text: // Ceres Solver - A fast non-linear least squares minimizer
  // Copyright 2014 Google Inc. All rights reserved.
  // http://code.google.com/p/ceres-solver/
  //
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  // modification, are permitted provided that the following conditions are met:
  //
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  //   this list of conditions and the following disclaimer.
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  //   this list of conditions and the following disclaimer in the documentation
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  // * Neither the name of Google Inc. nor the names of its contributors may be
  //   used to endorse or promote products derived from this software without
  //   specific prior written permission.
  //
  // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
  // AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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  // ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
  // LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
  // CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
  // SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
  // INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
  // CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
  // ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
  // POSSIBILITY OF SUCH DAMAGE.
  //
  // Author: keir@google.com (Keir Mierle)
  //         sameeragarwal@google.com (Sameer Agarwal)
  //
  // Templated functions for manipulating rotations. The templated
  // functions are useful when implementing functors for automatic
  // differentiation.
  //
  // In the following, the Quaternions are laid out as 4-vectors, thus:
  //
  //   q[0]  scalar part.
  //   q[1]  coefficient of i.
  //   q[2]  coefficient of j.
  //   q[3]  coefficient of k.
  //
  // where: i*i = j*j = k*k = -1 and i*j = k, j*k = i, k*i = j.
  
  #ifndef CERES_PUBLIC_ROTATION_H_
  #define CERES_PUBLIC_ROTATION_H_
  
  #include <algorithm>
  #include <cmath>
  #include <limits>
  #include "glog/logging.h"
  
  namespace ceres {
  
  // Trivial wrapper to index linear arrays as matrices, given a fixed
  // column and row stride. When an array "T* array" is wrapped by a
  //
  //   (const) MatrixAdapter<T, row_stride, col_stride> M"
  //
  // the expression  M(i, j) is equivalent to
  //
  //   arrary[i * row_stride + j * col_stride]
  //
  // Conversion functions to and from rotation matrices accept
  // MatrixAdapters to permit using row-major and column-major layouts,
  // and rotation matrices embedded in larger matrices (such as a 3x4
  // projection matrix).
  template <typename T, int row_stride, int col_stride>
  struct MatrixAdapter;
  
  // Convenience functions to create a MatrixAdapter that treats the
  // array pointed to by "pointer" as a 3x3 (contiguous) column-major or
  // row-major matrix.
  template <typename T>
  MatrixAdapter<T, 1, 3> ColumnMajorAdapter3x3(T* pointer);
  
  template <typename T>
  MatrixAdapter<T, 3, 1> RowMajorAdapter3x3(T* pointer);
  
  // Convert a value in combined axis-angle representation to a quaternion.
  // The value angle_axis is a triple whose norm is an angle in radians,
  // and whose direction is aligned with the axis of rotation,
  // and quaternion is a 4-tuple that will contain the resulting quaternion.
  // The implementation may be used with auto-differentiation up to the first
  // derivative, higher derivatives may have unexpected results near the origin.
  template<typename T>
  void AngleAxisToQuaternion(const T* angle_axis, T* quaternion);
  
  // Convert a quaternion to the equivalent combined axis-angle representation.
  // The value quaternion must be a unit quaternion - it is not normalized first,
  // and angle_axis will be filled with a value whose norm is the angle of
  // rotation in radians, and whose direction is the axis of rotation.
  // The implemention may be used with auto-differentiation up to the first
  // derivative, higher derivatives may have unexpected results near the origin.
  template<typename T>
  void QuaternionToAngleAxis(const T* quaternion, T* angle_axis);
  
  // Conversions between 3x3 rotation matrix (in column major order) and
  // quaternion rotation representations.  Templated for use with
  // autodifferentiation.
  template <typename T>
  void RotationMatrixToQuaternion(const T* R, T* quaternion);
  
  template <typename T, int row_stride, int col_stride>
  void RotationMatrixToQuaternion(
      const MatrixAdapter<const T, row_stride, col_stride>& R,
      T* quaternion);
  
  // Conversions between 3x3 rotation matrix (in column major order) and
  // axis-angle rotation representations.  Templated for use with
  // autodifferentiation.
  template <typename T>
  void RotationMatrixToAngleAxis(const T* R, T* angle_axis);
  
  template <typename T, int row_stride, int col_stride>
  void RotationMatrixToAngleAxis(
      const MatrixAdapter<const T, row_stride, col_stride>& R,
      T* angle_axis);
  
  template <typename T>
  void AngleAxisToRotationMatrix(const T* angle_axis, T* R);
  
  template <typename T, int row_stride, int col_stride>
  void AngleAxisToRotationMatrix(
      const T* angle_axis,
      const MatrixAdapter<T, row_stride, col_stride>& R);
  
