| 123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195196197198199200201202203204205206207208209210211212213214215216217218219220221222223224225226227228229230231232233234235236237238239240241242243244245246247248249250251252253254255256257258259260261262263264265266267268269270271272273274275276277278279280281282283284285286287288289290291292293294295296297298299300301302303304305306307308309310311312313314315316317318319320321322323324325326327328329330331332333334335336337338339340341342343344345346347348349350351352353354355356357358359360361362363364365366367368369370371372373374375376377378379380381382383384385386387388389390391392393394395396397398399400401402403404405406407408409410411412413414415416417418419420421422423424425426427428429430431432433434435436437438439440441442443444445446447448449450451452453454455456457458459460461462463464465466467468469470471472473474475476477478479480481482483484485486487488489490491492493494495496497498499500501502503504505506507508509510511512513514515516517518519520521522523524525526527528529530531532533534535536537538539540541542543544545546547548549550551552553554555556557558559560561562563564565566567568569570571572573574575576577578579580581582583584585586587588589590591592593594595596597598599600601602603604605606607608609610611612613614615616617618619620621622623624625626627628629630631632633634635636637638639640641642643644645646647648649650651652653654655656657658659660661662663664665666667668669670671672673674675676677678679680681682683684685686687688689690691692693694695696697698699700701702703704705706707708709710711712713714715716717718719720721722723724725726727728729730731732733734735736737738739740741742743744745746747748749750751752753754755756757758759760761762763764765766767768769770771772773774775776777778779780781782783784785786787788789790791792793794795796797798799800801802803804805806807808809810811812813814815816817818819820821822823824825826827828829830831832833834835836837838839840841842843844845846847848849850851852853854855856857858859860861862863864865866867868869870871872873874875876877878879880881882883884885886887888889890891892893894895896897898899900901902903904905906907908909910911912913914915916917918919920921922923924925926927928929930931932933934935936937938939940941942943944945946947948949950951952953954955956957958959960961962963964965966967968969970971972973974975976977978979980981982983984985986987988989990991992993994995996997998999100010011002100310041005100610071008100910101011101210131014101510161017101810191020102110221023102410251026102710281029103010311032103310341035103610371038103910401041104210431044104510461047104810491050105110521053105410551056105710581059106010611062106310641065106610671068106910701071107210731074107510761077107810791080108110821083108410851086108710881089109010911092109310941095109610971098109911001101110211031104110511061107110811091110111111121113111411151116111711181119112011211122112311241125112611271128112911301131 | // Ceres Solver - A fast non-linear least squares minimizer// Copyright 2015 Google Inc. All rights reserved.// http://ceres-solver.org///// Redistribution and use in source and binary forms, with or without// modification, are permitted provided that the following conditions are met://// * Redistributions of source code must retain the above copyright notice,//   this list of conditions and the following disclaimer.// * Redistributions in binary form must reproduce the above copyright notice,//   this list of conditions and the following disclaimer in the documentation//   and/or other materials provided with the distribution.// * 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// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE// 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: sameeragarwal@google.com (Sameer Agarwal)#include "ceres/rotation.h"#include <cmath>#include <limits>#include <string>#include "ceres/internal/eigen.h"#include "ceres/internal/port.h"#include "ceres/is_close.h"#include "ceres/jet.h"#include "ceres/stringprintf.h"#include "ceres/test_util.h"#include "glog/logging.h"#include "gmock/gmock.h"#include "gtest/gtest.h"namespace ceres {namespace internal {using std::max;using std::min;using std::numeric_limits;using std::string;using std::swap;const double kPi = 3.14159265358979323846;const double kHalfSqrt2 = 0.707106781186547524401;static double RandDouble() {  double r = rand();  return r / RAND_MAX;}// A tolerance value for floating-point comparisons.static double const kTolerance = numeric_limits<double>::epsilon() * 10;// Looser tolerance used for numerically unstable conversions.static double const kLooseTolerance = 1e-9;// Use as:// double quaternion[4];// EXPECT_THAT(quaternion, IsNormalizedQuaternion());MATCHER(IsNormalizedQuaternion, "") {  double norm2 =      arg[0] * arg[0] + arg[1] * arg[1] + arg[2] * arg[2] + arg[3] * arg[3];  if (fabs(norm2 - 1.0) > kTolerance) {    *result_listener << "squared norm is " << norm2;    return false;  }  return true;}// Use as:// double expected_quaternion[4];// double actual_quaternion[4];// EXPECT_THAT(actual_quaternion, IsNearQuaternion(expected_quaternion));MATCHER_P(IsNearQuaternion, expected, "") {  // Quaternions are equivalent upto a sign change. So we will compare  // both signs before declaring failure.  