q)
{
  q.w = T(1);
  q.i = q.j = q.k = T(0);
}

// returns a normalized unit quaternion
template<typename T>
quaternion<T> rot_matrix_to_quaternion(const matrix3x3<T>& m)
{
  const T m00 = m(0, 0);
  const T m11 = m(1, 1);
  const T m22 = m(2, 2);
  const T m01 = m(0, 1);
  const T m02 = m(0, 2);
  const T m10 = m(1, 0);
  const T m12 = m(1, 2);
  const T m20 = m(2, 0);
  const T m21 = m(2, 1);
  const T tr = m00 + m11 + m22;

  quaternion<T> ret;

  if (tr > 0)
  {
    T s = std::sqrt(tr + T(1)) * 2; // S=4*qw
    ret.w = 0.25 * s;
    ret.i = (m21 - m12) / s;
    ret.j = (m02 - m20) / s;
    ret.k = (m10 - m01) / s;
  }
  else if ((m00 > m11) & (m00 > m22))
  {
    const T s = std::sqrt(T(1) + m00 - m11 - m22) * 2; // S=4*qx
    ret.w = (m21 - m12) / s;
    ret.i = 0.25 * s;
    ret.j = (m01 + m10) / s;
    ret.k = (m02 + m20) / s;
  }
  else if (m11 > m22)
  {
    const T s = std::sqrt(T(1) + m11 - m00 - m22) * 2; // S=4*qy
    ret.w = (m02 - m20) / s;
    ret.i = (m01 + m10) / s;
    ret.j = 0.25 * s;
    ret.k = (m12 + m21) / s;
  }
  else
  {
    const T s = std::sqrt(T(1) + m22 - m00 - m11) * 2; // S=4*qz
    ret.w = (m10 - m01) / s;
    ret.i = (m02 + m20) / s;
    ret.j = (m12 + m21) / s;
    ret.k = 0.25 * s;
  }

  return ret;
}

// assumes a normalized unit quaternion
template <typename T>
matrix3x3<T> quaternion_to_rot_matrix(const quaternion<T>& q)
{
  matrix3x3<T> m;
  const T w = q.w, i = q.i, j = q.j, k = q.k;

  m(0, 0) = 1 - 2 * j * j - 2 * k * k;
  m(0, 1) = 2 * i * j - 2 * k * w;
  m(0, 2) = 2 * i * k + 2 * j * w;

  m(1, 0) = 2 * i * j + 2 * k * w;
  m(1, 1) = 1 - 2 * i * i - 2 * k * k;
  m(1, 2) = 2 * j * k - 2 * i * w;

  m(2, 0) = 2 * i * k - 2 * j * w;
  m(2, 1) = 2 * j * k + 2 * i * w;
  m(2, 2) = 1 - 2 * i * i - 2 * j * j;

  return m;
}

template <typename T>
inline void mult_quaternion(const quaternion<T>& a, const quaternion<T>& b, quaternion<T>& r)
{
  r.w = a.w * b.w - (a.i * b.i + a.j * b.j + a.k * b.k);
  r.i = a.w * b.i + b.w * a.i + a.j * b.k - a.k * b.j;
  r.j = a.w * b.j + b.w * a.j + a.k * b.i - a.i * b.k;
  r.k = a.w * b.k + b.w * a.k + a.i * b.j - a.j * b.i;
}

// ==============================================================
//
// ==============================================================

template <typename T>
void transpose(matrix<T>& m)
{
  matrix<T> old(m);
  const int w = m.width;
  const int h = m.height;
  m.resize(h, w);

  for (int row = 0; row < h; ++row)
  {
    for (int col = 0; col < w; ++col)
    {
      m(col, row) = old(row, col);
    }
  }
}

template <typename T>
inline void transpose(matrix3x3<T>& m)
{
  const T m01 = m.r01, m02 = m.r02, m12 = m.r12, m10 = m.r10, m21 = m.r21, m20 = m.r20;
  m.r01 = m10;
  m.r02 = m20;
  m.r10 = m01;
  m.r20 = m02;
  m.r12 = m21;
  m.r21 = m12;
}

template <typename T>
inline matrix3x3<T> get_transpose(const matrix3x3<T>& m)
{
  matrix3x3<T> ret;
  ret.r00 = m.r00;
  ret.r01 = m.r10;
  ret.r02 = m.r20;
  ret.r10 = m.r01;
  ret.r11 = m.r11;
  ret.r20 = m.r02;
  ret.r12 = m.r21;
  ret.r21 = m.r12;
  ret.r22 = m.r22;
  return ret;
}

