I have two matrices each of which contains two vectors (each row is a vector):
u = [1 0 0; 2 0 0]
v = [1 1 0; 2 2 0]
I want to calculate two angles between the vectors of the corresponding rows in the matrices (angle between [1 0 0] , [1 1 0] and angle between [2 0 0] , [2 2 0]). In this example, both angles will be 45 degrees. So what I want is a new matrix like this:
angles = [45; 45]
When I try this:
u = [1 0 0; 2 0 0]
v = [1 1 0; 2 2 0]
dp = u(:,1) .* v(:,1) + u(:,2) .* v(:,2) + u(:,3) .* v(:,3);
angles = atan2d(norm(cross(u,v)),dp)
The anwser will be:
angles = [76.3670 ; 45.8683]
and when I try this (change norm to normr):
u = [1 0 0; 2 0 0]
v = [1 1 0; 2 2 0]
dp = u(:,1) .* v(:,1) + u(:,2) .* v(:,2) + u(:,3) .* v(:,3);
angles = atan2d(norm(crossr(u,v)),dp)
The anwser will be:
angles = [0 0 45.0000 ; 0 0 14.0362]
How can I make it calculate the angle between the vectors of each row?
Try:
u=[1 0 0;2 0 0];
v = [1 1 0;2 2 0];
atan2(cross(u,v,2),dot(u,v,2)) % radians
atan2d(cross(u,v,2),dot(u,v,2)) % degrees
The ,2 in the cross and dot functions specifies the dimension to operate, since you are storing each vector in a row.
There a discussion here, with many other ways to calculate, and you may find one more suitable to your particular case of application.
Related
I'm looking for a simple method to transforming a vector of the following type
[1 1 0 1 1]
To this group of vectors:
[0 0 0 0 1]
[0 0 0 1 0]
[0 1 0 0 0]
[1 0 0 0 0]
The vector itself represents polynomial coefficients (x^4 + x^3 + x + 1)
Thanks
Here are some approaches. Input is a vector v, output is a matrix M:
[~, rr, vv] = find(v);
M = full(sparse(1:nnz(v), flip(rr), 1));
M = flip(eye(numel(v)));
M = M(logical(v),:)
M = zeros(nnz(v), numel(v));
M(sub2ind(size(M), 1:size(M,1), flip(find(v)))) = 1;
M = double(bsxfun(#eq, flip(find(v(:))), 1:numel(v)));
I have indices
I = [nGrid x 9] matrix % mesh on fine grid (9 point rectangle)
J = [nGrid x 4] matrix % mesh on coarse grid (4 point rectangle)
Here, nGrid is large number depending on the mesh (e.g. 1.e05)
Then I want to do like
R_ref = [4 x 9] matrix % reference restriction matrix from fine to coarse
P_ref = [9 x 4] matrix % reference prolongation matrix from coarse to fine
R = sparse(size) % n_Coarse x n_Fine
P = sparse(size) % n_Fine x n_Coarse
for k = 1 : nGrid % number of elements on coarse grid
R(I(k,:),J(k,:)) = R_ref;
P(J(k,:),I(k,:)) = P_ref;
end
size is predetermined number.
Note that even if there is same index in (I,J), I do not want to accumulate. I just want to put stencils Rref and Pref at each indices respectively.
I know that this is very slow due to the data structure of sparse.
Usually, I use
sparse(row,col,entry,n_row,n_col)
I can use this by manipulate I, J, R_ref, P_ref by bsxfun and repmat.
However, this cannot be done because of the accumulation of sparse function
(if there exists (i,j) such that [row(i),col(i)]==[row(j),col(j)], then R(row(i),row(j)) = entry(i)+entry(j))
Is there any suggestion for this kind of assembly procedure?
