Develop Reference

how to calculate velocity of leg using kinect

I have group of skeleton joints, for example 4 joints of leg. Each joint have (x, y, z) coordinates. So I want to find the combine speed or velocity of these group of joints because i believe I need to calculate velocity of each four joints of leg in order to find out the velocity of leg, please correct me If I'm wrong.
So what would be best possible way to calculate?
I'm using this formula
velocity= Sqrt((x[n]−x[n − 1] )2 + (y[n]−y[n − 1] )2 + (z[n]−z[n − 1] )2)

There is a simpler equation: Average Velocity = (Delta Distance)/(Delta Time)
If you want to know the average velocity at some point in the leg, calculate the location before it moved & the location on impact. Divide the magnitude of the vector of the difference between the 2 points by the amount of time it took to move between those points.

How to rotate a plane in 3D using Quaternion?

I am attempting to fuse skeletons from two separate Kinects.
From first Kinect I have skeleton1, from it I choose 4 points corresponding with 4 of it's joints. With those 4 points I can construct a plane on which they all are. The plane coefficients Ax + By + Cz = D are known. As I understand the planes Quaternion would be Q = (D; A; B; C)
From the second Kinect I have the same data, but in second Kinects coordinate system.
How can I rotate the plane from second Kinect so that it would have the same orientation as the plane from the first Kinect?

Unfortunately, quaternions don't work that way. They do not represent a plane, and cannot be created just by re-arranging the components of a plane equation (plane equation has two degrees of freedom, while quaternion is a non-parametric representation of rotation/orientation).
However, if you are looking for a rotation between two normal vectors (one for each plane), which will transform one plane to another, it is a relatively simple task:
if the two normal vectors are the same, there is no rotation needed and the result is identity,
if the two vectors are parallel (their dotproduct is zero) but not identical, the transformation between them is a rotation of 180 degrees around an arbitrary axis perpendicular to them,
in all other cases, you can compute the axis of rotation using a crossproduct (and normalization), and the angle using arc cosine of their dotproduct.
If you still need to create a quaternion out of them, you can do it with a bit of trigonometry.

How does one know if a normal vector is between two angles?

I have a Max Angle and a Min Angle, and also a unit vector pointing in some direction (2D).
How do I find out if this normal vector is between the two angles?

I don't know c#, but I know math:
Suppose the vector's coordinates are (x,y), and it is a unit vector, so |(x,y)| = 1.
The angle a between the positive x-axis and the vector is
a = atan2(y,x)
where atan2 is the four-quadrant arctangent. You can then check if this angle is between your max and min angles (provided they are also defined with respect to the positive x-axis).
Note that a is in radians; if your angles are given in degrees, you should first compute
a = a*180/pi;
where pi = 3.1415..... of course.
Does this help at all?

Calculate the angle from the dot product (this is easy in 2d) and then compare to your angle range.

Getting rotation axis from initial and final rotated quaternions

I have to find the axis and angle of rotation of a camera with an UP and Direction vector(They both are perpendicular to each other). I have the initial and final positions of the UP and direction vectors of the camera that is rotated. I want to find the axis and angle of the rotation for the camera. I am using C# for my project. I am new to this 3D rotation. So pardon my questions if you find them silly.

From the direction (forward) vector f and up vector u you can get the side vector s by performing a vector cross product (s = f x u). All three vectors are now orthogonal. You should also make them orthonormal by normalizing each one of them. Taken together, these vectors form an orthonormal basis.
You now have two such basis: the one from your initial camera orientation and the one from your final camera orientation. Both basis can be represented as a rotation matrix. A rotation matrix is simply a 3x3 matrix where the 3 rows are respectively:
The forward vector
The up vector
The side vector
For example, the matrix:
[[1 0 0]
[0 1 0]
[0 0 1]]
could be your initial camera orientation at start-up with its forward vector, up vector and side vector pointing towards the positive x axis, y axis and z axis, respectively.
You can now convert these two basis (M1 and M2) to two unit quaternions (Q1 and Q2) using this algorithm which takes care about potential problems like divides by zero.
At this point, you have two unit quaternions representing your initial and final camera orientation. You must now find the quaternion qT that transforms Q1 into Q2, that is:
q2 = qT * q1
q2 * q1^-1 = qT * (q1 * q1^-1) = qT
=> qT = q2 * q1^-1
Knowing that the inverse of a unit quaternion is equal to its conjugate:
q1^-1 = q1* iif ||q1|| = 1
qT = q2 * q1^-1 = q2 * q1*
There is a single step left: extracting the axis and angle from quaternion qT:
angle = 2 * acos(qw)
x = qx / sqrt(1-qw*qw)
y = qy / sqrt(1-qw*qw)
z = qz / sqrt(1-qw*qw)
The angle is, of course, given in radian. Beware of the divide by zero when calculating x, y and z. This situation would happen when there is no rotation or a very small one, so you should test if angle > epsilon where you would choose epsilon to be quite small an angle (say 1/10 of a degree) and not calculate the vector if that is the case.

