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Given a normal vector in 3D-Space, how can I rotate the vector, such it points to a point in 3D-Space.

I tried couple of ways doing this, which ended up looking completely wrong.

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    $\begingroup$ If the answer from @lightxbulb isn't what you are looking for then perhaps you could clarify your questions some. $\endgroup$
    – pmw1234
    Oct 11, 2022 at 18:27
  • $\begingroup$ Vectors do not have position, only direction and magnitude, so creating a vector that "points to a point in 3D-Space" isn't possible. $\endgroup$
    – pmw1234
    Nov 7, 2022 at 17:08

1 Answer 1

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Let $e,f$ be the two edges of the triangle. Let's construct an orthonormal system: $e = normalize(e)$, $n = normalize(e\times f)$, $f = n \times e$, where $\times$ is the cross product. Now compute:

$$e' = e - \frac{e \cdot n'}{1+(n\cdot n')}(n+n')$$

$$f' = f - \frac{f \cdot n'}{1+(n\cdot n')}(n+n')$$

Then the rotation matrix is:

$$R = \begin{bmatrix} e' & f' & n'\end{bmatrix} \begin{bmatrix} e^T \\ f^T \\ n^T \end{bmatrix}.$$

Note that the computation of $e'$ and $f'$ breaks for $n' = -n$.

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