text: #include "IDMath.hpp"
  
  #include <cmath>
  #include <limits>
  
  namespace btInverseDynamics {
  static const idScalar kIsZero = 5 * std::numeric_limits<idScalar>::epsilon();
  // requirements for axis length deviation from 1.0
  // experimentally set from random euler angle rotation matrices
  static const idScalar kAxisLengthEpsilon = 10 * kIsZero;
  
  void setZero(vec3 &v) {
  	v(0) = 0;
  	v(1) = 0;
  	v(2) = 0;
  }
  
  void setZero(vecx &v) {
  	for (int i = 0; i < v.size(); i++) {
  		v(i) = 0;
  	}
  }
  
  void setZero(mat33 &m) {
  	m(0, 0) = 0;
  	m(0, 1) = 0;
  	m(0, 2) = 0;
  	m(1, 0) = 0;
  	m(1, 1) = 0;
  	m(1, 2) = 0;
  	m(2, 0) = 0;
  	m(2, 1) = 0;
  	m(2, 2) = 0;
  }
  
  void skew(vec3& v, mat33* result) {
  	(*result)(0, 0) = 0.0;
  	(*result)(0, 1) = -v(2);
  	(*result)(0, 2) = v(1);
  	(*result)(1, 0) = v(2);
  	(*result)(1, 1) = 0.0;
  	(*result)(1, 2) = -v(0);
  	(*result)(2, 0) = -v(1);
  	(*result)(2, 1) = v(0);
  	(*result)(2, 2) = 0.0;
  }
  
  idScalar maxAbs(const vecx &v) {
  	idScalar result = 0.0;
  	for (int i = 0; i < v.size(); i++) {
  		const idScalar tmp = BT_ID_FABS(v(i));
  		if (tmp > result) {
  			result = tmp;
  		}
  	}
  	return result;
  }
  
  idScalar maxAbs(const vec3 &v) {
  	idScalar result = 0.0;
  	for (int i = 0; i < 3; i++) {
  		const idScalar tmp = BT_ID_FABS(v(i));
  		if (tmp > result) {
  			result = tmp;
  		}
  	}
  	return result;
  }
  
  #if (defined BT_ID_HAVE_MAT3X)
  idScalar maxAbsMat3x(const mat3x &m) {
      // only used for tests -- so just loop here for portability
      idScalar result = 0.0;
      for (idArrayIdx col = 0; col < m.cols(); col++) {
          for (idArrayIdx row = 0; row < 3; row++) {
              result = BT_ID_MAX(result, std::fabs(m(row, col)));
          }
      }
      return result;
  }
  
  void mul(const mat33 &a, const mat3x &b, mat3x *result) {
      if (b.cols() != result->cols()) {
          bt_id_error_message("size missmatch. b.cols()= %d, result->cols()= %d\n",
                        static_cast<int>(b.cols()), static_cast<int>(result->cols()));
          abort();
      }
  
      for (idArrayIdx col = 0; col < b.cols(); col++) {
          const idScalar x = a(0,0)*b(0,col)+a(0,1)*b(1,col)+a(0,2)*b(2,col);
          const idScalar y = a(1,0)*b(0,col)+a(1,1)*b(1,col)+a(1,2)*b(2,col);
          const idScalar z = a(2,0)*b(0,col)+a(2,1)*b(1,col)+a(2,2)*b(2,col);
          setMat3xElem(0, col, x, result);
          setMat3xElem(1, col, y, result);
          setMat3xElem(2, col, z, result);
      }
  }
  void add(const mat3x &a, const mat3x &b, mat3x *result) {
      if (a.cols() != b.cols()) {
          bt_id_error_message("size missmatch. a.cols()= %d, b.cols()= %d\n",
                        static_cast<int>(a.cols()), static_cast<int>(b.cols()));
          abort();
      }
      for (idArrayIdx col = 0; col < b.cols(); col++) {
          for (idArrayIdx row = 0; row < 3; row++) {
              setMat3xElem(row, col, a(row, col) + b(row, col), result);
          }
      }
  }
  void sub(const mat3x &a, const mat3x &b, mat3x *result) {
      if (a.cols() != b.cols()) {
          bt_id_error_message("size missmatch. a.cols()= %d, b.cols()= %d\n",
                        static_cast<int>(a.cols()), static_cast<int>(b.cols()));
          abort();
      }
      for (idArrayIdx col = 0; col < b.cols(); col++) {
          for (idArrayIdx row = 0; row < 3; row++) {
              setMat3xElem(row, col, a(row, col) - b(row, col), result);
          }
      }
  }
  #endif
  
