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I have a set of $N$ points $(x_i, y_i, z_i)$ and I need to find the best plane fit to them. So far I've been numerically minimizing the sum of the absolute valued point-plane distances:

$$f(a, b, c, d)= \sum_{i=1}^{N} \frac{|ax_i + by_i + cz_i + d|}{\sqrt{a^2 + b^2 + c^2}}$$

which works. But recently I found out about the singular value decomposition method, which as far as I understand, analytically minimizes the squared point-plane distances:

$$f(a, b, c, d)= \sum_{i=1}^{N} \frac{(ax_i + by_i + cz_i + d)^2}{a^2 + b^2 + c^2}$$

Keeping in mind that I am not a mathematician, could you explain step by step why (or how) the SVD method minimizes the squared point-plane distance?

Gabriel
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  • Have a look at https://math.stackexchange.com/questions/99299/best-fitting-plane-given-a-set-of-points – Claude Leibovici Nov 11 '17 at 07:03
  • @ClaudeLeibovici I actually come from that question (is linked in my question) – Gabriel Nov 11 '17 at 15:20
  • If you want, open a chat room and let us discuss. – Claude Leibovici Nov 12 '17 at 03:30
  • Sorry for taking so long to answer @ClaudeLeibovici, I was doing some reading on the topic. Here's the chat room if you are still interested in discussing: https://chat.stackexchange.com/rooms/68709/why-does-the-svd-method-minimize-the-sum-of-the-squared-point-plane-distances – Gabriel Nov 14 '17 at 17:45

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