# Passive and Active Transformations

• Septimra
In summary, the conversation discussed the difference between active and passive transformations in a 2D rotation matrix. The speaker was confused about why the two equations resulted in different matrices, but after further clarification, they understood that the convention is to rotate vectors counterclockwise. The conversation also referenced a helpful image and provided additional resources for understanding the concept.
Septimra
Alright. I was looking into a 2D rotation matrix and there are two equations: one is through the transformation of the component of p (always with respect to x,y), x,y into x',y' and the other is through the transformation of the unit vectors i,j into i',j'.

In a sense 1 is passive the other is an active transformation. I do understand the difference between the two, but my question is that why do we get the different matrices if when we derive them, we do so with a clockwise orientation?

Hope its clear, If not--I'd be happy to clarify.

Can you clarify. I guess I understand what you mean, but just to be sure.

LOL Okay. I got the answer. Welp it seems like they way you derive the matrix you have to say consistent, otherwise you fall in a a trap.
https://en.wikipedia.org/wiki/Active_and_passive_transformation#/media/File:PassiveActive.JPG
Staring at this image for a bit and making sense of this reasoning: http://math.stackexchange.com/quest...erent-representations-of-2d-rotation-matrices

helped. You need to use a point relative to the rotated basis once you rotate the basis. You cannot use a rotated basis on a point relative to a the old basis, but with rotational matrices you only an inverted answer and nothing any more alarming.

See this thread for some general comments about "active" vs. "passive" rotations.

The convention that I'm familiar with is to rotate vectors counterclockwise. A rotation is linear, so if we know how the basis vectors are rotated, we will know how all vectors are rotated. If you rotate (1,0) by an angle ##\theta## counterclockwise, the result is ##(\cos\theta,\sin\theta)##. The same rotation applied to (0,1) yields ##(-\sin\theta,\cos\theta)##. It follows immediately from this that the matrix of the rotation operator with respect to the standard ordered basis is
$$\begin{pmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{pmatrix}.$$ The rotated basis vectors end up as the columns of the matrix. If you don't understand this, I recommend the FAQ post that I linked to in the post I linked to above.

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Septimra
Hehe alright, thanks. Looking back at it, I was being pretty dense.

## 1. What are passive and active transformations?

Passive and active transformations are types of mathematical operations used to manipulate and transform geometric shapes. They are commonly used in computer graphics and computer vision applications.

## 2. What is the difference between passive and active transformations?

The main difference between passive and active transformations is in the order in which the operations are applied. In passive transformations, the operations are applied to the object first and then the coordinate system. In active transformations, the operations are applied to the coordinate system first and then the object. This results in a different final transformation.

## 3. What are some examples of passive transformations?

Examples of passive transformations include translations, rotations, and reflections. These operations are applied directly to the object, resulting in a new transformed object with the same coordinate system.

## 4. What are some examples of active transformations?

Examples of active transformations include shearing, scaling, and stretching. These operations are applied to the coordinate system, resulting in a new coordinate system that is then applied to the object to achieve the desired transformation.

## 5. How are passive and active transformations used in real-world applications?

Passive and active transformations are used in a variety of real-world applications, such as computer animation, video games, and medical imaging. They allow for the manipulation and transformation of objects and images in a precise and efficient manner.

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