  mat33 transformX(const idScalar &alpha) {
  	mat33 T;
  	const idScalar cos_alpha = BT_ID_COS(alpha);
  	const idScalar sin_alpha = BT_ID_SIN(alpha);
  	// [1  0 0]
  	// [0  c s]
  	// [0 -s c]
  	T(0, 0) = 1.0;
  	T(0, 1) = 0.0;
  	T(0, 2) = 0.0;
  
  	T(1, 0) = 0.0;
  	T(1, 1) = cos_alpha;
  	T(1, 2) = sin_alpha;
  
  	T(2, 0) = 0.0;
  	T(2, 1) = -sin_alpha;
  	T(2, 2) = cos_alpha;
  
  	return T;
  }
  
  mat33 transformY(const idScalar &beta) {
  	mat33 T;
  	const idScalar cos_beta = BT_ID_COS(beta);
  	const idScalar sin_beta = BT_ID_SIN(beta);
  	// [c 0 -s]
  	// [0 1  0]
  	// [s 0  c]
  	T(0, 0) = cos_beta;
  	T(0, 1) = 0.0;
  	T(0, 2) = -sin_beta;
  
  	T(1, 0) = 0.0;
  	T(1, 1) = 1.0;
  	T(1, 2) = 0.0;
  
  	T(2, 0) = sin_beta;
  	T(2, 1) = 0.0;
  	T(2, 2) = cos_beta;
  
  	return T;
  }
  
  mat33 transformZ(const idScalar &gamma) {
  	mat33 T;
  	const idScalar cos_gamma = BT_ID_COS(gamma);
  	const idScalar sin_gamma = BT_ID_SIN(gamma);
  	// [ c s 0]
  	// [-s c 0]
  	// [ 0 0 1]
  	T(0, 0) = cos_gamma;
  	T(0, 1) = sin_gamma;
  	T(0, 2) = 0.0;
  
  	T(1, 0) = -sin_gamma;
  	T(1, 1) = cos_gamma;
  	T(1, 2) = 0.0;
  
  	T(2, 0) = 0.0;
  	T(2, 1) = 0.0;
  	T(2, 2) = 1.0;
  
  	return T;
  }
  
  mat33 tildeOperator(const vec3 &v) {
  	mat33 m;
  	m(0, 0) = 0.0;
  	m(0, 1) = -v(2);
  	m(0, 2) = v(1);
  	m(1, 0) = v(2);
  	m(1, 1) = 0.0;
  	m(1, 2) = -v(0);
  	m(2, 0) = -v(1);
  	m(2, 1) = v(0);
  	m(2, 2) = 0.0;
  	return m;
  }
  
  void getVecMatFromDH(idScalar theta, idScalar d, idScalar a, idScalar alpha, vec3 *r, mat33 *T) {
  	const idScalar sa = BT_ID_SIN(alpha);
  	const idScalar ca = BT_ID_COS(alpha);
  	const idScalar st = BT_ID_SIN(theta);
  	const idScalar ct = BT_ID_COS(theta);
  
  	(*r)(0) = a;
  	(*r)(1) = -sa * d;
  	(*r)(2) = ca * d;
  