When to Normalize a Vector?

I am learning XNA and in almost all of the educational kits found on http://creators.xna.com/en-US/. I always see a call to Normalize() on a vector. I understand that normalize basically converts the vector into unit length, so all it gives is direction.
Now my question is When to normalize and what exactly does it help me in. I am doing 2D programming so please explain in 2D concepts and not 3D.
EDIT: Here is code in the XNA kit, so why is the Normalize being called?
if (currentKeyboardState.IsKeyDown(Keys.Left)
|| currentGamePadState.DPad.Left == ButtonState.Pressed)
{
catMovement.X -= 1.0f;
}
if (currentKeyboardState.IsKeyDown(Keys.Right)
|| currentGamePadState.DPad.Right == ButtonState.Pressed)
{
catMovement.X += 1.0f;
}
if (currentKeyboardState.IsKeyDown(Keys.Up)
|| currentGamePadState.DPad.Up == ButtonState.Pressed)
{
catMovement.Y -= 1.0f;
}
if (currentKeyboardState.IsKeyDown(Keys.Down)
|| currentGamePadState.DPad.Down == ButtonState.Pressed)
{
catMovement.Y += 1.0f;
}
float smoothStop = 1;
if (catMovement != Vector2.Zero)
{
catMovement.Normalize();
}
catPosition += catMovement * 10 * smoothStop;

In your example, the keyboard presses give you movement in X or Y, or both. In the case of both X and Y, as when you press right and down at the same time, your movement is diagonal. But where movement just in X or Y alone gives you a vector of length 1, the diagonal vector is longer than one. That is, about 1.4 (the square root of 2).
Without normalizing the movement vector, then diagonal movement would be faster than just X or Y movement. With normalizing, the speed is the same in all 8 directions, which I guess is what the game calls for.

One common use case of vector normalization when you need to move something by a number of units in a direction. For example, if you have a game where an entity A moves towards an entity B at a speed of 5 units/second, you'll get the vector from A to B (which is B - A), you'll normalize it so you only keep the direction toward the entity B from A's viewpoint, and then you'll multiply it by 5 units/second. The resulting vector will be the velocity of A and you can then simply multiply it by the elapsed time to get the displacement by which you can move the object.

It depends on what you're using the vector for, but if you're only using the vectors to give a direction then a number of algorithms and formulas are just simpler if your vectors are of unit length.
For example, angles: the angle theta between two unit vectors u and v is given by the formula cos(theta) = u.v (where . is the dot product). For non-unit vectors, you have to compute and divide out the lengths: cos(theta) = (u.v) / (|u| |v|).
A slightly more complicated example: projection of one vector onto another. If v is a unit vector, then the orthogonal projection of u onto v is given by (u.v) v, while if v is a non-unit vector then the formula is (u.v / v.v) v.
In other words: normalize if you know that all you need is the direction and if you're not certain the vector is a unit vector already. It helps you because you're probably going to end up dividing your direction vectors by their length all the time, so you might as well just do it once when you create the vector and get it over with.
EDIT: I assume that the reason Normalize is being called in your example is so that the direction of velocity can be distinguished from the speed. In the final line of the code, 10 * smoothStop is the speed of the object, which is handy to know. And to recover velocity from speed, you need to multiply by a unit direction vector.

You normalize any vector by dividing each component by the magnitude of the vector, which is given by the square root of the sum of squares of components. Each component then has a magnitude that varies between [-1, 1], and the magnitude is equal to one. This is the definition of a unit vector.