Rotation is combination of shearing and scaling

[phiby]

I have read at a lot of places that in 2D transformations rotation is a combination of scaling and simultaneous shear?

What exactly does this mean & what's the proof for this?

 

[Stephen Tashi]

Are you familiar with how linear tranformations are represented (one might also say "implemented") by matrices? The statement appears to be a claim that a matrix which represents a rotation can be expressed as the product of a matrix representing scaling times a matrix representing shear.

 

[phiby]

Are you familiar with how linear tranformations are represented (one might also say "implemented") by matrices?

Yes, absolutely.

The statement appears to be a claim that a matrix which represents a rotation can be expressed as the product of a matrix representing scaling times a matrix representing shear.

For rotation of angle θ,
you could use
Scaling factors
Sx = Sy = cosθ
Shear factors
Shx = -sinθ
Shy = sinθ

This would make rotation as a combination of scaling and shear.

But I don't see what's the relevance of this. i.e. what's the point of mentioning this?

As examples of different places where I read this statement

http://docstore.mik.ua/orelly/java-ent/jfc/ch04_11.htm

http://techbase.kde.org/Development/Architecture/KDE3/Low-level_Graphics

etc etc

 

[Deveno]

$$A = \begin{bmatrix}\cos\theta&0\\0&\cos\theta \end{bmatrix}$$

is a scaling matrix (from a factor of 0 to 1).

$$B = \begin{bmatrix}1&\tan\theta\\-\tan\theta&1\end{bmatrix}$$

is a (dual) shear matrix (with a distortion factor of 1 to ∞).

AB is a rotation matrix.

for some graphical applications of matrices (in computer games, for example), matrices are used to quickly manipulate objects in well-known ways (rotating, stretching to fit, shape distortion to accommodate perspective, etc.). the popular windows program "microsoft paint" illustrates how some of these are implemented (for 2-D operations, 2x2 matrices are often "block-embedded" in 3x3 matrices, so affine transformations (changing the origin), can be used as well, while maintaining linearity). the graphical "handles" or "hot spots" one sees in various kinds of graphics, are often keyed to specific linear (or affine) transformations.


linear algebra is a computer programmer's friend, and the different ways of describing different operations often depend on what particular subroutines are available in the software package.

 

[blue_raver22]

I can provide an explanation for the statement that rotation is a combination of shearing and scaling in 2D transformations. In mathematics and computer graphics, transformations refer to the process of changing the position, size, or orientation of an object. In 2D transformations, an object is represented by a set of points in a 2D coordinate system. Rotation, shearing, and scaling are three common types of transformations used to manipulate objects in 2D space.

Rotation is the transformation that involves changing the orientation of an object by rotating it around a fixed point. This can be achieved by multiplying the coordinates of each point in the object by a rotation matrix. The rotation matrix is a 2x2 matrix that represents the amount and direction of rotation. When a point is multiplied by this matrix, its coordinates are changed to reflect the rotation.

Shearing, on the other hand, is a transformation that involves changing the shape of an object by shifting one set of coordinates in relation to another set. This is achieved by multiplying the coordinates of each point by a shear matrix. The shear matrix is also a 2x2 matrix that represents the amount and direction of shearing. When a point is multiplied by this matrix, its coordinates are changed to reflect the shear.

Lastly, scaling is a transformation that involves changing the size of an object by multiplying the coordinates of each point by a scaling matrix. The scaling matrix is a 2x2 matrix that represents the amount and direction of scaling. When a point is multiplied by this matrix, its coordinates are changed to reflect the scaling.

Now, let's consider what happens when we apply rotation, shearing, and scaling in different orders. When we apply rotation followed by shearing, the result is a transformation that combines both the rotation and shearing effects. Similarly, when we apply shearing followed by scaling, the result is a transformation that combines both the shearing and scaling effects. However, when we apply rotation followed by scaling, the result is a transformation that only has the rotation effect. This is because the scaling matrix does not affect the orientation of the object, but only its size.

In summary, rotation can be seen as a combination of shearing and scaling because when applied in different orders, it produces different transformations that include both shearing and scaling effects. This can be visually observed by looking at the transformation of an object when applying rotation, shearing, and scaling in different orders.

The proof for