the "scale" and "transformation" values

[Philipp]

It's not entirely clear how the "scale" and "transformation" values are formed. For example, if I need to rotate a rectangle for 30° without distorting its angles and dimensions. By what formula should these quantities be calculated?

[Bhikkhu Pesala]

The values are trigonometric: try -0.5773 and 0.5773 (tan of 30°)

[Philipp]

 Thanks for the reply. However, I guessed that these are trigonometric functions. In this case, it's sin. But in order for the dimensions of the rectangle to be preserved, compensation for the "scale" values is necessary. I can't figure it out.
 Can you tell us the principle of the formula by which transformations occur by changes in these quantities?

[Philipp]

 Is my question clear enough? Maybe it should be rephrased and clarified? Or do you just not fully understand this mechanism of transformation either?

[Bhikkhu Pesala]

The help file does not elaborate. You will have to figure it out for yourself. Maybe this illustration will help you. It looks like the transformations skew the glyph in the x and y directions.

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[Philipp]

 Thank you, Bhikkhu Pesala, for Your work and illustration. I noticed this pattern. However, the essence of the question was to establish such a mathematical regularity in which a given figure retained not only its shape, but also its original dimensions. So in my drawing, the rectangle turned 30°.

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However, its length and width have changed. By changing the values of the "Scale X" "Scale Y" counters, you can achieve this manually, by selection method. However, this is not serious.
 So, for example, if there is a problem to rotate a triangle, the selection method will be very difficult. And if you need to rotate a triangle that is also scaled, it will be almost impossible to achieve this manually, by selection method.
 Therefore, the question is to describe a mathematical function, a formula, so as not to waste time on stupid inaccurate selection manually.
 And yes, experiments show that in this case this dependence is not a tangent, but a sinus.

[Philipp]

 Hurrah! Remembered! This is the matrix of rotation of the vector on the plane. https://en.wikipedia.org/wiki/Rotation_matrix
 

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 Now the triangle can be easily rotated without distortion. For example, an equilateral triangle rotates 60°.
 

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