// dest matrix must already be of the right size
template <typename T>
void transpose(const matrix<T>& src, matrix<T>& dest)
{
  if (src.width != dest.height || src.height != dest.width)
    throw math3d::invalid_vector("math3d::transpose(): Destination matrix must be of the right size.");

  const int w = src.width;
  const int h = src.height;

  for (int row = 0; row < h; ++row)
  {
    for (int col = 0; col < w; ++col)
    {
      dest(col, row) = src(row, col);
    }
  }
}

template <typename T>
inline void transpose(const matrix3x3<T>& src, matrix3x3<T>& dest)
{
  dest.r00 = src.r00;
  dest.r11 = src.r11;
  dest.r22 = src.r22;
  dest.r01 = src.r10;
  dest.r02 = src.r20;
  dest.r10 = src.r01;
  dest.r12 = src.r21;
  dest.r21 = src.r12;
  dest.r20 = src.r02;
}

template <typename T>
void set_identity(matrix<T>& m, T val = 1)
{
  if (m.width != m.height)
    throw invalid_vector("Cannot set identity on a rectangular matrix.");

  if (m.width == 0)
    return;

  const int n = m.width * m.height;
  const int w = m.width;

  int one = 0;
  for (int k = 0; k < n; ++k)
  {
    if (k == one)
    {
      m(k / w, k % w) = val;
      one += w + 1;
    }
    else
      m(k / w, k % w) = 0;
  }
}

template <typename T>
void set_identity(matrix3x3<T>& m, T val = 1)
{
  m.r00 = val;
  m.r01 = 0;
  m.r02 = 0;
  m.r10 = 0;
  m.r11 = val;
  m.r12 = 0;
  m.r20 = 0;
  m.r21 = 0;
  m.r22 = val;
}


// ==============================================================
//                         Rigid Motion
// ==============================================================

typedef std::pair<matrix3x3<double>, point3d> rigid_motion_t;

template <typename T>
void rotate(vec3d<T>& p, const matrix<T>& rot)
{
  if (rot.height != 3 || rot.width != 3)
    throw std::logic_error("Rotation matrix must be 3x3");

  T oldx = p.x, oldy = p.y, oldz = p.z;
  p.x = oldx * rot(0, 0) + oldy * rot(0, 1) + oldz * rot(0, 2);
  p.y = oldx * rot(1, 0) + oldy * rot(1, 1) + oldz * rot(1, 2);
  p.z = oldx * rot(2, 0) + oldy * rot(2, 1) + oldz * rot(2, 2);
}

template <typename T>
void rotate(vec3d<T>& p, const matrix3x3<T>& rot)
{
  T oldx = p.x, oldy = p.y, oldz = p.z;
  p.x = oldx * rot.r00 + oldy * rot.r01 + oldz * rot.r02;
  p.y = oldx * rot.r10 + oldy * rot.r11 + oldz * rot.r12;
  p.z = oldx * rot.r20 + oldy * rot.r21 + oldz * rot.r22;
}

template <typename T, typename S>
void rotate(vec3d<T>& p, const matrix<S>& rot)
{
  if (rot.height != 3 || rot.width != 3)
    throw std::logic_error("Rotation matrix must be 3x3");

  T oldx = p.x, oldy = p.y, oldz = p.z;
  p.x = T(oldx * rot(0, 0) + oldy * rot(0, 1) + oldz * rot(0, 2));
  p.y = T(oldx * rot(1, 0) + oldy * rot(1, 1) + oldz * rot(1, 2));
  p.z = T(oldx * rot(2, 0) + oldy * rot(2, 1) + oldz * rot(2, 2));
}

template <typename T, typename S>
inline void rotate(vec3d<T>& p, const matrix3x3<S>& rot)
{
  T oldx = p.x, oldy = p.y, oldz = p.z;
  p.x = T(oldx * rot.r00 + oldy * rot.r01 + oldz * rot.r02);
  p.y = T(oldx * rot.r10 + oldy * rot.r11 + oldz * rot.r12);
  p.z = T(oldx * rot.r20 + oldy * rot.r21 + oldz * rot.r22);
}

template <typename T>
inline void rotate(vec3d<T>& p, const quaternion<T>& rot)
{
  rotate(p, quaternion_to_rot_matrix(rot));
}


template <typename T>
inline vec3d<T> get_rotate(const vec3d<T>& v, const quaternion<T>& q)
{
  return get_rotate(v, quaternion_to_rot_matrix(q));
  /*
    const T
    a = q.w, b = q.i, c = q.j, d = q.k,
    t2 = a*b, t3 = a*c, t4 = a*d, t5 = -b*b, t6 = b*c,
    t7 = b*d, t8 = -c*c, t9 = c*d, t10 = -d*d,
    v1 = v.x, v2 = v.y, v3 = v.z;
    return vec3d<T>(
    2*( (t8 + t10)*v1 + (t6 -  t4)*v2 + (t3 + t7)*v3 ) + v1,
    2*( (t4 +  t6)*v1 + (t5 + t10)*v2 + (t9 - t2)*v3 ) + v2,
    2*( (t7 -  t3)*v1 + (t2 +  t9)*v2 + (t5 + t8)*v3 ) + v3);
  */
}

template <typename T>
inline vec3d<T> get_rotate(const vec3d<T>& p, const matrix3x3<T>& rot)
{
  return vec3d<T>(p.x * rot.r00 + p.y * rot.r01 + p.z * rot.r02,
                  p.x * rot.r10 + p.y * rot.r11 + p.z * rot.r12,
                  p.x * rot.r20 + p.y * rot.r21 + p.z * rot.r22);
}