  // Conversions between 3x3 rotation matrix (in row major order) and
  // Euler angle (in degrees) rotation representations.
  //
  // The {pitch,roll,yaw} Euler angles are rotations around the {x,y,z}
  // axes, respectively.  They are applied in that same order, so the
  // total rotation R is Rz * Ry * Rx.
  template <typename T>
  void EulerAnglesToRotationMatrix(const T* euler, int row_stride, T* R);
  
  template <typename T, int row_stride, int col_stride>
  void EulerAnglesToRotationMatrix(
      const T* euler,
      const MatrixAdapter<T, row_stride, col_stride>& R);
  
  // Convert a 4-vector to a 3x3 scaled rotation matrix.
  //
  // The choice of rotation is such that the quaternion [1 0 0 0] goes to an
  // identity matrix and for small a, b, c the quaternion [1 a b c] goes to
  // the matrix
  //
  //         [  0 -c  b ]
  //   I + 2 [  c  0 -a ] + higher order terms
  //         [ -b  a  0 ]
  //
  // which corresponds to a Rodrigues approximation, the last matrix being
  // the cross-product matrix of [a b c]. Together with the property that
  // R(q1 * q2) = R(q1) * R(q2) this uniquely defines the mapping from q to R.
  //
  // No normalization of the quaternion is performed, i.e.
  // R = ||q||^2 * Q, where Q is an orthonormal matrix
  // such that det(Q) = 1 and Q*Q' = I
  //
  // WARNING: The rotation matrix is ROW MAJOR
  template <typename T> inline
  void QuaternionToScaledRotation(const T q[4], T R[3 * 3]);
  
  template <typename T, int row_stride, int col_stride> inline
  void QuaternionToScaledRotation(
      const T q[4],
      const MatrixAdapter<T, row_stride, col_stride>& R);
  
  // Same as above except that the rotation matrix is normalized by the
  // Frobenius norm, so that R * R' = I (and det(R) = 1).
  //
  // WARNING: The rotation matrix is ROW MAJOR
  template <typename T> inline
  void QuaternionToRotation(const T q[4], T R[3 * 3]);
  
  template <typename T, int row_stride, int col_stride> inline
  void QuaternionToRotation(
      const T q[4],
      const MatrixAdapter<T, row_stride, col_stride>& R);
  
  // Rotates a point pt by a quaternion q:
  //
  //   result = R(q) * pt
  //
  // Assumes the quaternion is unit norm. This assumption allows us to
  // write the transform as (something)*pt + pt, as is clear from the
  // formula below. If you pass in a quaternion with |q|^2 = 2 then you
  // WILL NOT get back 2 times the result you get for a unit quaternion.
  template <typename T> inline
  void UnitQuaternionRotatePoint(const T q[4], const T pt[3], T result[3]);
  
  // With this function you do not need to assume that q has unit norm.
  // It does assume that the norm is non-zero.
  template <typename T> inline
  void QuaternionRotatePoint(const T q[4], const T pt[3], T result[3]);
  
  // zw = z * w, where * is the Quaternion product between 4 vectors.
  template<typename T> inline
  void QuaternionProduct(const T z[4], const T w[4], T zw[4]);
  
  // xy = x cross y;
  template<typename T> inline
  void CrossProduct(const T x[3], const T y[3], T x_cross_y[3]);
  
  template<typename T> inline
  T DotProduct(const T x[3], const T y[3]);
  
  // y = R(angle_axis) * x;
  template<typename T> inline
  void AngleAxisRotatePoint(const T angle_axis[3], const T pt[3], T result[3]);
  
  // --- IMPLEMENTATION
  
  template<typename T, int row_stride, int col_stride>
  struct MatrixAdapter {
    T* pointer_;
    explicit MatrixAdapter(T* pointer)
      : pointer_(pointer)
    {}
  
