I have been trying to find the difference between the 2 but to no luck minus this
The primary diff erence between
the two representations is that a quaternion’s axis of rotation is scaled
by the sine of the half angle of rotation, and instead of storing the angle in the
fourth component of the vector, we store the cosine of the half angle.
I have no idea what
sine of the half angle of rotation
or
cosine of the half angle
means?
Quaternios and Axis-angle are both 4D representations of 3D rotations/orientations and both have pro's and cons.
Axis-angle: represents the rotation by its angle a and the rotation axis n. For example, a rotation of 180 degrees around the Y-Axis would be represented as a = 180, n= {0,1,0}. The representation is very intuitive, but for actually applying the rotation, another representation is required, such as a quaternion or rotation matrix.
Quaternion: represents a rotation by a 4D vector. Requires more math and is less intuitive, but is a much more powerful representation. Quaternions are easily interpolated (blending) and it is easy to apply them on 3D point. These formula's can easily be found on the web. Given a rotation of a radians about a normalized axis n, the quaternion 4D vector will be {cos a/2, (sin a/2) n_x, (sin a/2) n_y, (sin a/2) n_z}. That's where the sine and cosine of the half angle come from.
It means that if you, for example, want to make a 180deg rotation around the Z axis (0,0,1), then the quaternion's real part will be cos(180deg/2)=0, and its imaginary part will be sin(180deg/2)*(0,0,1)=(0,0,1). That's q=0+0i+0j+1k. 90-degree rotation will give you q=cos(90deg/2)+sin(90deg/2)*(0i+0j+1k)=sqrt(2)/2+0i+0j+sqrt(2)/2*k, and so on.
OTOH, if you're asking what sine and cosine are, check if your languange provides sin() and cos() functions (their arguments will probably be in radians, though), and check out http://en.wikipedia.org/wiki/Sine.
Related
I have shapes of letters represented in polar coordinates. The centroid of the shape equals the center of the polar representation. I need the values of rho every 10 degrees of the object's outer margin.
Examples of shapes in polar coordinates I have:
I have 2 problems I can't solve:
Matlab's [theta,rho]=cart2pol(x,y) theta doesn't return the angles in degrees. When plotting the object with polar(theta,rho) the object is well represented, but I don't understand what's the equivalence between matlab's theta value and each angle in degrees.
As the margin is in discrete, probably I won't have a margin point for every angle I need, so I need a way to get the nearest margin point for a given angle (and the farthest point for the letters with interior and exterior margin).
I have nxn matrix in 2D space; I would like to rotate the matrix around the x-axis using matlab. Where the x-axis pass through the center of the matrix (pass through the point [n/2 n/2].
I found the Matlab function B = rot90(A) which rotate the matrix A by 90 degree. But I’m looking for a method that rotate matrix A by any given angle (e.g. 30, 45, 170 degree) around the x-axis.
You can as well try imrotate(). This function is from the Image Processing Toolbox, but since its main input is a matrix (either real or logical) it'll work also for non picture-related matrices (I've tried with a magic matrix).
The syntax is:
B=imrotate(A,theta);
where A is you matrix, B is the rotated version of A and theta is the rotation in degrees. The rotation is performed in counterclockwise direction around its center point; to rotate the matrix clockwise, specify a negative value for theta.
I have a very basic question. I am analyzing a movie. In the movie I am fitting a curve to define an object and computing the first derivative of a curve at it's starting point. The first derivative is changing in each frame. I want to measire the first derivative in the from of angle given in degrees or radians. Is there a faster way to do it in MATLAB.
I know its a very simple question, but if someone can explain me the concept then that would be very helpful. Thanks
The derivative of the curve is defined as dy/dx the ratio between the change in X direction and Y direction. This ratio is also the tan of the angle between the curve and the X axis. Therefore, if you want the angle (in radians) all you need it compute atan of the derivative.
My math is too rusty to figure this out. I want to derive the onscreen angle (the angle as seen on the 2d screen) of a 3d vector.
Given the x and y rotation of a vector (z rotation is zero and doesn't mstter), what does the angle on screen look like?
