solve on vector equation
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Michiel Mathijs
el 19 de Dic. de 2019
Editada: David Goodmanson
el 21 de Dic. de 2019
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
4 comentarios
darova
el 19 de Dic. de 2019
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]);
Respuesta aceptada
darova
el 20 de Dic. de 2019
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
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David Goodmanson
el 21 de Dic. de 2019
Editada: David Goodmanson
el 21 de Dic. de 2019
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
.
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