# Rotation parametrization of alignment of two vectors

• sodemus
In summary, the simplest way to express the unique rotation between two vectors using quaternions and the parameter p is to construct a unit quaternion using the dot product of the two vectors and then modify it using p. This method does not require the use of cross-products, which is an added bonus.
sodemus
Hello,
I'm looking for an appropriate rotation representation for the following situation.

I have two (always non-zero) vectors, v1, v2, that may or may not be parallel. The rotation relating the two vectors is obviously non-unique having one degree of freedom, parametrized by p. So my question is: Is there a "simple" way (in terms of quaternions, rotation matrices, Euler angles ... or whatever you prefer) to express the unique (2 DOF) rotation in terms of the two vectors and the "free" parameter p, parametrizing the one degree of freedom mentioned above? What is the simplest way you can come up with?

I can do this myself but whatever I've come up with is algebraically pretty messy.

In case I haven't made myself clear, what I want is the "simplest" possible form of
R = R(v1,v2,p)

Edit: If it could be done without using cross-products that is an extra bonus!

One approach to this problem is to use quaternions. Quaternions are a representation of rotations in 4D space that can be used to describe rotations between two vectors. To find the quaternion representation, you need to take the dot product of the two vectors and find the angle between them, then use this angle to construct a unit quaternion representing the rotation. The parameter p can also be used to modify the quaternion further. For example, if p is the angle between the two vectors, then you can use p to modify the quaternion by adding or subtracting from it. You can also use p to scale the quaternion by multiplying it by a scalar. This gives you a simple and efficient way of expressing the rotation between two vectors with an additional degree of freedom.

## 1. What is rotation parametrization?

Rotation parametrization is a method of representing the rotation of an object or system in three-dimensional space using a set of parameters. These parameters can be used to describe the orientation and alignment of two vectors.

## 2. How is rotation parametrization different from other methods of representing rotation?

Rotation parametrization is different from other methods such as Euler angles or quaternions because it allows for a more intuitive and direct representation of the alignment of two vectors. It also avoids the issue of gimbal lock, which can occur with other methods.

## 3. What is the purpose of aligning two vectors using rotation parametrization?

The purpose of aligning two vectors using rotation parametrization is to find the optimal orientation and alignment between the two vectors. This can be useful in various applications, such as computer graphics, robotics, and navigation systems.

## 4. What are some common parameters used in rotation parametrization?

Some common parameters used in rotation parametrization include angle-axis representation, rotation matrix, and quaternion. These parameters can be used to describe the rotation around a specific axis and the magnitude of the rotation.

## 5. How is rotation parametrization used in scientific research?

Rotation parametrization is used in various scientific research fields, such as molecular dynamics, crystallography, and geophysics. It allows for the accurate representation of the orientation and alignment of molecules, crystals, and geological structures, which can provide valuable insights into their properties and behavior.

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