We know when y is zero and x is positive, the angle is 90. When y is zero and x is negative the angle is -90. When y is 90, for any value of x, the angle is 180. When y is -90, for any value of x, the angle is 0.
So what the formula here so I can derive the angle for the other values of x and y rotation?
The problem, as stated, doesn't make sense. If you're holding z to zero rotation, you've converted a 3D problem to 2D already. Also, it seems the angle you're measuring is from the y-axis which is fine but will change the ultimate formula. Normally, the angle is measured from the x-axis and trigometric functions will assume that. Finally, if using Cartesian coordinates, holding y constant will not keep the angle constant (and from the system you described for x, the angle would be in the range from -90 to 90 - but exclusive of the end points).
The arctangent function mentioned above assumes an angle measured from the x-axis.
Angle can be calculated using the inverse tangent of the y/x ratio. On unity3d coordinated system (left-handed) you can get the angle by,
angle = Mathf.Rad2Deg * Mathf.Atan(y/x);
Your question is what will a 3-d vector look like.
(edit after posted added perspective info)
If you are looking at it isometrically from the z-axis, it would not matter what the z value of the vector is.
(Assuming a starting point of 0,0,0)
1,1,2 looks the same as 1,1,3.
all x,y,z1 looks the same as any x,y,z2 for any values of z1 and z2
You could create the illusion that something is coming "out of the page" by drawing higher values of z bigger. It would not change the angle, but it would be a visual hint of the z value.
Lastly, you can use Dinal24's method. You would apply the same technique twice, once for x/y, and then again with the z.
This page may be helpful: http://www.mathopenref.com/trigprobslantangle.html
Rather than code this up yourself, try to find a library that already does it, like https://processing.org/reference/PVector.html
Still need the math: I am trying to calculate the yxy rotation sequence given a quaternion transformation. I can easily do this using Matlab's quat2angle function. However, I need to calculate this by hand using a python script.
This part solved: Please look at this awesome presentation which helped me resolve these issues below:
https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=3&cad=rja&uact=8&ved=0CCoQFjAC&url=http%3A%2F%2Fwww.udel.edu%2Fbiology%2Frosewc%2Fkaap686%2Freserve%2Fshoulder%2Fshoulder%2FBluePresentation.ppt&ei=jgRAVLHfOsSrogTJiYHABQ&usg=AFQjCNGFmwh11jEZen80jc3tM4f7HUQcNw&sig2=Dlr8_7TIFPLyUfJy6-pSJA&bvm=bv.77648437,d.cGU
Also, with Matlab, I am seeing strange results with the way they calculate yxy. I have a quaternion transformation of [1.0000 -0.0002 -0.0011 -0.0006] and I get y = 112.4291 x = -0.0719 y1 = -112.5506 (in degrees).
I don't expect to see any rotations here (my sensors aren't rotating). Why is Matlab showing me rotation? And when I try to just move in the x rotation, I see y and y1 also rotate, however, I don't expect y or y1 to be rotating. Any thoughts?
UPDATE:
When I add y + y1 I seem to get the value for the first y (when doing simple rotation around the first y), and this smooths out the data. However, when I combine the three rotations of the shoulder, the data doesn't make sense. I am trying to define shoulder movement based on plane of elevation, elevation and rotation (yxy) in a way that's easy to interpret. When I rotate around x, then the second y, I get "clipping" (data goes to 180 then -180 following positive trend for y1 and opposite happens for y), even though I start my sensors at the zero position. Also, If I try to rotate only around the second y, I see rotation in the x. That doesn't make any sense either. Any additional thoughts?
Note:
I am using 2 IMU sensors, taring them in the same orientation, holding one constant and rotating the other, calculating the relative rotation between them using quaternions, and then calculating the yxy rotation sequence angles.
In case anyone is interested in quaternion calculations and transformations. I solved it using this transformations library:
http://www.lfd.uci.edu/~gohlke/code/transformations.py.html
There are several functions in here using matrices, quaternions, and Euler rotations. And you can convert quaternions to several different Euler rotation sequences. Give thanks to the person who created this script.