solve on vector equation

Dear

i want to solve a vector equation to find the N vector. The equation is :

eq1 = nt/ni*cross(N,cross(-1*N,cv1))-N*sqrt(1-(nt/ni)^2*dot(cross(N,cv1),cross(N,cv1)))==[1;0;0];

Normal solve yields thre empty arrays>

Thanks in advance

With kind regards

[darova]

Do you have original equation?

[Michiel Mathijs]

Thank you for answering so soon.

This is the original equation.

[darova]

Did you try fsolve?

eq1 = @(N) nt/ni*cross(N,cross(-1*N,cv1))-N*sqrt(1-(nt/ni)^2*dot(cross(N,cv1),cross(N,cv1))) - [1 0 0];
n1 = fsolve(eq1,[1 1 1]);

[Michiel Mathijs]

Yes it does not give the right answer. It gives following message:

No solution found.

fsolve stopped because the relative size of the current step is less than the

value of the step size tolerance, but the vector of function values

is not near zero as measured by the value of the function tolerance.

<stopping criteria details>

Answer 1

[darova]

Try this (solution exists not for any s1 vector)

function main

nt = 1;
ni = 2;
s1 = [1 2 2];
s1 = s1/norm(s1);

    function y = F(Nx)
        p = cross(-Nx,s1);
        f = nt/ni*cross(Nx,p) - Nx*sqrt(1-(nt/ni)^2*dot(p,p));
        y = f' - [1;0;0];
    end

N = fsolve(@F,[1 1 1]);
p = cross(-N,s1);
t = [s1
    N*sqrt(1-(nt/ni)^2*dot(p,p))
    nt/ni*cross(N,p)];
v0 = zeros(3,1);
quiver3(v0,v0,v0,t(:,1),t(:,2),t(:,3),1)        % all vectors together
hold on
quiver3(t(2,1),t(2,2),t(2,3),1,0,0, 1,'r')      % vector [1 0 0]
% quiver3(0,0,0,N(1),N(2),N(3),'g')
hold off
text(s1(1),s1(2),s1(3),'s1 vector')
view(0,0)
axis equal

end

Answer 2

[David Goodmanson]

Hi Michiel,

The unit vector N has to lie in the plane defined by s1 and s2. The result for any s1,s2 is

N  = s2 - (n1/n2)*s1;
N = N/norm(N);
% have to determine whether N points in +N direction or -N direction
sgn = sign((n1/n2)*dot(s1,s2)-1);
N = sgn*N;

N has to lie in the plane defined by s1,s2 because from the bac-cab rule (letting n1/n2 = n12)

s2 = n12 (N x(-N x s1)) - N sqrt(...)
s2 = n12 (-N (N.s1) + s1) - N sqrt(...)
s2 - n12 s1 =  -N ( n12 (N.s1) + sqrt(...) )
const N = s2 -n21 s1

.