  	(*T)(0, 0) = ct;
  	(*T)(0, 1) = -st;
  	(*T)(0, 2) = 0.0;
  
  	(*T)(1, 0) = st * ca;
  	(*T)(1, 1) = ct * ca;
  	(*T)(1, 2) = -sa;
  
  	(*T)(2, 0) = st * sa;
  	(*T)(2, 1) = ct * sa;
  	(*T)(2, 2) = ca;
  }
  
  void bodyTParentFromAxisAngle(const vec3 &axis, const idScalar &angle, mat33 *T) {
  	const idScalar c = BT_ID_COS(angle);
  	const idScalar s = -BT_ID_SIN(angle);
  	const idScalar one_m_c = 1.0 - c;
  
  	const idScalar &x = axis(0);
  	const idScalar &y = axis(1);
  	const idScalar &z = axis(2);
  
  	(*T)(0, 0) = x * x * one_m_c + c;
  	(*T)(0, 1) = x * y * one_m_c - z * s;
  	(*T)(0, 2) = x * z * one_m_c + y * s;
  
  	(*T)(1, 0) = x * y * one_m_c + z * s;
  	(*T)(1, 1) = y * y * one_m_c + c;
  	(*T)(1, 2) = y * z * one_m_c - x * s;
  
  	(*T)(2, 0) = x * z * one_m_c - y * s;
  	(*T)(2, 1) = y * z * one_m_c + x * s;
  	(*T)(2, 2) = z * z * one_m_c + c;
  }
  
  bool isPositiveDefinite(const mat33 &m) {
  	// test if all upper left determinants are positive
  	if (m(0, 0) <= 0) {  // upper 1x1
  		return false;
  	}
  	if (m(0, 0) * m(1, 1) - m(0, 1) * m(1, 0) <= 0) {  // upper 2x2
  		return false;
  	}
  	if ((m(0, 0) * (m(1, 1) * m(2, 2) - m(1, 2) * m(2, 1)) -
  		 m(0, 1) * (m(1, 0) * m(2, 2) - m(1, 2) * m(2, 0)) +
  		 m(0, 2) * (m(1, 0) * m(2, 1) - m(1, 1) * m(2, 0))) < 0) {
  		return false;
  	}
  	return true;
  }
  
  bool isPositiveSemiDefinite(const mat33 &m) {
  	// test if all upper left determinants are positive
  	if (m(0, 0) < 0) {  // upper 1x1
  		return false;
  	}
  	if (m(0, 0) * m(1, 1) - m(0, 1) * m(1, 0) < 0) {  // upper 2x2
  		return false;
  	}
  	if ((m(0, 0) * (m(1, 1) * m(2, 2) - m(1, 2) * m(2, 1)) -
  		 m(0, 1) * (m(1, 0) * m(2, 2) - m(1, 2) * m(2, 0)) +
  		 m(0, 2) * (m(1, 0) * m(2, 1) - m(1, 1) * m(2, 0))) < 0) {
  		return false;
  	}
  	return true;
  }
  
  bool isPositiveSemiDefiniteFuzzy(const mat33 &m) {
  	// test if all upper left determinants are positive
  	if (m(0, 0) < -kIsZero) {  // upper 1x1
  		return false;
  	}
  	if (m(0, 0) * m(1, 1) - m(0, 1) * m(1, 0) < -kIsZero) {  // upper 2x2
  		return false;
  	}
  	if ((m(0, 0) * (m(1, 1) * m(2, 2) - m(1, 2) * m(2, 1)) -
  		 m(0, 1) * (m(1, 0) * m(2, 2) - m(1, 2) * m(2, 0)) +
  		 m(0, 2) * (m(1, 0) * m(2, 1) - m(1, 1) * m(2, 0))) < -kIsZero) {
  		return false;
  	}
  	return true;
  }
  
