text: # Rigid motion in curvilinear coordinates
  
  I would like someone to clarify this since it has bedazzled me and can't seem to get a grip on it. Consider a 3D real space and Euclidean coordinates ($x_1,x_2,x_3$), with an associated standard basis E={$e_1,e_2,e_3$}. Now also consider new curvilinear coordinates ($y_1,y_2,y_3$) such that $y_i=f_i(x_1,x_2,x_3)$. To each point y there is a local covariant basis $g_i=\frac{\partial x_j}{\partial y_i}e_j$ (summation convention in use. In matrix notation just consider G=J E, with J a $3\times 3$ matrix). The equation relates the Euclidean frame and the local frame.
  
  First question: It is my understanding that vectors live on the tangent space of the base space at point y. Thus a vector at y (or the respective x) lives in a different space than a vector at the origin. In other words, the local basis and the Euclidean basis are defined at different points. How can then, they be associated? i.e. how can G=J E hold since the two frames are defined at different points?
  
  (In a flat space space, one could transport parallely the E frame, in a very trivial way, from the origin to the point x and then associated the two frames at the same point. In general manifolds one should define a connection.)
  
  Second question: Why does this matter? Because in order to describe a rigid motion, you also need the coordinates of the origin i.e. you need an affine space. Thus, should the relation between the E frame (at the origin) and the local frame G (at x) be something like:
  
  $$g_i=\frac{\partial x_j}{\partial y_i}e_j'$$
  
  where,
  
  $$e_j'=e_j+T$$
  
  that is, the transported E at the point x?
  
  EDIT:
  
  Regarding my second question, should the transformation from global to local coordinates be defined as, $$\begin{bmatrix} u_1\\ u_2\\ u_3\\ 0 \end{bmatrix}=\begin{bmatrix} J & \mathbf{x}\\ \mathbf{0}& 1 \end{bmatrix}\begin{bmatrix} v_1\\ v_2\\ v_3\\ 0 \end{bmatrix}$$
  
  where $v_i$ are the components of a vector located at x in the global frame , and $u$ the components in the local frame at x, and $J=[\frac{\partial x_j}{\partial y_i}]$ (i.e. G=J E), using of course homogenous coordinates.
  
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distilabel_metadata.role: system
distilabel_metadata.content: # Rigid motion in curvilinear coordinates
  
  I would like someone to clarify this since it has bedazzled me and can't seem to get a grip on it. Consider a 3D real space and Euclidean coordinates ($x_1,x_2,x_3$), with an associated standard basis E={$e_1,e_2,e_3$}. Now also consider new curvilinear coordinates ($y_1,y_2,y_3$) such that $y_i=f_i(x_1,x_2,x_3)$. To each point y there is a local covariant basis $g_i=\frac{\partial x_j}{\partial y_i}e_j$ (summation convention in use. In matrix notation just consider G=J E, with J a $3\times 3$ matrix). The equation relates the Euclidean frame and the local frame.
  
  First question: It is my understanding that vectors live on the tangent space of the base space at point y. Thus a vector at y (or the respective x) lives in a different space than a vector at the origin. In other words, the local basis and the Euclidean basis are defined at different points. How can then, they be associated? i.e. how can G=J E hold since the two frames are defined at different points?
  
  (In a flat space space, one could transport parallely the E frame, in a very trivial way, from the origin to the point x and then associated the two frames at the same point. In general manifolds one should define a connection.)
  
  Second question: Why does this matter? Because in order to describe a rigid motion, you also need the coordinates of the origin i.e. you need an affine space. Thus, should the relation between the E frame (at the origin) and the local frame G (at x) be something like:
  
  $$g_i=\frac{\partial x_j}{\partial y_i}e_j'$$
  
  where,
  
  $$e_j'=e_j+T$$
  
  that is, the transported E at the point x?
  
  EDIT:
  
  Regarding my second question, should the transformation from global to local coordinates be defined as, $$\begin{bmatrix} u_1\\ u_2\\ u_3\\ 0 \end{bmatrix}=\begin{bmatrix} J & \mathbf{x}\\ \mathbf{0}& 1 \end{bmatrix}\begin{bmatrix} v_1\\ v_2\\ v_3\\ 0 \end{bmatrix}$$
  
  where $v_i$ are the components of a vector located at x in the global frame , and $u$ the components in the local frame at x, and $J=[\frac{\partial x_j}{\partial y_i}]$ (i.e. G=J E), using of course homogenous coordinates.
  
  -
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The text extract discusses the transformation between Euclidean coordinates and curvilinear coordinates in a 3D real space. It introduces the concept of a local covariant basis and raises two specific questions regarding the relationship between the Euclidean and local frames, and the implications for describing rigid motions.
  
  1. **First Question**: The author asks how the Euclidean basis and the local basis can be associated, given that they are defined at different points. The author suggests that in a flat space, parallel transport can trivially align the bases, but in a general manifold, a connection is needed.
  
  2. **Second Question**: The author inquires why this matters for describing rigid motions, suggesting that the transformation should account for the coordinates of the origin and the transported Euclidean frame.
  
  The text also includes an edit where the author proposes a transformation using homogeneous coordinates to address the second question.
  
  **Label**: multiple-question-and-answer