This may be a basic question but I am having trouble figuring out the correct answer. I am trying to interpolate the local coordinate charts between two points $p_1$, $p_2 \in \mathbb{R}^N$. The data I have is the ambient intrinsic dimension of the manifold (dimension d), the two points $p_1, p_2$, and an orthonormal basis for the tangent space at each point. I would like to find the coordinates $(f_1(x_1, x_2, \ldots, x_d), f_2(x_1,x_2,\dots,x_d), \ldots, f_N(x_1,x_2,\ldots,x_d))$ that could fit these restrictions. I would like for each $f_i$ to be a polynomial function in $(x_1,x_2,\ldots, x_d)$ that best fits the data. There is obvious ambiguity in this problem because a choice for $(x_1, x_2, \ldots, x_d)$ has to be made for $p_1$ and $p_2$. I am wondering if there is a solution to this problem despite this ambiguity. If anyone knows of any references to for this type of problem that would also be appreciated.

Edit: I have reworded the question.

This may be a basic question but I am having trouble figuring out the correct answer. I want to find a local coordinate chart that fits a d-dimensional submanifold in $\mathbb{R}^N$. I am given two points $p_1, p_2 \in \mathbb{R}^N$ and corresponding orthonormal bases $(\phi_1, \phi_2, \ldots, \phi_d)$, $(\tau_1, \tau_2, \ldots, \tau_d) \subseteq \mathbb{R}^N$ for their tangent spaces. I would like to find an algorithmic method for finding functions $(f_1(x_1,x_2,\ldots,x_d), f_2(x_1,x_2,\ldots,x_d),\ldots f_N(x_1,x_2,\ldots,x_d))$ that satisfy these conditions.