Let there be a triangle with coordinates A(2,2) , B(5,2) and C(5,4)(adsbygoogle = window.adsbygoogle || []).push({});

I have learned two ways to shear an object(x-axis invariant) in the Cartesian coordinate system.

The first way is to find the coordinate vector of a point and multiply the y-component of the vector by the shear factor(2)(Don't change the value) and replace the x-component with it.

The second way is to use the transformation matrix and multiply it with the coordinate matrix.

So the transformation matrix is ##\begin{pmatrix}1 & 2\\ 0 & 1 \end{pmatrix}## and the cordinate matrix: ##\begin{pmatrix}2 & 5 & 5\\ 2 & 2 & 4 \end{pmatrix}## and the result is :

$$\begin{pmatrix}6 & 9 & 13\\ 2 & 2 & 4 \end{pmatrix}$$

This gives the same answer as the first method.However,the first method is easy to understand.

I realised that the matrix method does the same thing.It leaves the y-components as it is and add the ##\text{Shear factor} \times \text{Y-component}## to the x-component.

So when the matrix theory(or whatever) was made,that person must have considered these.The multiplication rule:Row and column made this transformation matrices useful.

My teacher asked to replace the ##0## with the shear factor.

I was wondering how did someone find this rule about replacing the zero?By guessing?

So can I make a transformation matrix which would translate,reflect,rotate,enlarge,stretch and shear it at the same time?

My main question here is how to make a transformation matrix which would transform the object in the way I like.

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# Matrix and transformation problem

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