<p>I wasn't familiar with the &quot;Jacobi transformation&quot;, and after googling, it seems there are multiple things with that name; but given the mention of eigenvalues, I'm guessing they were referring to <a href="https://mathworld.wolfram.com/JacobiTransformation.html" rel="nofollow noreferrer">a method of matrix diagonalization</a> using &quot;Jacobi rotation matrices&quot;, which are better known as <a href="https://en.wikipedia.org/wiki/Givens_rotation" rel="nofollow noreferrer">Givens rotations</a>.</p>
<p>In other words, the Jacobi transformation would be a method of finding eigenvalues and eigenvectors of a matrix. (There are many methods for doing that, and you can read more elsewhere about exactly what those methods are and when one should use one method or another.)</p>
<p>As for the use of eigenvalues/vectors in graphics, here are a few examples:</p>
<ul>
<li>Decomposing a transformation matrix into rotation and scaling parts</li>
<li>Finding directions of principal curvature on a mesh</li>
<li>Fitting an oriented bounding box to a mesh</li>
<li>In texture compression: finding a line segment in RGB space to best match a block of pixels</li>
<li>In animation: finding a set of blend shapes to best match a captured or simulated vertex deformation</li>
</ul>

CC BY-SA 4.0
2021-11-27T07:20:35.543Z
2021-11-27T07:20:35.543Z
Nathan Reed
<p>I have been going through scratchapixel.com lessons. Unfortunately some of the lessons are incomplete and one of the missing chapters is titled &quot;Jacobi Transformation and Eigenvalue Algorithm&quot;. Can someone explain its usages in CG (not physics) as well as give an example or two?</p>

<linear-algebra>
Usage of Jacobi transformation in computer graphics
PentaKon
# Usage of Jacobi transformation in computer graphics

(Asked by: username_0 on 2021-11-26)

I have been going through scratchapixel.com lessons. Unfortunately some of the lessons are incomplete and one of the missing chapters is titled "Jacobi Transformation and Eigenvalue Algorithm". Can someone explain its usages in CG (not physics) as well as give an example or two?

**Tags:** linear-algebra


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## 1 answers

### Answer by username_1 on 2021-11-27. Score: 2 | Accepted answer

I wasn't familiar with the "Jacobi transformation", and after googling, it seems there are multiple things with that name; but given the mention of eigenvalues, I'm guessing they were referring to [a method of matrix diagonalization](https://mathworld.wolfram.com/JacobiTransformation.html) using "Jacobi rotation matrices", which are better known as [Givens rotations](https://en.wikipedia.org/wiki/Givens_rotation).

In other words, the Jacobi transformation would be a method of finding eigenvalues and eigenvectors of a matrix. (There are many methods for doing that, and you can read more elsewhere about exactly what those methods are and when one should use one method or another.)

As for the use of eigenvalues/vectors in graphics, here are a few examples:

* Decomposing a transformation matrix into rotation and scaling parts
* Finding directions of principal curvature on a mesh
* Fitting an oriented bounding box to a mesh
* In texture compression: finding a line segment in RGB space to best match a block of pixels
* In animation: finding a set of blend shapes to best match a captured or simulated vertex deformation


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