About Unit3 and Point3

[BLB0803]

I already know that Point3 represents a 3D point, and from the name, Unit3 seems to represent a three-dimensional unit vector.
So I thought that Unit3 means:

                   v       <- A Point3
A Unit3 ->  u = -------
                || v ||

But I saw this function in Unit3.cpp:

// in Unit3.cpp
Unit3 Unit3::FromPoint3(const Point3& point, OptionalJacobian<2, 3> H) {
  // 3*3 Derivative of representation with respect to point is 3*3:
  Matrix3 D_p_point;
  Unit3 direction;
  direction.p_ = normalize(point, H ? &D_p_point : 0);
  if (H)
    *H << direction.basis().transpose() * D_p_point;
  return direction;
}

I thought the Jacobian matrix was only the D_p_point, but there is also direction.basis().transpose() in the code.
Is it my understanding of Unit3 that is wrong or my understanding of the derivative of Unit3 to Point3 is wrong?
Thanks very much if anyone can answer my doubts.

---

I found a document and got what I want, the Unit3 is not a simple 3D-vector but a point on sphere S^2, so I closed the question.
If someone have the same question could check the link below, and see 14-The Sphere S^2:
/gtsam/blob/develop/doc/math.pdf