    T& operator()(int r, int c) const {
      return pointer_[r * row_stride + c * col_stride];
    }
  };
  
  template <typename T>
  MatrixAdapter<T, 1, 3> ColumnMajorAdapter3x3(T* pointer) {
    return MatrixAdapter<T, 1, 3>(pointer);
  }
  
  template <typename T>
  MatrixAdapter<T, 3, 1> RowMajorAdapter3x3(T* pointer) {
    return MatrixAdapter<T, 3, 1>(pointer);
  }
  
  template<typename T>
  inline void AngleAxisToQuaternion(const T* angle_axis, T* quaternion) {
    const T& a0 = angle_axis[0];
    const T& a1 = angle_axis[1];
    const T& a2 = angle_axis[2];
    const T theta_squared = a0 * a0 + a1 * a1 + a2 * a2;
  
    // For points not at the origin, the full conversion is numerically stable.
    if (theta_squared > T(0.0)) {
      const T theta = sqrt(theta_squared);
      const T half_theta = theta * T(0.5);
      const T k = sin(half_theta) / theta;
      quaternion[0] = cos(half_theta);
      quaternion[1] = a0 * k;
      quaternion[2] = a1 * k;
      quaternion[3] = a2 * k;
    } else {
      // At the origin, sqrt() will produce NaN in the derivative since
      // the argument is zero.  By approximating with a Taylor series,
      // and truncating at one term, the value and first derivatives will be
      // computed correctly when Jets are used.
      const T k(0.5);
      quaternion[0] = T(1.0);
      quaternion[1] = a0 * k;
      quaternion[2] = a1 * k;
      quaternion[3] = a2 * k;
    }
  }
  
  template<typename T>
  inline void QuaternionToAngleAxis(const T* quaternion, T* angle_axis) {
    const T& q1 = quaternion[1];
    const T& q2 = quaternion[2];
    const T& q3 = quaternion[3];
    const T sin_squared_theta = q1 * q1 + q2 * q2 + q3 * q3;
  
    // For quaternions representing non-zero rotation, the conversion
    // is numerically stable.
    if (sin_squared_theta > T(0.0)) {
      const T sin_theta = sqrt(sin_squared_theta);
      const T& cos_theta = quaternion[0];
  
      // If cos_theta is negative, theta is greater than pi/2, which
      // means that angle for the angle_axis vector which is 2 * theta
      // would be greater than pi.
      //
      // While this will result in the correct rotation, it does not
      // result in a normalized angle-axis vector.
      //
      // In that case we observe that 2 * theta ~ 2 * theta - 2 * pi,
      // which is equivalent saying
      //
      //   theta - pi = atan(sin(theta - pi), cos(theta - pi))
      //              = atan(-sin(theta), -cos(theta))
      //
      const T two_theta =
          T(2.0) * ((cos_theta < 0.0)
                    ? atan2(-sin_theta, -cos_theta)
                    : atan2(sin_theta, cos_theta));
      const T k = two_theta / sin_theta;
      angle_axis[0] = q1 * k;
      angle_axis[1] = q2 * k;
      angle_axis[2] = q3 * k;
    } else {
      // For zero rotation, sqrt() will produce NaN in the derivative since
      // the argument is zero.  By approximating with a Taylor series,
      // and truncating at one term, the value and first derivatives will be
      // computed correctly when Jets are used.
      const T k(2.0);
      angle_axis[0] = q1 * k;
      angle_axis[1] = q2 * k;
      angle_axis[2] = q3 * k;
    }
  }
  
  template <typename T>
  void RotationMatrixToQuaternion(const T* R, T* angle_axis) {
    RotationMatrixToQuaternion(ColumnMajorAdapter3x3(R), angle_axis);
  }
  
  // This algorithm comes from "Quaternion Calculus and Fast Animation",
  // Ken Shoemake, 1987 SIGGRAPH course notes
  template <typename T, int row_stride, int col_stride>
  void RotationMatrixToQuaternion(
      const MatrixAdapter<const T, row_stride, col_stride>& R,
      T* quaternion) {
    const T trace = R(0, 0) + R(1, 1) + R(2, 2);
    if (trace >= 0.0) {
      T t = sqrt(trace + T(1.0));
      quaternion[0] = T(0.5) * t;
      t = T(0.5) / t;
      quaternion[1] = (R(2, 1) - R(1, 2)) * t;
      quaternion[2] = (R(0, 2) - R(2, 0)) * t;
      quaternion[3] = (R(1, 0) - R(0, 1)) * t;
    } else {
      int i = 0;
      if (R(1, 1) > R(0, 0)) {
        i = 1;
      }
  
      if (R(2, 2) > R(i, i)) {
        i = 2;
      }
  
      const int j = (i + 1) % 3;
      const int k = (j + 1) % 3;
      T t = sqrt(R(i, i) - R(j, j) - R(k, k) + T(1.0));
      quaternion[i + 1] = T(0.5) * t;
      t = T(0.5) / t;
      quaternion[0] = (R(k, j) - R(j, k)) * t;
      quaternion[j + 1] = (R(j, i) + R(i, j)) * t;
      quaternion[k + 1] = (R(k, i) + R(i, k)) * t;
    }
  }
  