template <typename T, typename RotationType>
inline void rotate_translate(vec3d<T>& v, const RotationType& rot, const point3d& trans)
{
  rotate(v, rot);
  v += trans;
}

template <typename T>
inline vec3d<T> get_rotate_translate(const vec3d<T>& p, const matrix3x3<T>& rot, const point3d& t)
{
  return vec3d<T>(p.x * rot.r00 + p.y * rot.r01 + p.z * rot.r02 + t.x,
                  p.x * rot.r10 + p.y * rot.r11 + p.z * rot.r12 + t.y,
                  p.x * rot.r20 + p.y * rot.r21 + p.z * rot.r22 + t.z);
}

template <typename T>
inline vec3d<T> get_rotate_translate(const vec3d<T>& p, const matrix<T>& rot, const point3d& t)
{
  if (rot.height != 3 || rot.width != 3)
    throw std::logic_error("Rotation matrix must be 3x3");

  return vec3d<T>(p.x * rot(0, 0) + p.y * rot(0, 1) + p.z * rot(0, 2) + t.x,
                  p.x * rot(1, 0) + p.y * rot(1, 1) + p.z * rot(1, 2) + t.y,
                  p.x * rot(2, 0) + p.y * rot(2, 1) + p.z * rot(2, 2) + t.z);
}

template <typename T>
inline vec3d<T> get_rotate_translate(const vec3d<T>& p, const T* rot, const T* t)
{
  return get_rotate_translate(p, matrix3x3<T>(rot), vec3d<T>(t[0], t[1], t[2]));
}

template <typename T>
inline vec3d<T> get_rotate_translate(const vec3d<T>& v, const quaternion<T>& rot, const point3d& t)
{
  return (get_rotate(v, rot) + t);
}


template <typename R, typename T>
inline void invert(R& r, T& t)
{
  transpose(r);
  rotate(t, r);
  t.x = -t.x;
  t.y = -t.y;
  t.z = -t.z;
}


template <typename T>
void relative_motion(

  const matrix3x3<T>& Ri, const point3d& Ti,
  const matrix3x3<T>& Rj, const point3d& Tj,
  matrix3x3<T>& Rij, point3d& Tij

)
{
  matrix3x3<T> Ri_inv = Ri;
  point3d Ti_inv = Ti;
  invert(Ri_inv, Ti_inv);

  mult_matrix(Ri_inv, Rj, Rij);
  Tij = get_rotate_translate(Tj, Ri_inv, Ti_inv);
}


// ==============================================================
//                      Vector Operations
// ==============================================================

template <typename T>
inline double normalize(vec3d<T>& p)
{
  const double n = magnitude(p);
  if (n == 0.)
  {
    p.x = 0;
    p.y = 0;
    p.z = 0;
  }
  else
  {
    p.x /= n;
    p.y /= n;
    p.z /= n;
  }
  return n;
}

template <typename T>
inline vec3d<T> get_normalize(const vec3d<T>& p)
{
  vec3d<T> q(p);
  normalize(q);
  return q;
}

template <typename T>
inline double dist(const T& p1, const T& p2)
{
  const double sqdist = squared_dist(p1, p2);
  return (sqdist == 0. ? 0. : std::sqrt(sqdist));
}

// ||p1 - p2||^2
template <typename T>
inline double squared_dist(const vec3d<T>& p1, const vec3d<T>& p2)
{
  T x = p1.x - p2.x;
  T y = p1.y - p2.y;
  T z = p1.z - p2.z;
  return ((x * x) + (y * y) + (z * z));
}

template <typename T>
inline T dot_product(const vec3d<T>& v1, const vec3d<T>& v2)
{
  return (v1.x * v2.x) + (v1.y * v2.y) + (v1.z * v2.z);
}

template <typename T, typename S>
inline double dot_product(const vec3d<T>& v1, const vec3d<S>& v2)
{
  return (v1.x * v2.x) + (v1.y * v2.y) + (v1.z * v2.z);
}

template <typename T>
inline T dot_product(const quaternion<T>& p, const quaternion<T>& q)
{
  return (p.w * q.w + p.i * q.i + p.j * q.j + p.k * q.k);
}

template <typename T>
inline double norm2(const T& v)
{
  return dot_product(v, v);
}

template <typename T>
inline double magnitude(const T& p)
{
  return std::sqrt(dot_product(p, p));
}

template <typename T>
inline vec3d<T> cross_product(const vec3d<T>& v1, const vec3d<T>& v2)
{
  return vec3d<T>(
           (v1.y * v2.z) - (v1.z * v2.y),
           (v1.z * v2.x) - (v1.x * v2.z),
           (v1.x * v2.y) - (v1.y * v2.x)
         );
}


template <typename Iterator>
inline double median(Iterator start, Iterator end)
{
  const typename Iterator::difference_type n = end - start;
  if (n <= 0) return 0.;

  if (n % 2 == 0)
    return (*(start + (n / 2)) + * (start + (n / 2 - 1))) / 2.;
  else
    return *(start + ((n - 1) / 2));
}

}

#endif