# Exploring the Mathematics of Rotation: A Mathematician's Perspective

• aaaa202
In summary, rotations are defined as linear maps in SO_n(\mathbb{R}) which preserve length and orientation, but there are other linear transformations in O_n(\mathbb{R}) that also preserve lengths but do not qualify as rotations.
aaaa202
For a mathematician, how is rotation defined in the most general sense?

Question arose to me because it occurred to me that an essential property of the rotation matrix is that it preserves lengths. Is this the only mapping (if not please give me a counterexample) that has this property (identity is also a special case of a rotation of course) and can this be shown?

Might seem like a very lose question and indeed it is, so speak freely about anything that you think could improve my understanding of the fundamental mathematics behind rotation.

I would personally define a rotation as everything in $SO_n(\mathbb{R})$. So the linear maps which preserve length and which preserve the orientation.

We had some threads about this before though. I remember Fredrik say that he would consider $O_n(\mathbb{R})$ as the rotations for convenience. So my answer is definitely not the only possible answer you can give.

I'll see if I can dig up some old threads.

micromass said:
I remember Fredrik say that he would consider $O_n(\mathbb{R})$ as the rotations for convenience.
I have seen at least two conventions in the literature:

1. Members of O(n) are called rotations (or orthogonal transformations), and members of SO(n) are called proper rotations.
2. Members of O(n) are called orthogonal transformations, and members of SO(n) are called rotations.

In one of those threads I said that I prefer convention 1. Lately I've been thinking that I prefer convention 2.

aaaa202 said:
For a mathematician, how is rotation defined in the most general sense?

Question arose to me because it occurred to me that an essential property of the rotation matrix is that it preserves lengths. Is this the only mapping (if not please give me a counterexample) that has this property (identity is also a special case of a rotation of course) and can this be shown?

Might seem like a very lose question and indeed it is, so speak freely about anything that you think could improve my understanding of the fundamental mathematics behind rotation.
Here is a simple concrete example in ##\mathbb{R}^2##: let ##T:\mathbb{R}^2 \rightarrow \mathbb{R}^2## be the map defined by
$$T\left(\begin{bmatrix}x \\ y \end{bmatrix} \right) = \begin{bmatrix}y \\ x \end{bmatrix}$$
i.e. we interchange the ##x## and ##y## axes. This is a linear map which preserves lengths, but it is not a rotation in the geometric sense. (There is also a "flip" involved.) Note that the matrix representation of ##T## with respect to the usual basis is
$$[T] = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$
so the determinant is -1, which shows that ##T \in O(\mathbb{R}^2)## but ##T \not\in SO(\mathbb{R}^2)##.

## 1. What is rotation in mathematics?

Rotation in mathematics is a transformation that turns a figure or object around a fixed point called the center of rotation. This transformation involves changing the position of each point of the figure by a certain angle and direction.

## 2. How is rotation represented mathematically?

In mathematics, rotation is represented using a coordinate system and the use of trigonometric functions such as sine, cosine, and tangent. The coordinates of the points on the figure are changed according to the angle of rotation and the direction in which it is rotated.

## 3. What are some real-life applications of rotation in mathematics?

Rotation is used in many real-life applications such as engineering, computer graphics, and astronomy. It is used to design and analyze 3D objects, create animations and special effects, and calculate the movements of celestial bodies.

## 4. How does the concept of rotation relate to other mathematical concepts?

Rotation is closely related to other mathematical concepts such as symmetry, transformations, and matrices. It is also used in the study of geometry, trigonometry, and calculus.

## 5. What are some challenges in exploring the mathematics of rotation?

One of the main challenges in exploring the mathematics of rotation is visualizing the transformation in 3D space. It can also be challenging to accurately calculate the coordinates of points after rotation, especially with complex figures. Additionally, understanding the relationship between rotation and other mathematical concepts can be difficult for some students.

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