Sample code
%% INPUTS
% N and M could be much larger
N = 2^5+1; % number of fine grid in x direction
M = 2^5+1; % number of fine grid in y direction
% [nOx * nOy] == nGrid
nOx = floor((M)/2)+1; % number of coarse grid on x direction
nOy = floor((N)/2)+1; % number of coarse grid on y direction
Rref = [4 4 -1 4 -2 0 -1 0 0
-1 -1 -2 4 4 4 -1 4 4
-1 -1 4 -2 4 -2 4 4 -1
0 4 4 0 0 4 -1 -2 -1]/8;
Pref = [2 1 0 1 0 0 0 0 0
0 0 0 1 1 1 0 1 2
0 0 1 0 1 0 2 1 0
0 2 1 0 0 1 0 0 0]'/2;
%% INDEX GENERATION
tri_ref = reshape(bsxfun(#plus,[0,1,2]',[0,N,2*N]),[],1)';
TRI_ref = reshape(bsxfun(#plus,[0,1]',[0,nOy]),[],1)';
I = reshape(bsxfun(#plus,(1:2:N-2)',0:2*N:(M-2)*N),[],1);
J = reshape(bsxfun(#plus,(1:nOy-1)',0:nOy:(nOx-2)*nOy),[],1);
I = bsxfun(#plus,I,tri_ref);
J = bsxfun(#plus,J,TRI_ref);
%% THIS PART IS WHAT I WANT TO CHANGE
R = sparse(nOx*nOy,N*M);
P = R';
for k = 1 : size(I,1)
R(J(k,:),I(k,:)) = Rref;
P(I(k,:),J(k,:)) = Pref;
end
I am using Matlab and Euler Angles in order to reorient a 3axes coordinate system. Specifically,
Rz = [cos(ψ) sin(ψ) 0;-sin(ψ) cos(ψ) 0;0 0 1];
Ry = [cos(φ) 0 -sin(φ);0 1 0;sin(φ) 0 cos(φ)];
Rx = [1 0 0;0 cos(θ) -sin(θ);0 sin(θ) cos(θ)];
Rtotal = Rz*Ry*Rz
Then I loop through my old system coordinates (x,y,z) and make a vector coord_old. Then I get the reoriented system with (xn,yn,zn)
for i=1:size(num,1)
coord_old = [x(i,1);y(i,1);z(i,1)];
coord_new = Rtotal*coord_old;
xn(i,1) = coord_new(1,1);
yn(i,1) = coord_new(2,1);
zn(i,1) = coord_new(3,1);
end
My issue is that when θ,φ,ψ≃0 then x->-y and y->x and when θ,φ≃0 and ψ=90 then x and y will not rotate! That means that when x,y should rotate they don't and when they shouldn't rotate they stay as they were!
--EDIT--
For example, when ψ=20.0871, φ=0.0580 and θ=0.0088 I get these results
See that x->-y and y->x while z doesn't change at all!
Any thoughts?
Ok, I see two main problems here:
Rtotal = Rz*Ry*Rz is probably not what you want since Rz is multiplied twice. I think you mean Rtotal = Rz*Ry*Rx.
Your rotation matrix seems to be incorrect. Check this Wikipedia artice to get the correct signs.
Here a corrected rotation matrix:
Rz = [cos(psi) -sin(psi) 0; sin(psi) cos(psi) 0; 0 0 1];
Ry = [cos(phi) 0 sin(phi); 0 1 0; -sin(phi) 0 cos(phi)];
Rx = [1 0 0; 0 cos(theta) -sin(theta); 0 sin(theta) cos(theta)];
Rtotal = Rz*Ry*Rx;
With this matrix I get the correct results:
x=1; y=2; z=3;
psi=0; phi=0; theta=0;
[xn,yn,zn] >> 1 2 3
x=1; y=2; z=3;
psi=90/180*pi; phi=0; theta=0;
[xn,yn,zn] >> -2 1 3
And here a full graphical example of a cube in 3d-space:
% Create cube (not in origin)
DVert = [0 0 0; 0 1 0; 1 1 0; 1 0 0 ; ...
0 0 1; 0 1 1; 1 1 1; 1 0 1];
DSide = [1 2 3 4; 2 6 7 3; 4 3 7 8; ...
1 5 8 4; 1 2 6 5; 5 6 7 8];
DCol = [0 0 1; 0 0.33 1; 0 0.66 1; ...
0 1 0.33; 0 1 0.66; 0 1 1];
% Rotation angles
psi = 20 /180*pi; % Z
phi = 45 /180*pi; % Y
theta = 0 /180*pi; % X
% Rotation matrix
Rz = [cos(psi) -sin(psi) 0; sin(psi) cos(psi) 0; 0 0 1];
Ry = [cos(phi) 0 sin(phi); 0 1 0; -sin(phi) 0 cos(phi)];
Rx = [1 0 0; 0 cos(theta) -sin(theta); 0 sin(theta) cos(theta)];
Rtotal = Rz*Ry*Rz;
% Apply rotation
DVertNew = Rtotal * DVert';
% Plot cubes
figure;
patch('Faces',DSide,'Vertices',DVert,'FaceColor','flat','FaceVertexCData',DCol);
patch('Faces',DSide,'Vertices',DVertNew','FaceColor','flat','FaceVertexCData',DCol);
% Customize view
grid on;
axis equal;
view(30,30);
When I use your code and insert 0 for all angles, I get Rtotal:
Rtotal =
1 0 0
0 1 0
0 0 1
This is the identity matrix and will not change your values.
You have an error in your matrix multiplication. I think you should multiply: Rtotal*coord_old. I think you are missing the _old. depending on what is in you coordvariable, this may be the bug.
When I run:
for i=1:size(1,1)
coord_old = [1;2;3];
coord_new = Rtotal*coord_old;
xn(i,1) = coord_new(1,1);
yn(i,1) = coord_new(2,1);
zn(i,1) = coord_new(3,1);
end
I get the correct result:
coord_new =
1
2
3
Thank you both #Steffen and #Matt. Unfortunately, my reputation is not high enough to vote Up your answers!
The problem was not with Rtotal as #Matt correctly stated. It should be as it was Rz*Ry*Rx. However, both your ideas helped me test my code with simple examples (5 sets of coordinates and right hand rule), and realize where my (amateur) mistake was.