bool near = true;  for (int i = 0; i < 4; i++) {    if (fabs(arg[i] - expected[i]) > kTolerance) {      near = false;      break;    }  }  if (near) {    return true;  }  near = true;  for (int i = 0; i < 4; i++) {    if (fabs(arg[i] + expected[i]) > kTolerance) {      near = false;      break;    }  }  if (near) {    return true;  }  // clang-format off  *result_listener << "expected : "                   << expected[0] << " "                   << expected[1] << " "                   << expected[2] << " "                   << expected[3] << " "                   << "actual : "                   << arg[0] << " "                   << arg[1] << " "                   << arg[2] << " "                   << arg[3];  // clang-format on  return false;}// Use as:// double expected_axis_angle[3];// double actual_axis_angle[3];// EXPECT_THAT(actual_axis_angle, IsNearAngleAxis(expected_axis_angle));MATCHER_P(IsNearAngleAxis, expected, "") {  Eigen::Vector3d a(arg[0], arg[1], arg[2]);  Eigen::Vector3d e(expected[0], expected[1], expected[2]);  const double e_norm = e.norm();  double delta_norm = numeric_limits<double>::max();  if (e_norm > 0) {    // Deal with the sign ambiguity near PI. Since the sign can flip,    // we take the smaller of the two differences.    if (fabs(e_norm - kPi) < kLooseTolerance) {      delta_norm = min((a - e).norm(), (a + e).norm()) / e_norm;    } else {      delta_norm = (a - e).norm() / e_norm;    }  } else {    delta_norm = a.norm();  }  if (delta_norm <= kLooseTolerance) {    return true;  }  // clang-format off  *result_listener << " arg:"                   << " " << arg[0]                   << " " << arg[1]                   << " " << arg[2]                   << " was expected to be:"                   << " " << expected[0]                   << " " << expected[1]                   << " " << expected[2];  // clang-format on  return false;}// Use as:// double matrix[9];// EXPECT_THAT(matrix, IsOrthonormal());MATCHER(IsOrthonormal, "") {  for (int c1 = 0; c1 < 3; c1++) {    for (int c2 = 0; c2 < 3; c2++) {      double v = 0;      for (int i = 0; i < 3; i++) {        v += arg[i + 3 * c1] * arg[i + 3 * c2];      }      double expected = (c1 == c2) ? 1 : 0;      if (fabs(expected - v) > kTolerance) {        *result_listener << "Columns " << c1 << " and " << c2                         << " should have dot product " << expected                         << " but have " << v;        return false;      }    }  }  return true;}// Use as:// double matrix1[9];// double matrix2[9];// EXPECT_THAT(matrix1, IsNear3x3Matrix(matrix2));MATCHER_P(IsNear3x3Matrix, expected, "") {  for (int i = 0; i < 9; i++) {    if (fabs(arg[i] - expected[i]) > kTolerance) {      *result_listener << "component " << i << " should be " << expected[i];      return false;    }  }  return true;}// Transforms a zero axis/angle to a quaternion.TEST(Rotation, ZeroAngleAxisToQuaternion) {  double axis_angle[3] = {0, 0, 0};  double quaternion[4];  double expected[4] = {1, 0, 0, 0};  AngleAxisToQuaternion(axis_angle, quaternion);  EXPECT_THAT(quaternion, IsNormalizedQuaternion());  EXPECT_THAT(quaternion, IsNearQuaternion(expected));}// Test that exact conversion works for small angles.TEST(Rotation, SmallAngleAxisToQuaternion) {  // Small, finite value to test.  double theta = 1.0e-2;  double axis_angle[3] = {theta, 0, 0};  double quaternion[4];  double expected[4] = {cos(theta / 2), sin(theta / 2.0), 0, 0};  AngleAxisToQuaternion(axis_angle, quaternion);  EXPECT_THAT(quaternion, IsNormalizedQuaternion());  EXPECT_THAT(quaternion, IsNearQuaternion(expected));}// Test that approximate conversion works for very small angles.TEST(Rotation, TinyAngleAxisToQuaternion) {  // Very small value that could potentially cause underflow.  