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Thread: 3D Unit Normal, Finding the 3rd Component

  1. #1
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    3D Unit Normal, Finding the 3rd Component

    Hi,
    I'm trying to find the z component of a unit normal of a plane given the x and y components.

    Since it's a unit normal I know that I have $\displaystyle n_x^2 + n_y^2+n_z^2 = 1$, therefore, $\displaystyle n_z = \sqrt{1 - n_x^2 - n_y^2}$.

    This seems fine in testing, until a case where $\displaystyle n_z$ should be negative, at which point the world breaks! Am I doing something stupid here? It's worth mentioning that I'm doing this as part of an image-plane to world-plane correction and as such don't have any coordinates of points on the world-plane itself. The only information I have is a set of image-plane coordinates.

    Thanks!
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  2. #2
    A Plied Mathematician
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    Actually, you must have $\displaystyle n_{z}=\pm\sqrt{1-n_{x}^{2}-n_{y}^{2}}.$ Do you know ahead of time when you need the positive and when you need the negative?
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  3. #3
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    Sorry, I forgot the +/-. I don't unfortunately, this is my problem. I was wondering if there was some way I'd missed to determine the sign.

    Given that I'm going to be using Broyden's method to approximate $\displaystyle n_x$ and $\displaystyle n_y $ as well as 2 other variables, I starting to think it might just be worth putting $\displaystyle n_z$ in as a 5th variable to approximate.
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  4. #4
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    I just realised, $\displaystyle n_z$ is going to be a half plane anyway, as you can't look behind the camera from a camera!
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