  idScalar determinant(const mat33 &m) {
  	return m(0, 0) * m(1, 1) * m(2, 2) + m(0, 1) * m(1, 2) * m(2, 0) + m(0, 2) * m(1, 0) * m(2, 1) -
  		   m(0, 2) * m(1, 1) * m(2, 0) - m(0, 0) * m(1, 2) * m(2, 1) - m(0, 1) * m(1, 0) * m(2, 2);
  }
  
  bool isValidInertiaMatrix(const mat33 &I, const int index, bool has_fixed_joint) {
  	// TODO(Thomas) do we really want this?
  	//			  in cases where the inertia tensor about the center of mass is zero,
  	//			  the determinant of the inertia tensor about the joint axis is almost
  	//			  zero and can have a very small negative value.
  	if (!isPositiveSemiDefiniteFuzzy(I)) {
  		bt_id_error_message("invalid inertia matrix for body %d, not positive definite "
  					  "(fixed joint)\n",
  					  index);
  		bt_id_error_message("matrix is:\n"
  					  "[%.20e %.20e %.20e;\n"
  					  "%.20e %.20e %.20e;\n"
  					  "%.20e %.20e %.20e]\n",
  					  I(0, 0), I(0, 1), I(0, 2), I(1, 0), I(1, 1), I(1, 2), I(2, 0), I(2, 1),
  					  I(2, 2));
  
  		return false;
  	}
  
  	// check triangle inequality, must have I(i,i)+I(j,j)>=I(k,k)
  	if (!has_fixed_joint) {
  		if (I(0, 0) + I(1, 1) < I(2, 2)) {
  			bt_id_error_message("invalid inertia tensor for body %d, I(0,0) + I(1,1) < I(2,2)\n", index);
  			bt_id_error_message("matrix is:\n"
  						  "[%.20e %.20e %.20e;\n"
  						  "%.20e %.20e %.20e;\n"
  						  "%.20e %.20e %.20e]\n",
  						  I(0, 0), I(0, 1), I(0, 2), I(1, 0), I(1, 1), I(1, 2), I(2, 0), I(2, 1),
  						  I(2, 2));
  			return false;
  		}
  		if (I(0, 0) + I(1, 1) < I(2, 2)) {
  			bt_id_error_message("invalid inertia tensor for body %d, I(0,0) + I(1,1) < I(2,2)\n", index);
  			bt_id_error_message("matrix is:\n"
  						  "[%.20e %.20e %.20e;\n"
  						  "%.20e %.20e %.20e;\n"
  						  "%.20e %.20e %.20e]\n",
  						  I(0, 0), I(0, 1), I(0, 2), I(1, 0), I(1, 1), I(1, 2), I(2, 0), I(2, 1),
  						  I(2, 2));
  			return false;
  		}
  		if (I(1, 1) + I(2, 2) < I(0, 0)) {
  			bt_id_error_message("invalid inertia tensor for body %d, I(1,1) + I(2,2) < I(0,0)\n", index);
  			bt_id_error_message("matrix is:\n"
  						  "[%.20e %.20e %.20e;\n"
  						  "%.20e %.20e %.20e;\n"
  						  "%.20e %.20e %.20e]\n",
  						  I(0, 0), I(0, 1), I(0, 2), I(1, 0), I(1, 1), I(1, 2), I(2, 0), I(2, 1),
  						  I(2, 2));
  			return false;
  		}
  	}
  	// check positive/zero diagonal elements
  	for (int i = 0; i < 3; i++) {
  		if (I(i, i) < 0) {  // accept zero
  			bt_id_error_message("invalid inertia tensor, I(%d,%d)= %e <0\n", i, i, I(i, i));
  			return false;
  		}
  	}
  	// check symmetry
  	if (BT_ID_FABS(I(1, 0) - I(0, 1)) > kIsZero) {
  		bt_id_error_message("invalid inertia tensor for body %d I(1,0)!=I(0,1). I(1,0)-I(0,1)= "
  					  "%e\n",
  					  index, I(1, 0) - I(0, 1));
  		return false;
  	}
  	if (BT_ID_FABS(I(2, 0) - I(0, 2)) > kIsZero) {
  		bt_id_error_message("invalid inertia tensor for body %d I(2,0)!=I(0,2). I(2,0)-I(0,2)= "
  					  "%e\n",
  					  index, I(2, 0) - I(0, 2));
  		return false;
  	}
  	if (BT_ID_FABS(I(1, 2) - I(2, 1)) > kIsZero) {
  		bt_id_error_message("invalid inertia tensor body %d I(1,2)!=I(2,1). I(1,2)-I(2,1)= %e\n", index,
  					  I(1, 2) - I(2, 1));
  		return false;
  	}
  	return true;
  }
  