double theta = pow(numeric_limits<double>::min(), 0.75);  double axis_angle[3] = {theta, 0, 0};  double quaternion[4];  double expected[4] = {cos(theta / 2), sin(theta / 2.0), 0, 0};  AngleAxisToQuaternion(axis_angle, quaternion);  EXPECT_THAT(quaternion, IsNormalizedQuaternion());  EXPECT_THAT(quaternion, IsNearQuaternion(expected));}// Transforms a rotation by pi/2 around X to a quaternion.TEST(Rotation, XRotationToQuaternion) {  double axis_angle[3] = {kPi / 2, 0, 0};  double quaternion[4];  double expected[4] = {kHalfSqrt2, kHalfSqrt2, 0, 0};  AngleAxisToQuaternion(axis_angle, quaternion);  EXPECT_THAT(quaternion, IsNormalizedQuaternion());  EXPECT_THAT(quaternion, IsNearQuaternion(expected));}// Transforms a unit quaternion to an axis angle.TEST(Rotation, UnitQuaternionToAngleAxis) {  double quaternion[4] = {1, 0, 0, 0};  double axis_angle[3];  double expected[3] = {0, 0, 0};  QuaternionToAngleAxis(quaternion, axis_angle);  EXPECT_THAT(axis_angle, IsNearAngleAxis(expected));}// Transforms a quaternion that rotates by pi about the Y axis to an axis angle.TEST(Rotation, YRotationQuaternionToAngleAxis) {  double quaternion[4] = {0, 0, 1, 0};  double axis_angle[3];  double expected[3] = {0, kPi, 0};  QuaternionToAngleAxis(quaternion, axis_angle);  EXPECT_THAT(axis_angle, IsNearAngleAxis(expected));}// Transforms a quaternion that rotates by pi/3 about the Z axis to an axis// angle.TEST(Rotation, ZRotationQuaternionToAngleAxis) {  double quaternion[4] = {sqrt(3) / 2, 0, 0, 0.5};  double axis_angle[3];  double expected[3] = {0, 0, kPi / 3};  QuaternionToAngleAxis(quaternion, axis_angle);  EXPECT_THAT(axis_angle, IsNearAngleAxis(expected));}// Test that exact conversion works for small angles.TEST(Rotation, SmallQuaternionToAngleAxis) {  // Small, finite value to test.  double theta = 1.0e-2;  double quaternion[4] = {cos(theta / 2), sin(theta / 2.0), 0, 0};  double axis_angle[3];  double expected[3] = {theta, 0, 0};  QuaternionToAngleAxis(quaternion, axis_angle);  EXPECT_THAT(axis_angle, IsNearAngleAxis(expected));}// Test that approximate conversion works for very small angles.TEST(Rotation, TinyQuaternionToAngleAxis) {  // Very small value that could potentially cause underflow.  double theta = pow(numeric_limits<double>::min(), 0.75);  double quaternion[4] = {cos(theta / 2), sin(theta / 2.0), 0, 0};  double axis_angle[3];  double expected[3] = {theta, 0, 0};  QuaternionToAngleAxis(quaternion, axis_angle);  EXPECT_THAT(axis_angle, IsNearAngleAxis(expected));}TEST(Rotation, QuaternionToAngleAxisAngleIsLessThanPi) {  double quaternion[4];  double angle_axis[3];  const double half_theta = 0.75 * kPi;  quaternion[0] = cos(half_theta);  quaternion[1] = 1.0 * sin(half_theta);  quaternion[2] = 0.0;  quaternion[3] = 0.0;  QuaternionToAngleAxis(quaternion, angle_axis);  const double angle =      sqrt(angle_axis[0] * angle_axis[0] + angle_axis[1] * angle_axis[1] +           angle_axis[2] * angle_axis[2]);  EXPECT_LE(angle, kPi);}static constexpr int kNumTrials = 10000;// Takes a bunch of random axis/angle values, converts them to quaternions,// and back again.TEST(Rotation, AngleAxisToQuaterionAndBack) {  srand(5);  for (int i = 0; i < kNumTrials; i++) {    double axis_angle[3];    // Make an axis by choosing three random numbers in [-1, 1) and    // normalizing.    double norm = 0;    for (int i = 0; i < 3; i++) {      axis_angle[i] = RandDouble() * 2 - 1;      norm += axis_angle[i] * axis_angle[i];    }    norm = sqrt(norm);    // Angle in [-pi, pi).    double theta = kPi * 2 * RandDouble() - kPi;    for (int i = 0; i < 3; i++) {      axis_angle[i] = axis_angle[i] * theta / norm;    }    double quaternion[4];    double round_trip[3];    // We use ASSERTs here because if there's one failure, there are    // probably many and spewing a million failures doesn't make anyone's    // day.    AngleAxisToQuaternion(axis_angle, quaternion);    ASSERT_THAT(quaternion, IsNormalizedQuaternion());    QuaternionToAngleAxis(quaternion, round_trip);    ASSERT_THAT(round_trip, IsNearAngleAxis(axis_angle));  }}// Takes a bunch of random quaternions, converts them to axis/angle,// and back again.TEST(Rotation, QuaterionToAngleAxisAndBack) {  srand(5);  for (int i = 0; i < kNumTrials; i++) {    double quaternion[4];    // Choose four random numbers in [-1, 1) and normalize.    