I have a 3d object which is free to rotate along x,y and z axis and it is then saved as a transform matrix. In a case where the sequence of rotation is not known and the object is rotated for more than 3 times (eg :-if i rotate the object x-60degress, y-30 degrees, z-45 degrees then again x->30 degrees), is it possible to extract the angles rotated from the transform matrix?.I know that it is possible to get angles if the sequence of rotation is known, but if I have only the final transform matrix with me and nothing else, is it possible to get the angles rotated(x,y,and z) from the transform matrix ?
Euler angle conversion is a pretty well known topic. Just normalize the matrix orientation vectors and then use something like this c source code.
The matrix is the current state of things it has no knowledge of what the transformation has been in the past. It does not know how the matrix was built. You can just take the matrix into and decompose it into any pieces you like, as long as:
The data do not overlap. For example:Two X turns after each other is indistinguishable form each other (no way to know if its 1 2 or three different rotations summed).
The sequence order is known
A decomposition can be built out of the data (for example scale can be measured)
Related
I have a 2D shape (a circle) that I want to extrude along a 3D curve to create a 3D tube mesh.
Currently the way I generate cross-sections along the curve (which form the basis of the resulting mesh) is to take every control point along the curve, create a 3D transform matrix for it, then multiply the 2D points of my circle by those curve-point matrices to determine their location in 3D space along the curve.
To create the matrix (from 3 vectors), I use the tangent on the curve as the up vector, world-up ([0,1,0]) as the forward vector, and the cross product of the up/forward vectors as the right vector. All three vectors are also orthogonalized during the process to create the final matrix.
The problem comes when my curve tangent is identical to the world-up axis. Ie, my tangent vector is [0,1,0] and the world-up is [0,1,0]....since the cross product of two parallel vectors is not explicit....the resulting extruded mesh has artifacts along those areas of the curve (pinching, twisting, etc).
I thought a potential solution would be to use the dot product of the curve tangent and the world-up as an interpolation value to shift my forward vector from world-up to world-right...in other words, as a curve tangent approaches [0,1,0], my forward vector approaches [1,0,0]...but that results in unwanted twisting along the final mesh as well.
How can I extrude my shape along a curve in a consistent manner that has no flipping/artifacts/twisting? I know it's possible since various off-the-shelf 3D applications can do it...I'm just not sure how.
One way I would approach this is to consider my tangent vector to the 3D curve as actually being a normal vector of the plane I am interested into.
Let's say, the tangent vector is
All you need now is two other vectors that are othoghonal to it, so let's.
Let's construct v like so:
(rotating the coordinates). Because v is the result of the cross product of u and something else, you know that v is orthogonal to u.
(This method will not work if u have equal x,y,z coordinates, in that case, construct the other vector by adding random numbers to at least two variables, rince&repeat).
Then you can simply construct w like before:
normalize and go.
I am using matlab's built in function called Procrustes to see the rotation translation and scale between two images. But, I am just using coordinates of the brightest points in the image and rotating these coordinates about the center of the image. Procrustes compares two matrices and gives you the rotation, translation, and scale. However, procrustes only works correctly if the matrices are in the same order for comparison.
I am given an image and a separate comparison coordinate matrix. The end goal is to find how much the image has been rotated, translated, and scaled compared to the coordinate matrix. I can just use Procrustes for this, but I need to correctly order the coordinates found from the image to match the order in the comparison coordinate matrix. My thought was to compare the distance between every possible combination of points in the coordinate matrix and compare it to the coordinates that I find in the picture. I just do not know how to write this code due to the fact if there is n coordinates, there will be n! possible combinations.
Just searching for the shortest distance is not so hard.
A = rand(1E4,2);
B = rand(1E4,2);
tic
idx = nan(1,1E4);
for ct = 1:size(A,1)
d = sum((A(ct,:)-B).^2,2);
idx(ct) = find(d==min(d));
end
toc
plot(A(1:10,1),A(1:10,2),'.r',B(idx(1:10),1),B(idx(1:10),2),'.b')
takes half a second on my PC.
The problems can start when two points in set A are matched to the same location in set B.
length(unique(idx))==length(idx)
This can be solved in several ways. The best (imho) is to determine a probability that point B matches with point A based on the distance (usually something that decreases exponentially), and solve for the most probable situation.
A simpler method (but more error prone) is to remove the matched point from set B.
I have a given 3x3 rotation matrix and I want to calculate the rotation angle around z axis. How do I get there?