  // The conversion of a rotation matrix to the angle-axis form is
  // numerically problematic when then rotation angle is close to zero
  // or to Pi. The following implementation detects when these two cases
  // occurs and deals with them by taking code paths that are guaranteed
  // to not perform division by a small number.
  template <typename T>
  inline void RotationMatrixToAngleAxis(const T* R, T* angle_axis) {
    RotationMatrixToAngleAxis(ColumnMajorAdapter3x3(R), angle_axis);
  }
  
  template <typename T, int row_stride, int col_stride>
  void RotationMatrixToAngleAxis(
      const MatrixAdapter<const T, row_stride, col_stride>& R,
      T* angle_axis) {
    T quaternion[4];
    RotationMatrixToQuaternion(R, quaternion);
    QuaternionToAngleAxis(quaternion, angle_axis);
    return;
  }
  
  template <typename T>
  inline void AngleAxisToRotationMatrix(const T* angle_axis, T* R) {
    AngleAxisToRotationMatrix(angle_axis, ColumnMajorAdapter3x3(R));
  }
  
  template <typename T, int row_stride, int col_stride>
  void AngleAxisToRotationMatrix(
      const T* angle_axis,
      const MatrixAdapter<T, row_stride, col_stride>& R) {
    static const T kOne = T(1.0);
    const T theta2 = DotProduct(angle_axis, angle_axis);
    if (theta2 > T(std::numeric_limits<double>::epsilon())) {
      // We want to be careful to only evaluate the square root if the
      // norm of the angle_axis vector is greater than zero. Otherwise
      // we get a division by zero.
      const T theta = sqrt(theta2);
      const T wx = angle_axis[0] / theta;
      const T wy = angle_axis[1] / theta;
      const T wz = angle_axis[2] / theta;
  
      const T costheta = cos(theta);
      const T sintheta = sin(theta);
  
      R(0, 0) =     costheta   + wx*wx*(kOne -    costheta);
      R(1, 0) =  wz*sintheta   + wx*wy*(kOne -    costheta);
      R(2, 0) = -wy*sintheta   + wx*wz*(kOne -    costheta);
      R(0, 1) =  wx*wy*(kOne - costheta)     - wz*sintheta;
      R(1, 1) =     costheta   + wy*wy*(kOne -    costheta);
      R(2, 1) =  wx*sintheta   + wy*wz*(kOne -    costheta);
      R(0, 2) =  wy*sintheta   + wx*wz*(kOne -    costheta);
      R(1, 2) = -wx*sintheta   + wy*wz*(kOne -    costheta);
      R(2, 2) =     costheta   + wz*wz*(kOne -    costheta);
    } else {
      // Near zero, we switch to using the first order Taylor expansion.
      R(0, 0) =  kOne;
      R(1, 0) =  angle_axis[2];
      R(2, 0) = -angle_axis[1];
      R(0, 1) = -angle_axis[2];
      R(1, 1) =  kOne;
      R(2, 1) =  angle_axis[0];
      R(0, 2) =  angle_axis[1];
      R(1, 2) = -angle_axis[0];
      R(2, 2) = kOne;
    }
  }
  
  template <typename T>
  inline void EulerAnglesToRotationMatrix(const T* euler,
                                          const int row_stride_parameter,
                                          T* R) {
    CHECK_EQ(row_stride_parameter, 3);
    EulerAnglesToRotationMatrix(euler, RowMajorAdapter3x3(R));
  }
  
  template <typename T, int row_stride, int col_stride>
  void EulerAnglesToRotationMatrix(
      const T* euler,
      const MatrixAdapter<T, row_stride, col_stride>& R) {
    const double kPi = 3.14159265358979323846;
    const T degrees_to_radians(kPi / 180.0);
  
    const T pitch(euler[0] * degrees_to_radians);
    const T roll(euler[1] * degrees_to_radians);
    const T yaw(euler[2] * degrees_to_radians);
  
    const T c1 = cos(yaw);
    const T s1 = sin(yaw);
    const T c2 = cos(roll);
    const T s2 = sin(roll);
    const T c3 = cos(pitch);
    const T s3 = sin(pitch);
  
