MATLAB: Creating a Random Vector with Several Conditions

MATLABrandom number generatorvectors

Hello everyone,

I have vector with known values. Z1=[x1,y1,z1]. I want to create a vector (Y1=[x2,y2,z2]) with several conditions. Conditions are;

0<y2<1, -1<z2<0, -1<x2<1

Also Z1 and Y1 must be perpendicular so that their dot product must be zero x1*x2+y1*y2+z1*z2=0 and Y1 should be a unit vector so that x2*x2+y2*y2+z2*z2=1.

I can not use the symbolic toolbox,fsolve or anonymous functions since the code won't be able to suitable to compile before run-time in simulink. I really appreciate any ideas.

Thanks.

Best Answer

Um, trivial? I suppose trivial is in the eyes of the beholder.

The equation dot(Z1,Y1) == 0 implies that Y1 lies in a plane in R^3, one that contains the origin. (That dot product is the orthogonality requirement.)

Next, the requirement that Y1 has unit length is just the equation of a sphere, with center at the origin.

The intersection of the sphere and plane will always be a unit circle, that lies in the plane of interest. You can parameterize that circle as a function of polar angle, in the rotated coordinate system.

So all you need do is choose a random point on the circle, one that satisfies the requirements on x,y,z. The constraints are just a constraint that specifies which octant the point can lie in. All of this can be done simply enough, if you understand how to transform the problem by rotating into the plane of interest. For example...

I would refuse to name the vectors Z1 and Y1, by the way. That is just poor coding practice, screaming to see you have bugs later on. You have a vector Z1, with elements [x1,y1,z1]? Then another vector named Y1, with elements [x2,y2,z2]? This is just asking for bugs. A bad idea. Learn to use more descriptive variable names. Your code will improve.

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MATLAB: How to properly sum 3D matrixes

Use this function:

function M=sumnd(M,dim)
s=size(M);
M=permute(M,[setdiff(1:ndims(M),dim),dim]);
M=reshape(M,[],s(dim));
M=sum(M,2);
s(dim)=1;
M=reshape(M,s);
end

Code:

A=sumnd(M,3) ;
A = simplify(A) ;

Result:

[ x1*y1*(z1 + z2 + z3), x1*y2*(z1 + z2 + z3), x1*y3*(z1 + z2 + z3)]
[ x2*y1*(z1 + z2 + z3), x2*y2*(z1 + z2 + z3), x2*y3*(z1 + z2 + z3)]
[ x3*y1*(z1 + z2 + z3), x3*y2*(z1 + z2 + z3), x3*y3*(z1 + z2 + z3)]

MATLAB: I want to show intersection of these two spheres. How should I do it

The spheres you describe have equations

1.) x^2 + y^2 + z^2 = 1

2.) (x-0.25)^2 + (y+1/5)^2 + (z-0.1)^2 = 1

Since equations 1 and 2 have same right-hand-side (equal to one), set the left-hand sides equal and you'll end up getting rid of the squared terms to be left with

3.) 5*x+4*y+2*z = 9/8

Equation 3 is the equation for the plane that contains the intersection of the two spheres. To see it add these lines of code to the end of your code above.

[X,Y] = meshgrid(linspace(-1,1,20));
Z = -5/2*X + 2*Y + 9/16;
surf(X,Y,Z)

That gives you the plane that contains the intersection. Realize that the intersection of the spheres is actually a curve that is a circle in this plane. Parametrizing that circle is more complicated.

Best Answer

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MATLAB: How to properly sum 3D matrixes

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