double norm = 0;    for (int i = 0; i < 4; i++) {      quaternion[i] = RandDouble() * 2 - 1;      norm += quaternion[i] * quaternion[i];    }    norm = sqrt(norm);    for (int i = 0; i < 4; i++) {      quaternion[i] = quaternion[i] / norm;    }    double axis_angle[3];    double round_trip[4];    QuaternionToAngleAxis(quaternion, axis_angle);    AngleAxisToQuaternion(axis_angle, round_trip);    ASSERT_THAT(round_trip, IsNormalizedQuaternion());    ASSERT_THAT(round_trip, IsNearQuaternion(quaternion));  }}// Transforms a zero axis/angle to a rotation matrix.TEST(Rotation, ZeroAngleAxisToRotationMatrix) {  double axis_angle[3] = {0, 0, 0};  double matrix[9];  double expected[9] = {1, 0, 0, 0, 1, 0, 0, 0, 1};  AngleAxisToRotationMatrix(axis_angle, matrix);  EXPECT_THAT(matrix, IsOrthonormal());  EXPECT_THAT(matrix, IsNear3x3Matrix(expected));}TEST(Rotation, NearZeroAngleAxisToRotationMatrix) {  double axis_angle[3] = {1e-24, 2e-24, 3e-24};  double matrix[9];  double expected[9] = {1, 0, 0, 0, 1, 0, 0, 0, 1};  AngleAxisToRotationMatrix(axis_angle, matrix);  EXPECT_THAT(matrix, IsOrthonormal());  EXPECT_THAT(matrix, IsNear3x3Matrix(expected));}// Transforms a rotation by pi/2 around X to a rotation matrix and back.TEST(Rotation, XRotationToRotationMatrix) {  double axis_angle[3] = {kPi / 2, 0, 0};  double matrix[9];  // The rotation matrices are stored column-major.  double expected[9] = {1, 0, 0, 0, 0, 1, 0, -1, 0};  AngleAxisToRotationMatrix(axis_angle, matrix);  EXPECT_THAT(matrix, IsOrthonormal());  EXPECT_THAT(matrix, IsNear3x3Matrix(expected));  double round_trip[3];  RotationMatrixToAngleAxis(matrix, round_trip);  EXPECT_THAT(round_trip, IsNearAngleAxis(axis_angle));}// Transforms an axis angle that rotates by pi about the Y axis to a// rotation matrix and back.TEST(Rotation, YRotationToRotationMatrix) {  double axis_angle[3] = {0, kPi, 0};  double matrix[9];  double expected[9] = {-1, 0, 0, 0, 1, 0, 0, 0, -1};  AngleAxisToRotationMatrix(axis_angle, matrix);  EXPECT_THAT(matrix, IsOrthonormal());  EXPECT_THAT(matrix, IsNear3x3Matrix(expected));  double round_trip[3];  RotationMatrixToAngleAxis(matrix, round_trip);  EXPECT_THAT(round_trip, IsNearAngleAxis(axis_angle));}TEST(Rotation, NearPiAngleAxisRoundTrip) {  double in_axis_angle[3];  double matrix[9];  double out_axis_angle[3];  srand(5);  for (int i = 0; i < kNumTrials; i++) {    // Make an axis by choosing three random numbers in [-1, 1) and    // normalizing.    double norm = 0;    for (int i = 0; i < 3; i++) {      in_axis_angle[i] = RandDouble() * 2 - 1;      norm += in_axis_angle[i] * in_axis_angle[i];    }    norm = sqrt(norm);    // Angle in [pi - kMaxSmallAngle, pi).    const double kMaxSmallAngle = 1e-8;    double theta = kPi - kMaxSmallAngle * RandDouble();    for (int i = 0; i < 3; i++) {      in_axis_angle[i] *= (theta / norm);    }    AngleAxisToRotationMatrix(in_axis_angle, matrix);    RotationMatrixToAngleAxis(matrix, out_axis_angle);    EXPECT_THAT(in_axis_angle, IsNearAngleAxis(out_axis_angle));  }}TEST(Rotation, AtPiAngleAxisRoundTrip) {  // A rotation of kPi about the X axis;  // clang-format off  static constexpr double kMatrix[3][3] = {    {1.0,  0.0,  0.0},    {0.0,  -1.0,  0.0},    {0.0,  0.0,  -1.0}  };  // clang-format on  double in_matrix[9];  // Fill it from kMatrix in col-major order.  for (int j = 0, k = 0; j < 3; ++j) {    for (int i = 0; i < 3; ++i, ++k) {      in_matrix[k] = kMatrix[i][j];    }  }  const double expected_axis_angle[3] = {kPi, 0, 0};  double out_matrix[9];  double axis_angle[3];  RotationMatrixToAngleAxis(in_matrix, axis_angle);  AngleAxisToRotationMatrix(axis_angle, out_matrix);  LOG(INFO) << "AngleAxis = " << axis_angle[0] << " " << axis_angle[1] << " "            << axis_angle[2];  LOG(INFO) << "Expected AngleAxis = " << kPi << " 0 0";  double out_rowmajor[3][3];  for (int j = 0, k = 0; j < 3; ++j) {    for (int i = 0; i < 3; ++i, ++k) {      out_rowmajor[i][j] = out_matrix[k];    }  }  LOG(INFO) << "Rotation:";  LOG(INFO) << "EXPECTED        |        ACTUAL";  for (int i = 0; i < 3; ++i) {    string line;    for (int j = 0; j < 3; ++j) {      StringAppendF(&line, "%g ", kMatrix[i][j]);    }    line += "         |        ";    for (int j = 0; j < 3; ++j) {      StringAppendF(&line, "%g ", out_rowmajor[i][j]);    }    LOG(INFO) << line;  }  EXPECT_THAT(axis_angle, IsNearAngleAxis(expected_axis_angle));  EXPECT_THAT(out_matrix, IsNear3x3Matrix(in_matrix));}// Transforms an axis angle that rotates by pi/3 about the Z axis to a// rotation matrix.TEST(Rotation, ZRotationToRotationMatrix) {  double axis_angle[3] = {0, 0, kPi / 3};  double matrix[9];  // This is laid-out row-major on the screen but is actually stored  // column-major.  // clang-format off  double expected[9] = { 0.5, sqrt(3) / 2, 0,   // Column 1                         -sqrt(3) / 2, 0.5, 0,  // Column 2                         0, 0, 1 };             // Column 3  // clang-format on  AngleAxisToRotationMatrix(axis_angle, matrix);  EXPECT_THAT(matrix, IsOrthonormal());  EXPECT_THAT(matrix, IsNear3x3Matrix(expected));  double round_trip[3];  RotationMatrixToAngleAxis(matrix, round_trip);  EXPECT_THAT(round_trip, IsNearAngleAxis(axis_angle));}// Takes a bunch of random axis/angle values, converts them to rotation// matrices, and back again.TEST(Rotation, AngleAxisToRotationMatrixAndBack) {  srand(5);  for (int i = 0; i < kNumTrials; i++) {    double axis_angle[3];    // Make an axis by choosing three random numbers in [-1, 1) and    // normalizing.    