    R(0, 0) = c1*c2;
    R(0, 1) = -s1*c3 + c1*s2*s3;
    R(0, 2) = s1*s3 + c1*s2*c3;
  
    R(1, 0) = s1*c2;
    R(1, 1) = c1*c3 + s1*s2*s3;
    R(1, 2) = -c1*s3 + s1*s2*c3;
  
    R(2, 0) = -s2;
    R(2, 1) = c2*s3;
    R(2, 2) = c2*c3;
  }
  
  template <typename T> inline
  void QuaternionToScaledRotation(const T q[4], T R[3 * 3]) {
    QuaternionToScaledRotation(q, RowMajorAdapter3x3(R));
  }
  
  template <typename T, int row_stride, int col_stride> inline
  void QuaternionToScaledRotation(
      const T q[4],
      const MatrixAdapter<T, row_stride, col_stride>& R) {
    // Make convenient names for elements of q.
    T a = q[0];
    T b = q[1];
    T c = q[2];
    T d = q[3];
    // This is not to eliminate common sub-expression, but to
    // make the lines shorter so that they fit in 80 columns!
    T aa = a * a;
    T ab = a * b;
    T ac = a * c;
    T ad = a * d;
    T bb = b * b;
    T bc = b * c;
    T bd = b * d;
    T cc = c * c;
    T cd = c * d;
    T dd = d * d;
  
    R(0, 0) = aa + bb - cc - dd; R(0, 1) = T(2) * (bc - ad);  R(0, 2) = T(2) * (ac + bd);  // NOLINT
    R(1, 0) = T(2) * (ad + bc);  R(1, 1) = aa - bb + cc - dd; R(1, 2) = T(2) * (cd - ab);  // NOLINT
    R(2, 0) = T(2) * (bd - ac);  R(2, 1) = T(2) * (ab + cd);  R(2, 2) = aa - bb - cc + dd; // NOLINT
  }
  
  template <typename T> inline
  void QuaternionToRotation(const T q[4], T R[3 * 3]) {
    QuaternionToRotation(q, RowMajorAdapter3x3(R));
  }
  
  template <typename T, int row_stride, int col_stride> inline
  void QuaternionToRotation(const T q[4],
                            const MatrixAdapter<T, row_stride, col_stride>& R) {
    QuaternionToScaledRotation(q, R);
  
    T normalizer = q[0]*q[0] + q[1]*q[1] + q[2]*q[2] + q[3]*q[3];
    CHECK_NE(normalizer, T(0));
    normalizer = T(1) / normalizer;
  
    for (int i = 0; i < 3; ++i) {
      for (int j = 0; j < 3; ++j) {
        R(i, j) *= normalizer;
      }
    }
  }
  
  template <typename T> inline
  void UnitQuaternionRotatePoint(const T q[4], const T pt[3], T result[3]) {
    const T t2 =  q[0] * q[1];
    const T t3 =  q[0] * q[2];
    const T t4 =  q[0] * q[3];
    const T t5 = -q[1] * q[1];
    const T t6 =  q[1] * q[2];
    const T t7 =  q[1] * q[3];
    const T t8 = -q[2] * q[2];
    const T t9 =  q[2] * q[3];
    const T t1 = -q[3] * q[3];
    result[0] = T(2) * ((t8 + t1) * pt[0] + (t6 - t4) * pt[1] + (t3 + t7) * pt[2]) + pt[0];  // NOLINT
    result[1] = T(2) * ((t4 + t6) * pt[0] + (t5 + t1) * pt[1] + (t9 - t2) * pt[2]) + pt[1];  // NOLINT
    result[2] = T(2) * ((t7 - t3) * pt[0] + (t2 + t9) * pt[1] + (t5 + t8) * pt[2]) + pt[2];  // NOLINT
  }
  
  template <typename T> inline
  void QuaternionRotatePoint(const T q[4], const T pt[3], T result[3]) {
    // 'scale' is 1 / norm(q).
    const T scale = T(1) / sqrt(q[0] * q[0] +
                                q[1] * q[1] +
                                q[2] * q[2] +
                                q[3] * q[3]);
  
