Rotation<Frenet<ThisFrame>, ThisFrame>
DynamicFrame<InertialFrame, ThisFrame>::FrenetFrame(
Instant const& t,
DegreesOfFreedom<ThisFrame> const& degrees_of_freedom) const {
Velocity<ThisFrame> const& velocity = degrees_of_freedom.velocity();
Vector<Acceleration, ThisFrame> const acceleration =
GeometricAcceleration(t, degrees_of_freedom);
Vector<Acceleration, ThisFrame> normal_acceleration = acceleration;
velocity.template Orthogonalize<Acceleration>(&normal_acceleration);
Vector<double, ThisFrame> tangent = Normalize(velocity);
Vector<double, ThisFrame> normal = Normalize(normal_acceleration);
Bivector<double, ThisFrame> binormal = Wedge(tangent, normal);
// Maps |tangent| to {1, 0, 0}, |normal| to {0, 1, 0}, and |binormal| to
// {0, 0, 1}.
return Rotation<Frenet<ThisFrame>, ThisFrame>(
R3x3Matrix(tangent.coordinates(),
normal.coordinates(),
binormal.coordinates()).Transpose());
}
Bivector<Scalar, ToFrame> Identity<FromFrame, ToFrame>::operator()(
Bivector<Scalar, FromFrame> const& bivector) const {
return Bivector<Scalar, ToFrame>(bivector.coordinates());
}
Vector<quantities::Product<LScalar, RScalar>, Frame> operator*(
Trivector<LScalar, Frame> const& left,
Bivector<RScalar, Frame> const& right) {
return Vector<quantities::Product<LScalar, RScalar>, Frame>(
left.coordinates() * right.coordinates());
}
Bivector<quantities::Product<LScalar, RScalar>, Frame> Commutator(
Bivector<LScalar, Frame> const& left,
Bivector<RScalar, Frame> const& right) {
return Bivector<quantities::Product<LScalar, RScalar>, Frame>(
Cross(left.coordinates(), right.coordinates()));
}
Trivector<quantities::Product<LScalar, RScalar>, Frame> Wedge(
Vector<LScalar, Frame> const& left,
Bivector<RScalar, Frame> const& right) {
return Trivector<quantities::Product<LScalar, RScalar>, Frame>(
Dot(left.coordinates(), right.coordinates()));
}
quantities::Product<LScalar, RScalar> InnerProduct(
Bivector<LScalar, Frame> const& left,
Bivector<RScalar, Frame> const& right) {
return Dot(left.coordinates(), right.coordinates());
}
inline Vector<quantities::Product<LScalar, RScalar>, Frame> operator*(
Bivector<LScalar, Frame> const& left,
Vector<RScalar, Frame> const& right) {
return Vector<quantities::Product<LScalar, RScalar>, Frame>(
Cross(left.coordinates(), right.coordinates()));
}
Bivector<Scalar, ToFrame> Permutation<FromFrame, ToFrame>::operator()(
Bivector<Scalar, FromFrame> const& bivector) const {
return Bivector<Scalar, ToFrame>(
Determinant() * (*this)(bivector.coordinates()));
}