For example, in this case below, how did they calculated the "-30deg rotation around the x axis"? Or how did they get to the "-74deg" value around that axis?
This is my original matrix:
Thank you!
It is simple if the rotation matrix is just a rotation matrix and there is no scaling. Here is a site that explains in more pretty terms then I am willing to diagram here. Basically the rotation matrix is composed of sinf(x) and cosf(x) of euler angles (well you can think of it like that at least). You can therefore use values within it to back calculate the euler angles.
http://nghiaho.com/?page_id=846
If you have scaling involved you will need to normalize each row of the matrix first. Then apply the above method.
I would like to find a solution for taking a rotation represented as a matrix and then resetting one of it's components. Basically I want to be able to multiply a vector by this matrix and get a direction that is rotation around x and z axis and be constant along the y axis (up). I want to take object rotation and get the vector that represents gravity but in object local space and disregarding the yaw. So I want to reset the yaw.
I don't want to convert this to euler angles. I would prefer using a quaternion or doing some sequence of operations on the rotation matrix directly in order to avoid possible bugs with certain angles.
Ok, so I have the follwoing:
btTransform t;
mBody->getMotionState()->getWorldTransform(t);
btMatrix3x3 trans = t.getBasis().inverse();
btVector3 up = (trans * btVector3(0, 1, 0));
I realized that if I used quaternion then I got completely wrong results (why?). Now I'm getting a vector in object space that represents up vector in world space. BUT I want to rotate this vector so that it represents global up vector in object space WHEN MODEL HAS ZERO rotation around Y axis. So I have to somehow rotate this vector back. How?
You can use quaternion swing twist decomposition with passed "Y" axis. It will decompose quaternion to rotation around Y axis and rotation around axis that is perpendicular to Y.
It is described here, in my answer.
Component of a quaternion rotation around an axis
As a followup to my previous question about determining camera parameters I have formulated a new problem.
I have two pictures of the same rectangle:
The first is an image without any transformations and shows the rectangle as it is.
The second image shows the rectangle after some 3d transformation (XYZ-rotation, scaling, XY-translation) is applied. This has caused the rectangle to look a trapezoid.
I hope the following picture describes my problem:
alt text http://wilco.menge.nl/application.data/cms/upload/transformation%20matrix.png
How do determine what transformations (more specifically: what transformation matrix) have caused this tranformation?
I know the pixel locations of the corners in both images, hence i also know the distances between the corners.
I'm confused. Is this a 2d or a 3d problem?
The way I understand it, you have a flat rectangle embedded in 3d space, and you're looking at two 2d "pictures" of it - one of the original version and one based on the transformed version. Is this correct?
If this is correct, then there is not enough information to solve the problem. For example, suppose the two pictures look exactly the same. This could be because the translation is the identity, or it could be because the translation moves the rectangle twice as far away from the camera and doubles its size (thus making it look exactly the same).
This is a math problem, not programming ..
you need to define a set of equations (your transformation matrix, my guess is 3 equations) and then solve it for the 4 transformations of the corner-points.
I've only ever described this using German words ... so the above will sound strange ..
Based on the information you have, this is not that easy. I will give you some ideas to play with, however. If you had the 3D coordinates of the corners, you'd have an easier time. Here's the basic idea.
Move a corner to the origin. Thereafter, rotations will take place about the origin.
Determine vectors of the axes. Do this by subtracting the adjacent corners from the origin point. These will be a local x and y axis for your world.
Determine angles using the vectors. You can use the dot and cross products to determine the angle between the local x axis and the global x axis (1, 0, 0).
Rotate by the angle in step 3. This will give you a new x axis which should match the global x axis and a new local y axis. You can then determine another rotation about the x axis which will bring the y axis into alignment with the global y axis.
Without the z coordinates, you can see that this will be difficult, but this is the general process. I hope this helps.
The solution will not be unique, as Alex319 points out.
If the second image is really a trapezoid as you say, then this won't be too hard. It is a trapezoid (not a parallelogram) because of perspective, so it must be an isosceles trapezoid.
Draw the two diagonals. They intersect at the center of the rectangle, so that takes care of the translation.
Rotate the trapezoid until its parallel sides are parallel to two sides of the original rectangle. (Which two? It doesn't matter.)
Draw a third parallel through the center. Scale this to the sides of the rectangle you chose.
Now for the rotation out of the plane. Measure the distance from the center to one of the parallel sides and use the law of sines.
If it's not a trapezoid, just a quadralateral, then it'll be harder, you'll have to use the angles between the diagonals to find the axis of rotation.