    // Make unit-norm version of q.
    const T unit[4] = {
      scale * q[0],
      scale * q[1],
      scale * q[2],
      scale * q[3],
    };
  
    UnitQuaternionRotatePoint(unit, pt, result);
  }
  
  template<typename T> inline
  void QuaternionProduct(const T z[4], const T w[4], T zw[4]) {
    zw[0] = z[0] * w[0] - z[1] * w[1] - z[2] * w[2] - z[3] * w[3];
    zw[1] = z[0] * w[1] + z[1] * w[0] + z[2] * w[3] - z[3] * w[2];
    zw[2] = z[0] * w[2] - z[1] * w[3] + z[2] * w[0] + z[3] * w[1];
    zw[3] = z[0] * w[3] + z[1] * w[2] - z[2] * w[1] + z[3] * w[0];
  }
  
  // xy = x cross y;
  template<typename T> inline
  void CrossProduct(const T x[3], const T y[3], T x_cross_y[3]) {
    x_cross_y[0] = x[1] * y[2] - x[2] * y[1];
    x_cross_y[1] = x[2] * y[0] - x[0] * y[2];
    x_cross_y[2] = x[0] * y[1] - x[1] * y[0];
  }
  
  template<typename T> inline
  T DotProduct(const T x[3], const T y[3]) {
    return (x[0] * y[0] + x[1] * y[1] + x[2] * y[2]);
  }
  
  template<typename T> inline
  void AngleAxisRotatePoint(const T angle_axis[3], const T pt[3], T result[3]) {
    const T theta2 = DotProduct(angle_axis, angle_axis);
    if (theta2 > T(std::numeric_limits<double>::epsilon())) {
      // Away from zero, use the rodriguez formula
      //
      //   result = pt costheta +
      //            (w x pt) * sintheta +
      //            w (w . pt) (1 - costheta)
      //
      // We want to be careful to only evaluate the square root if the
      // norm of the angle_axis vector is greater than zero. Otherwise
      // we get a division by zero.
      //
      const T theta = sqrt(theta2);
      const T costheta = cos(theta);
      const T sintheta = sin(theta);
      const T theta_inverse = 1.0 / theta;
  
      const T w[3] = { angle_axis[0] * theta_inverse,
                       angle_axis[1] * theta_inverse,
                       angle_axis[2] * theta_inverse };
  
      // Explicitly inlined evaluation of the cross product for
      // performance reasons.
      const T w_cross_pt[3] = { w[1] * pt[2] - w[2] * pt[1],
                                w[2] * pt[0] - w[0] * pt[2],
                                w[0] * pt[1] - w[1] * pt[0] };
      const T tmp =
          (w[0] * pt[0] + w[1] * pt[1] + w[2] * pt[2]) * (T(1.0) - costheta);
  
      result[0] = pt[0] * costheta + w_cross_pt[0] * sintheta + w[0] * tmp;
      result[1] = pt[1] * costheta + w_cross_pt[1] * sintheta + w[1] * tmp;
      result[2] = pt[2] * costheta + w_cross_pt[2] * sintheta + w[2] * tmp;
    } else {
      // Near zero, the first order Taylor approximation of the rotation
      // matrix R corresponding to a vector w and angle w is
      //
      //   R = I + hat(w) * sin(theta)
      //
      // But sintheta ~ theta and theta * w = angle_axis, which gives us
      //
      //  R = I + hat(w)
      //
      // and actually performing multiplication with the point pt, gives us
      // R * pt = pt + w x pt.
      //
      // Switching to the Taylor expansion near zero provides meaningful
      // derivatives when evaluated using Jets.
      //
      // Explicitly inlined evaluation of the cross product for
      // performance reasons.
      const T w_cross_pt[3] = { angle_axis[1] * pt[2] - angle_axis[2] * pt[1],
                                angle_axis[2] * pt[0] - angle_axis[0] * pt[2],
                                angle_axis[0] * pt[1] - angle_axis[1] * pt[0] };
  
      result[0] = pt[0] + w_cross_pt[0];
      result[1] = pt[1] + w_cross_pt[1];
      result[2] = pt[2] + w_cross_pt[2];
    }
  }
  
  }  // namespace ceres
  
  #endif  // CERES_PUBLIC_ROTATION_H_
  