  bool isValidTransformMatrix(const mat33 &m) {
  #define print_mat(x)																			   \
  	bt_id_error_message("matrix is [%e, %e, %e; %e, %e, %e; %e, %e, %e]\n", x(0, 0), x(0, 1), x(0, 2),   \
  				  x(1, 0), x(1, 1), x(1, 2), x(2, 0), x(2, 1), x(2, 2))
  
  	// check for unit length column vectors
  	for (int i = 0; i < 3; i++) {
  		const idScalar length_minus_1 =
  			BT_ID_FABS(m(0, i) * m(0, i) + m(1, i) * m(1, i) + m(2, i) * m(2, i) - 1.0);
  		if (length_minus_1 > kAxisLengthEpsilon) {
  			bt_id_error_message("Not a valid rotation matrix (column %d not unit length)\n"
  						  "column = [%.18e %.18e %.18e]\n"
  						  "length-1.0= %.18e\n",
  						  i, m(0, i), m(1, i), m(2, i), length_minus_1);
  			print_mat(m);
  			return false;
  		}
  	}
  	// check for orthogonal column vectors
  	if (BT_ID_FABS(m(0, 0) * m(0, 1) + m(1, 0) * m(1, 1) + m(2, 0) * m(2, 1)) > kAxisLengthEpsilon) {
  		bt_id_error_message("Not a valid rotation matrix (columns 0 and 1 not orthogonal)\n");
  		print_mat(m);
  		return false;
  	}
  	if (BT_ID_FABS(m(0, 0) * m(0, 2) + m(1, 0) * m(1, 2) + m(2, 0) * m(2, 2)) > kAxisLengthEpsilon) {
  		bt_id_error_message("Not a valid rotation matrix (columns 0 and 2 not orthogonal)\n");
  		print_mat(m);
  		return false;
  	}
  	if (BT_ID_FABS(m(0, 1) * m(0, 2) + m(1, 1) * m(1, 2) + m(2, 1) * m(2, 2)) > kAxisLengthEpsilon) {
  		bt_id_error_message("Not a valid rotation matrix (columns 0 and 2 not orthogonal)\n");
  		print_mat(m);
  		return false;
  	}
  	// check determinant (rotation not reflection)
  	if (determinant(m) <= 0) {
  		bt_id_error_message("Not a valid rotation matrix (determinant <=0)\n");
  		print_mat(m);
  		return false;
  	}
  	return true;
  }
  
  bool isUnitVector(const vec3 &vector) {
  	return BT_ID_FABS(vector(0) * vector(0) + vector(1) * vector(1) + vector(2) * vector(2) - 1.0) <
  		   kIsZero;
  }
  
  vec3 rpyFromMatrix(const mat33 &rot) {
  	vec3 rpy;
  	rpy(2) = BT_ID_ATAN2(-rot(1, 0), rot(0, 0));
  	rpy(1) = BT_ID_ATAN2(rot(2, 0), BT_ID_COS(rpy(2)) * rot(0, 0) - BT_ID_SIN(rpy(0)) * rot(1, 0));
  	rpy(0) = BT_ID_ATAN2(-rot(2, 0), rot(2, 2));
  	return rpy;
  }
  }
  