double norm = 0;    for (int i = 0; i < 3; i++) {      axis_angle[i] = RandDouble() * 2 - 1;      norm += axis_angle[i] * axis_angle[i];    }    norm = sqrt(norm);    // Angle in [-pi, pi).    double theta = kPi * 2 * RandDouble() - kPi;    for (int i = 0; i < 3; i++) {      axis_angle[i] = axis_angle[i] * theta / norm;    }    double matrix[9];    double round_trip[3];    AngleAxisToRotationMatrix(axis_angle, matrix);    ASSERT_THAT(matrix, IsOrthonormal());    RotationMatrixToAngleAxis(matrix, round_trip);    for (int i = 0; i < 3; ++i) {      EXPECT_NEAR(round_trip[i], axis_angle[i], kLooseTolerance);    }  }}// Takes a bunch of random axis/angle values near zero, converts them// to rotation matrices, and back again.TEST(Rotation, AngleAxisToRotationMatrixAndBackNearZero) {  srand(5);  for (int i = 0; i < kNumTrials; i++) {    double axis_angle[3];    // Make an axis by choosing three random numbers in [-1, 1) and    // normalizing.    double norm = 0;    for (int i = 0; i < 3; i++) {      axis_angle[i] = RandDouble() * 2 - 1;      norm += axis_angle[i] * axis_angle[i];    }    norm = sqrt(norm);    // Tiny theta.    double theta = 1e-16 * (kPi * 2 * RandDouble() - kPi);    for (int i = 0; i < 3; i++) {      axis_angle[i] = axis_angle[i] * theta / norm;    }    double matrix[9];    double round_trip[3];    AngleAxisToRotationMatrix(axis_angle, matrix);    ASSERT_THAT(matrix, IsOrthonormal());    RotationMatrixToAngleAxis(matrix, round_trip);    for (int i = 0; i < 3; ++i) {      EXPECT_NEAR(          round_trip[i], axis_angle[i], numeric_limits<double>::epsilon());    }  }}// Transposes a 3x3 matrix.static void Transpose3x3(double m[9]) {  swap(m[1], m[3]);  swap(m[2], m[6]);  swap(m[5], m[7]);}// Convert Euler angles from radians to degrees.static void ToDegrees(double euler_angles[3]) {  for (int i = 0; i < 3; ++i) {    euler_angles[i] *= 180.0 / kPi;  }}// Compare the 3x3 rotation matrices produced by the axis-angle// rotation 'aa' and the Euler angle rotation 'ea' (in radians).static void CompareEulerToAngleAxis(double aa[3], double ea[3]) {  double aa_matrix[9];  AngleAxisToRotationMatrix(aa, aa_matrix);  Transpose3x3(aa_matrix);  // Column to row major order.  double ea_matrix[9];  ToDegrees(ea);  // Radians to degrees.  const int kRowStride = 3;  EulerAnglesToRotationMatrix(ea, kRowStride, ea_matrix);  EXPECT_THAT(aa_matrix, IsOrthonormal());  EXPECT_THAT(ea_matrix, IsOrthonormal());  EXPECT_THAT(ea_matrix, IsNear3x3Matrix(aa_matrix));}// Test with rotation axis along the x/y/z axes.// Also test zero rotation.TEST(EulerAnglesToRotationMatrix, OnAxis) {  int n_tests = 0;  for (double x = -1.0; x <= 1.0; x += 1.0) {    for (double y = -1.0; y <= 1.0; y += 1.0) {      for (double z = -1.0; z <= 1.0; z += 1.0) {        if ((x != 0) + (y != 0) + (z != 0) > 1) continue;        double axis_angle[3] = {x, y, z};        double euler_angles[3] = {x, y, z};        CompareEulerToAngleAxis(axis_angle, euler_angles);        ++n_tests;      }    }  }  CHECK_EQ(7, n_tests);}// Test that a random rotation produces an orthonormal rotation// matrix.TEST(EulerAnglesToRotationMatrix, IsOrthonormal) {  srand(5);  for (int trial = 0; trial < kNumTrials; ++trial) {    double euler_angles_degrees[3];    for (int i = 0; i < 3; ++i) {      euler_angles_degrees[i] = RandDouble() * 360.0 - 180.0;    }    double rotation_matrix[9];    EulerAnglesToRotationMatrix(euler_angles_degrees, 3, rotation_matrix);    EXPECT_THAT(rotation_matrix, IsOrthonormal());  }}// Tests using Jets for specific behavior involving auto differentiation// near singularity points.typedef Jet<double, 3> J3;typedef Jet<double, 4> J4;namespace {J3 MakeJ3(double a, double v0, double v1, double v2) {  J3 j;  j.a = a;  j.v[0] = v0;  j.v[1] = v1;  j.v[2] = v2;  return j;}J4 MakeJ4(double a, double v0, double v1, double v2, double v3) {  J4 j;  j.a = a;  j.v[0] = v0;  j.v[1] = v1;  j.v[2] = v2;  j.v[3] = v3;  return j;}bool IsClose(double x, double y) {  EXPECT_FALSE(IsNaN(x));  EXPECT_FALSE(IsNaN(y));  return internal::IsClose(x, y, kTolerance, NULL, NULL);}}  // namespacetemplate <int N>bool IsClose(const Jet<double, N>& x, const Jet<double, N>& y) {  if (!IsClose(x.a, y.a)) {    return false;  }  for (int i = 0; i < N; i++) {    if (!IsClose(x.v[i], y.v[i])) {      return false;    }  }  return true;}template <int M, int N>void ExpectJetArraysClose(const Jet<double, N>* x, const Jet<double, N>* y) {  for (int i = 0; i < M; i++) {    if (!IsClose(x[i], y[i])) {      LOG(ERROR) << "Jet " << i << "/" << M << " not equal";      LOG(ERROR) << "x[" << i << "]: " << x[i];      LOG(ERROR) << "y[" << i << "]: " << y[i];      Jet<double, N> d, zero;      d.a = y[i].a - x[i].a;      for (int j = 0; j < N; j++) {        d.v[j] = y[i].v[j] - x[i].v[j];      }      LOG(ERROR) << "diff: " << d;      EXPECT_TRUE(IsClose(x[i], y[i]));    }  }}// Log-10 of a value well below machine precision.static const int kSmallTinyCutoff =    static_cast<int>(2 * log(numeric_limits<double>::epsilon()) / log(10.0));// Log-10 of a value just below values representable by double.static const int kTinyZeroLimit =    static_cast<int>(1 + log(numeric_limits<double>::min()) / log(10.0));// Test that exact conversion works for small angles when jets are used.TEST(Rotation, SmallAngleAxisToQuaternionForJets) {  // Examine small x rotations that are still large enough  // to be well within the range represented by doubles.  for (int i = -2; i >= kSmallTinyCutoff; i--) {    double theta = pow(10.0, i);    J3 axis_angle[3] = {J3(theta, 0), J3(0, 1), J3(0, 2)};    J3 quaternion[4];    J3 expected[4] = {        MakeJ3(cos(theta / 2), -sin(theta / 2) / 2, 0, 0),        MakeJ3(sin(theta / 2), cos(theta / 2) / 2, 0, 0),        MakeJ3(0, 0, sin(theta / 2) / theta, 0),        MakeJ3(0, 0, 0, sin(theta / 2) / theta),    };    AngleAxisToQuaternion(axis_angle, quaternion);    ExpectJetArraysClose<4, 3>(quaternion, expected);  }}// Test that conversion works for very small angles when jets are used.TEST(Rotation, TinyAngleAxisToQuaternionForJets) {  // Examine tiny x rotations that extend all the way to where  // underflow occurs.  for (int i = kSmallTinyCutoff; i >= kTinyZeroLimit; i--) {    double theta = pow(10.0, i);    J3 axis_angle[3] = {J3(theta, 0), J3(0, 1), J3(0, 2)};    J3 quaternion[4];    // To avoid loss of precision in the test itself,    // a finite expansion is used here, which will    // be exact up to machine precision for the test values used.    J3 expected[4] = {        MakeJ3(1.0, 0, 0, 0),        MakeJ3(0, 0.5, 0, 0),        MakeJ3(0, 0, 0.5, 0),        MakeJ3(0, 0, 0, 0.5),    };    AngleAxisToQuaternion(axis_angle, quaternion);    ExpectJetArraysClose<4, 3>(quaternion, expected);  }}// Test that derivatives are correct for zero rotation.TEST(Rotation, ZeroAngleAxisToQuaternionForJets) {  J3 axis_angle[3] = {J3(0, 0), J3(0, 1), J3(0, 2)};  J3 quaternion[4];  J3 expected[4] = {      MakeJ3(1.0, 0, 0, 0),      MakeJ3(0, 0.5, 0, 0),      MakeJ3(0, 0, 0.5, 0),      MakeJ3(0, 0, 0, 0.5),  };  AngleAxisToQuaternion(axis_angle, quaternion);  ExpectJetArraysClose<4, 3>(quaternion, expected);}// Test that exact conversion works for small angles.TEST(Rotation, SmallQuaternionToAngleAxisForJets) {  // Examine small x rotations that are still large enough  // to be well within the range represented by doubles.  for (int i = -2; i >= kSmallTinyCutoff; i--) {    double theta = pow(10.0, i);    double s = sin(theta);    double c = cos(theta);    J4 quaternion[4] = {J4(c, 0), J4(s, 1), J4(0, 2), J4(0, 3)};    J4 axis_angle[3];    // clang-format off    J4 expected[3] = {        MakeJ4(2*theta, -2*s, 2*c,  0,         0),        MakeJ4(0,        0,   0,    2*theta/s, 0),        MakeJ4(0,        0,   0,    0,         2*theta/s),    };    // clang-format on    QuaternionToAngleAxis(quaternion, axis_angle);    ExpectJetArraysClose<3, 4>(axis_angle, expected);  }}// Test that conversion works for very small angles.TEST(Rotation, TinyQuaternionToAngleAxisForJets) {  // Examine tiny x rotations that extend all the way to where  // underflow occurs.  for (int i = kSmallTinyCutoff; i >= kTinyZeroLimit; i--) {    double theta = pow(10.0, i);    double s = sin(theta);    double c = cos(theta);    J4 quaternion[4] = {J4(c, 0), J4(s, 1), J4(0, 2), J4(0, 3)};    J4 axis_angle[3];    // To avoid loss of precision in the test itself,    // a finite expansion is used here, which will    // be exact up to machine precision for the test values used.    // clang-format off    J4 expected[3] = {        MakeJ4(2*theta, -2*s, 2.0, 0,   0),        MakeJ4(0,        0,   0,   2.0, 0),        MakeJ4(0,        0,   0,   0,   2.0),    };    // clang-format on    QuaternionToAngleAxis(quaternion, axis_angle);    ExpectJetArraysClose<3, 4>(axis_angle, expected);  }}// Test that conversion works for no rotation.TEST(Rotation, ZeroQuaternionToAngleAxisForJets) {  J4 quaternion[4] = {J4(1, 0), J4(0, 1), J4(0, 2), J4(0, 3)};  J4 axis_angle[3];  J4 expected[3] = {      MakeJ4(0, 0, 2.0, 0, 0),      MakeJ4(0, 0, 0, 2.0, 0),      MakeJ4(0, 0, 0, 0, 2.0),  };  QuaternionToAngleAxis(quaternion, axis_angle);  ExpectJetArraysClose<3, 4>(axis_angle, expected);}TEST(Quaternion, RotatePointGivesSameAnswerAsRotationByMatrixCanned) {  // Canned data generated in octave.  