I had forgotten I had erased parts of codes where I was expressing my angles to degrees... I should be using sind & cosd instead of sin and cos.
I have a matrix corresponding to 8 vortex of a cube,
CubeVortex = [3 3 0;
0 3 0;
0 3 3;
3 3 3;
0 0 3;
3 0 3;
3 0 0;
0 0 0];
Now I want to get the coordinates of all the edges divided in 3, like,
As you can see, there will be 12x2 = 24 coordinates.
It would be a little hard to write them.
Is there a way to calculate them from CubeVortex?
One way to do this:
Cube = [
3 3 0;
0 3 0;
0 3 3;
3 3 3;
0 0 3;
3 0 3;
3 0 0;
0 0 0];
% find edges by looking for all combinations of points on cube that
% differ by only one coordinate
sections_per_edge = 3;
weights = ((1:sections_per_edge-1) / sections_per_edge).';
edges = []; % indices into Cube
points = [];
n = size(Cube, 1);
for i = 1:n-1
pointA = Cube(i, :);
for j = i+1:n
pointB = Cube(j, :);
if nnz(pointA - pointB) == 1
edges = [edges; i, j];
% find points along edge as weighted average of point A and B
points = [points; weights * pointA + (1 - weights) * pointB];
end
end
end
% plot corners
plot3(Cube(:,1), Cube(:,2), Cube(:,3), '.r', 'markersize', 20)
hold on
% plot points along edges
plot3(points(:,1), points(:,2), points(:,3), '.b', 'markersize', 20)
% draw edges
line([Cube(edges(:,1), 1), Cube(edges(:,2), 1)].', ...
[Cube(edges(:,1), 2), Cube(edges(:,2), 2)].', ...
[Cube(edges(:,1), 3), Cube(edges(:,2), 3)].', 'color', 'k')
axis([-1,4,-1,4])
Result:
Increasing sections_per_edge to 10, you get
One approach could be this -
n = 3; %// number of IDs
m = 3; %// number of columns
combs = dec2base(0:(n+1)^m-1,n+1,m)-'0' %// form repeated combinations
out = c1(sum(ismember(combs,[1 2]),2)==1,:) %// combinations for intermediate points
You can make this generic for a N-point case and more efficient one, with this -
N = 3;
[x,y,z] = ndgrid(0:N,0:N,0:N)
combs = [z(:) y(:) x(:)]
out = combs(sum(combs~=0 & combs~=N,2)==1,:)
Thus, for your 3-point (0 to 3 that is) case, you would have -
out =
0 0 1
0 0 2
0 1 0
0 1 3
0 2 0
0 2 3
0 3 1
0 3 2
1 0 0
1 0 3
1 3 0
1 3 3
2 0 0
2 0 3
2 3 0
2 3 3
3 0 1
3 0 2
3 1 0
3 1 3
3 2 0
3 2 3
3 3 1
3 3 2
This should do it:
NewVortex=[];
for i=1:3
NewVortex=[CubeVortex*i/3;NewVortex];
end
NewVortex
I have the following vector:
y = [1; 3; 2; 3; 1];
All its values are between 1 and n (in this case, 3) and denote different options.
I want to create a matrix of size size(y, 1) x n whose rows correpond to y values:
1 0 0 % because y(1) = 1
0 0 1 % because y(2) = 3
0 1 0 % because y(3) = 2
0 0 1
1 0 0
One way to do this would be
Y = zeros(size(y, 1), num_labels);
for i = 1:m
Y(i, y(i)) = 1;
end
Is there a better way to do this, maybe in a single expression?
Basically, what I need is to generate a matrix with boolean predicate (i, j) => j == y(i).
You can try this if a is a column vector
a = [1; 3; 2; 3; 1];
bsxfun(#eq, a, [1:max(a)])
and this if it is a row vector
a = [1; 3; 2; 3; 1]';
bsxfun(#eq, a', [1:max(a)])
If you have access to Statistics Toolbox, the command dummyvar does exactly what you need.
>> y = [1; 3; 2; 3; 1];
>> dummyvar(y)
ans =
1 0 0
0 0 1
0 1 0
0 0 1
1 0 0
You can use sub2ind after initializing the matrix as follows:
y = [1; 3; 2; 3; 1];
m = length(y);
n = max(y);
Y = zeros(m, n);
Y(sub2ind(size(Y), 1:m, y')) = 1
Y =
1 0 0
0 0 1
0 1 0
0 0 1
1 0 0
The trick here is to know that the corresponding rows of y go from 1 to m one by one.
accumarray([(1:length(y)).' y], 1)
As suggested by Dmitri Bouianov on Coursera discussion forum, this also works:
Y = eye(num_labels)(y, :);
This solution uses elements of y to as indices to select rows from an identity matrix.
In Octave (at least as of 3.6.3, not sure when it was introduced), you can use broadcasting to do this extremely easily. It works like this:
Y = y==1:3
(if y is a row matrix, you need to transpose it first - if you want to have Y transposed instead, use y==(1:3)')