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# How to calculate normal to a line?

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Shivakumar on 27 Aug 2013
Commented: ddd ppp on 4 May 2018
I plot a line in MATLAB using the points. Please tell me how to obtain the normal of that line? Can I get these plots in a single plot?
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ddd ppp on 4 May 2018
normal vector to image?

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### Accepted Answer

Jan on 27 Aug 2013
It is easier to answer, if you explain any details. At first I assume you mean 2D-lines, because for 3D-lines a normal line is not defined.
If you have a vector with the coordinates [x, y], the vectors [y, -x] and [-y, x] are orthogonal. When the line is defined by the coordinates of two points A and B, create the vector B-A at first, determine the orientation by the above simple formula, decide for one of the both vectors, and the midpoint between the points (A+B) * 0.5 might be a nice point to start from. Adjusting the length of the normal vector to either 1 or e.g. the distance norm(B-A) might be nice also.
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Jan on 21 Sep 2013
@Shivakumar: The normal of a line is a vector. Vectors can be move freely in the space, because they have a direction and a length. but no start point. Therefore the normal of the line at the midpoint is the normal at any other point also. If you want to draw the normal, it looks nice, when you start it at the midpoint of the line segment. but in a mathematical sense it is correct to start it from any other point of the X-Y-plane as well, e.g. at the origin.

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### More Answers (2)

Image Analyst on 27 Aug 2013
A perpendicular line has a negative inverse slope. So if you used polyfit
coeffs = polyfit(x, y, 1);
then coeffs(1) is the slope. The new slope is -1/coeffs(1). Now you simply use the point-slope formula of a line to draw it. Obviously you need to know at least one point that that line goes through since there are an infinite number of lines at that slope (all parallel to each other of course).
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Shivakumar on 30 Aug 2013
I want the perpendicular line to cross in the middle. Thank you.

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Shashank Prasanna on 28 Aug 2013
Edited: Shashank Prasanna on 28 Aug 2013
If this is a homework, please spend some time familiarizing yourself with basics of MATLAB. You can start by going through the Getting Started guide
There are several ways you could do this and all of the already suggested approaches are good. Here is how you can think about it in terms of linear algebra.
Answer: Normal lies in the null space of the the matrix A-B
A = [-0.6779, -0.7352]; B = [0.7352, -0.6779];
null(A-B)
Proof:
(A-B)*null(A-B) % Should yield a number close to zero
If you are looking to plot:
x = [A(1);B(1)];
y = [A(2);B(2)];
line(x,y,'color','k','LineWidth',2)
normal = [mean(x),mean(y)] + null(A-B)';
line([mean(x),normal(1)],[mean(y),normal(2)],'color','r','LineWidth',2)
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ddd ppp on 4 May 2018
is the null space a vector or line? does it have direction?

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