double const q[4] = {      +0.1956830471754074,      -0.0150618562474847,      +0.7634572982788086,      -0.3019454777240753,  };  double const Q[3][3] = {      // Scaled rotation matrix.      {-0.6355194033477252, +0.0951730541682254, +0.3078870197911186},      {-0.1411693904792992, +0.5297609702153905, -0.4551502574482019},      {-0.2896955822708862, -0.4669396571547050, -0.4536309793389248},  };  double const R[3][3] = {      // With unit rows and columns.      {-0.8918859164053080, +0.1335655625725649, +0.4320876677394745},      {-0.1981166751680096, +0.7434648665444399, -0.6387564287225856},      {-0.4065578619806013, -0.6553016349046693, -0.6366242786393164},  };  // Compute R from q and compare to known answer.  double Rq[3][3];  QuaternionToScaledRotation<double>(q, Rq[0]);  ExpectArraysClose(9, Q[0], Rq[0], kTolerance);  // Now do the same but compute R with normalization.  QuaternionToRotation<double>(q, Rq[0]);  ExpectArraysClose(9, R[0], Rq[0], kTolerance);}TEST(Quaternion, RotatePointGivesSameAnswerAsRotationByMatrix) {  // Rotation defined by a unit quaternion.  double const q[4] = {      +0.2318160216097109,      -0.0178430356832060,      +0.9044300776717159,      -0.3576998641394597,  };  double const p[3] = {      +0.11,      -13.15,      1.17,  };  double R[3 * 3];  QuaternionToRotation(q, R);  double result1[3];  UnitQuaternionRotatePoint(q, p, result1);  double result2[3];  VectorRef(result2, 3) = ConstMatrixRef(R, 3, 3) * ConstVectorRef(p, 3);  ExpectArraysClose(3, result1, result2, kTolerance);}// Verify that (a * b) * c == a * (b * c).TEST(Quaternion, MultiplicationIsAssociative) {  double a[4];  double b[4];  double c[4];  for (int i = 0; i < 4; ++i) {    a[i] = 2 * RandDouble() - 1;    b[i] = 2 * RandDouble() - 1;    c[i] = 2 * RandDouble() - 1;  }  double ab[4];  double ab_c[4];  QuaternionProduct(a, b, ab);  QuaternionProduct(ab, c, ab_c);  double bc[4];  double a_bc[4];  QuaternionProduct(b, c, bc);  QuaternionProduct(a, bc, a_bc);  ASSERT_NEAR(ab_c[0], a_bc[0], kTolerance);  ASSERT_NEAR(ab_c[1], a_bc[1], kTolerance);  ASSERT_NEAR(ab_c[2], a_bc[2], kTolerance);  ASSERT_NEAR(ab_c[3], a_bc[3], kTolerance);}TEST(AngleAxis, RotatePointGivesSameAnswerAsRotationMatrix) {  double angle_axis[3];  double R[9];  double p[3];  double angle_axis_rotated_p[3];  double rotation_matrix_rotated_p[3];  for (int i = 0; i < 10000; ++i) {    double theta = (2.0 * i * 0.0011 - 1.0) * kPi;    for (int j = 0; j < 50; ++j) {      double norm2 = 0.0;      for (int k = 0; k < 3; ++k) {        angle_axis[k] = 2.0 * RandDouble() - 1.0;        p[k] = 2.0 * RandDouble() - 1.0;        norm2 = angle_axis[k] * angle_axis[k];      }      const double inv_norm = theta / sqrt(norm2);      for (int k = 0; k < 3; ++k) {        angle_axis[k] *= inv_norm;      }      AngleAxisToRotationMatrix(angle_axis, R);      rotation_matrix_rotated_p[0] = R[0] * p[0] + R[3] * p[1] + R[6] * p[2];      rotation_matrix_rotated_p[1] = R[1] * p[0] + R[4] * p[1] + R[7] * p[2];      rotation_matrix_rotated_p[2] = R[2] * p[0] + R[5] * p[1] + R[8] * p[2];      AngleAxisRotatePoint(angle_axis, p, angle_axis_rotated_p);      for (int k = 0; k < 3; ++k) {        // clang-format off        EXPECT_NEAR(rotation_matrix_rotated_p[k],                    angle_axis_rotated_p[k],                    kTolerance) << "p: " << p[0]                                << " " << p[1]                                << " " << p[2]                                << " angle_axis: " << angle_axis[0]                                << " " << angle_axis[1]                                << " " << angle_axis[2];        // clang-format on      }    }  }}TEST(AngleAxis, NearZeroRotatePointGivesSameAnswerAsRotationMatrix) {  double angle_axis[3];  double R[9];  double p[3];  double angle_axis_rotated_p[3];  double rotation_matrix_rotated_p[3];  for (int i = 0; i < 10000; ++i) {    double norm2 = 0.0;    for (int k = 0; k < 3; ++k) {      angle_axis[k] = 2.0 * RandDouble() - 1.0;      p[k] = 2.0 * RandDouble() - 1.0;      norm2 = angle_axis[k] * angle_axis[k];    }    double theta = (2.0 * i * 0.0001 - 1.0) * 1e-16;    const double inv_norm = theta / sqrt(norm2);    for (int k = 0; k < 3; ++k) {      angle_axis[k] *= inv_norm;    }    AngleAxisToRotationMatrix(angle_axis, R);    rotation_matrix_rotated_p[0] = R[0] * p[0] + R[3] * p[1] + R[6] * p[2];    rotation_matrix_rotated_p[1] = R[1] * p[0] + R[4] * p[1] + R[7] * p[2];    rotation_matrix_rotated_p[2] = R[2] * p[0] + R[5] * p[1] + R[8] * p[2];    