Rotate Normal Around Tangent

Question

Paul Huter 2012-10-5

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I want to rotate a normal vector around a tangent vector to create a circle. I have not been able to find anything to do this. Does such a function exist? Or how would I generate one?

Thanks.

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Matt J 2012-10-5

Clarify what this is supposed to do. What data are you given and in what form? What will the output data be, and in what form?

Paul Huter 2012-10-5

I have a 3D curve, and I have calculated normal and tangent vectors at each point on the curve based on a velocity (I am using Frenet-frame here). I want to be able to draw a 3D circle that is normal to the curve at some points (not all...there are a lot of points...). I figured the best way would be to rotate the normal vector about the tangent vector.

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Answer 1

Matt J 2012-10-6

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编辑：Matt J 2012-10-6

在 MATLAB Online 中打开

I'm assuming you're choosing the radius, R, of this circle. Then if T and N are the tangent and normal vectors at point P on the curve (all in column vector form):

n=1000;
theta=linspace(0,2*pi,n+1); 
  theta(end)=[];
refcircle = [R*cos(theta);R*sin(theta);zeros(1,n);ones(1,n)] ;
T=T/norm(T);
N=N/norm(N);
E=cross(N,T);
A=[0 0 0; R 0 0; 0 R 0].';
B=[P,P+R*N,P+R*E];
params=absor(A,B);  %get this function from FEX
C = params.M*refcircle; %circle points at 3D curve
plot3(C(1,:), C(2,:), C(3,:)) %plot the circle

The above uses ABSOR, available here

http://www.mathworks.com/matlabcentral/fileexchange/26186-absolute-orientation-horns-method

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Matt J 2012-10-7

编辑：Matt J 2012-10-7

在 MATLAB Online 中打开

Clarify whether the plot you're talking about is from the code as I gave it to you, or the result of you adapting/inserting it into your larger problem. If the latter, I'd have to see what you did.

However, when I run it in isolation with the sample data P,T,N,R data below, I definitely get a plot of a circle floating in 3D space. Verify first that you can reproduce this.

P=[1;1;1];
T=[1;1;1];
N=[-1;2;-1];
R=3;
    n=1000;
      theta=linspace(0,2*pi,n+1); 
        theta(end)=[];
      refcircle = [R*cos(theta);R*sin(theta)] ;
      T=T/norm(T);
      N=N/norm(N);
      E=cross(N,T);
      C=bsxfun(@plus, [N,E]*refcircle, P);
      plot3(C(1,:), C(2,:), C(3,:)) %plot the circle

Paul Huter 2012-10-7

Turns out the way I was calculating my tangent/normal vectors was doing them as row-vectors, not columns. Looking at the way you did it (with columns) got it to work.

Thanks for your help.

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Answer 2

Muthu Annamalai 2012-10-5

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Paul, You need to find the points of a 2D rotation transform using the equations, for example affine transformation http://en.wikipedia.org/wiki/Rotation_(mathematics), and then you may visualize it using plot() commands. HTH, -Muthu

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Paul Huter 2012-10-5

As stated above, I am calculating Frenet-frame vectors, which I have (in the past) used to generate a rotation matrix (3D). Is this same rotation matrix going to allow me to draw a circle perpendicular to the curve (i.e. by multiplying the x, y, z points of a circle by the rotation matrix)?

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Answer 3

Image Analyst 2012-10-6

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编辑：Image Analyst 2012-10-6

Sounds like the streamtube() function. Could that be used? Or maybe morphological dilation, imdilate(). For morphological dilation, imagine a sphere whose center is tracing out your 3D curve. The dilated volumetric image is the volume swept out by that sphere as it travels along your curve.