AngleAxisRotatePoint(angle_axis, p, angle_axis_rotated_p);    for (int k = 0; k < 3; ++k) {      // clang-format off      EXPECT_NEAR(rotation_matrix_rotated_p[k],                  angle_axis_rotated_p[k],                  kTolerance) << "p: " << p[0]                              << " " << p[1]                              << " " << p[2]                              << " angle_axis: " << angle_axis[0]                              << " " << angle_axis[1]                              << " " << angle_axis[2];      // clang-format on    }  }}TEST(MatrixAdapter, RowMajor3x3ReturnTypeAndAccessIsCorrect) {  double array[9] = {1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0};  const float const_array[9] = {      1.0f, 2.0f, 3.0f, 4.0f, 5.0f, 6.0f, 7.0f, 8.0f, 9.0f};  MatrixAdapter<double, 3, 1> A = RowMajorAdapter3x3(array);  MatrixAdapter<const float, 3, 1> B = RowMajorAdapter3x3(const_array);  for (int i = 0; i < 3; ++i) {    for (int j = 0; j < 3; ++j) {      // The values are integers from 1 to 9, so equality tests are appropriate      // even for float and double values.      EXPECT_EQ(A(i, j), array[3 * i + j]);      EXPECT_EQ(B(i, j), const_array[3 * i + j]);    }  }}TEST(MatrixAdapter, ColumnMajor3x3ReturnTypeAndAccessIsCorrect) {  double array[9] = {1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0};  const float const_array[9] = {      1.0f, 2.0f, 3.0f, 4.0f, 5.0f, 6.0f, 7.0f, 8.0f, 9.0f};  MatrixAdapter<double, 1, 3> A = ColumnMajorAdapter3x3(array);  MatrixAdapter<const float, 1, 3> B = ColumnMajorAdapter3x3(const_array);  for (int i = 0; i < 3; ++i) {    for (int j = 0; j < 3; ++j) {      // The values are integers from 1 to 9, so equality tests are      // appropriate even for float and double values.      EXPECT_EQ(A(i, j), array[3 * j + i]);      EXPECT_EQ(B(i, j), const_array[3 * j + i]);    }  }}TEST(MatrixAdapter, RowMajor2x4IsCorrect) {  const int expected[8] = {1, 2, 3, 4, 5, 6, 7, 8};  int array[8];  MatrixAdapter<int, 4, 1> M(array);  // clang-format off  M(0, 0) = 1; M(0, 1) = 2; M(0, 2) = 3; M(0, 3) = 4;  M(1, 0) = 5; M(1, 1) = 6; M(1, 2) = 7; M(1, 3) = 8;  // clang-format on  for (int k = 0; k < 8; ++k) {    EXPECT_EQ(array[k], expected[k]);  }}TEST(MatrixAdapter, ColumnMajor2x4IsCorrect) {  const int expected[8] = {1, 5, 2, 6, 3, 7, 4, 8};  int array[8];  MatrixAdapter<int, 1, 2> M(array);  // clang-format off  M(0, 0) = 1; M(0, 1) = 2; M(0, 2) = 3; M(0, 3) = 4;  M(1, 0) = 5; M(1, 1) = 6; M(1, 2) = 7; M(1, 3) = 8;  // clang-format on  for (int k = 0; k < 8; ++k) {    EXPECT_EQ(array[k], expected[k]);  }}TEST(RotationMatrixToAngleAxis, NearPiExampleOneFromTobiasStrauss) {  // Example from Tobias Strauss  // clang-format off  const double rotation_matrix[] = {    -0.999807135425239,    -0.0128154391194470,   -0.0148814136745799,    -0.0128154391194470,   -0.148441438622958,     0.988838158557669,    -0.0148814136745799,    0.988838158557669,     0.148248574048196  };  // clang-format on  double angle_axis[3];  RotationMatrixToAngleAxis(RowMajorAdapter3x3(rotation_matrix), angle_axis);  double round_trip[9];  AngleAxisToRotationMatrix(angle_axis, RowMajorAdapter3x3(round_trip));  EXPECT_THAT(rotation_matrix, IsNear3x3Matrix(round_trip));}static void CheckRotationMatrixToAngleAxisRoundTrip(const double theta,                                                    const double phi,                                                    const double angle) {  double angle_axis[3];  angle_axis[0] = angle * sin(phi) * cos(theta);  angle_axis[1] = angle * sin(phi) * sin(theta);  angle_axis[2] = angle * cos(phi);  double rotation_matrix[9];  AngleAxisToRotationMatrix(angle_axis, rotation_matrix);  double angle_axis_round_trip[3];  RotationMatrixToAngleAxis(rotation_matrix, angle_axis_round_trip);  EXPECT_THAT(angle_axis_round_trip, IsNearAngleAxis(angle_axis));}TEST(RotationMatrixToAngleAxis, ExhaustiveRoundTrip) {  const double kMaxSmallAngle = 1e-8;  const int kNumSteps = 1000;  for (int i = 0; i < kNumSteps; ++i) {    const double theta = static_cast<double>(i) / kNumSteps * 2.0 * kPi;    for (int j = 0; j < kNumSteps; ++j) {      const double phi = static_cast<double>(j) / kNumSteps * kPi;      // Rotations of angle Pi.      CheckRotationMatrixToAngleAxisRoundTrip(theta, phi, kPi);      // Rotation of angle approximately Pi.      CheckRotationMatrixToAngleAxisRoundTrip(          theta, phi, kPi - kMaxSmallAngle * RandDouble());      // Rotations of angle approximately zero.      CheckRotationMatrixToAngleAxisRoundTrip(          theta, phi, kMaxSmallAngle * 2.0 * RandDouble() - 1.0);    }  }}}  // namespace internal}  // namespace ceres
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