Why shearing multiplying by y? [closed]

We know that shear is transformation that distorts the shape of the object. Shear relative to X-axis that is y=0 line can be written as: x′=x+shx.y, y′=y.My question is why shx is multiplying by y,why not any other constant ? I want to understand intuition behind this. Every book, website don't mention any reason.

• Seems like a math question, not a design question. Commented Oct 25, 2021 at 18:42
• @Scott In designing shearing is also used.. Commented Oct 25, 2021 at 18:56
• Yeah.. but so is a computer.. that doesn't make questions about how install Windows/MacOS on topic. Just my opinion.. I may be wrong. `:)` Commented Oct 25, 2021 at 19:09

We have a problem: Your formulas look like you have coordinate system and math notation which differs radically from that versions which most of us saw in the elementary school. But we can check what shearing means in the common elementary school geometry.

In the next image the blue rectangle must be mapped to the sheared version which is the interior of the dashed polygon:

S is the vertical shearing angle.

One should see that a point (x,y) in the blue rectangle should be moved only vertically. So, the mapped coordinate point (x',y') must have the original X-ccordinate i.e. x'= x

The Y-coordinate of the mapped point must be modified. It must get an addition (=displacement) which is proportional to x. The formula for a thing which is proportional to x is A * x where the multiplier A is a constant. So we have formula for the mapped Y-coordinate:

y' = y + (A * x)

From elementary trigonometry we should recall that A is the tangent of the shearing angle S.

Line y=0 is the X-axis. It will be mapped onto the lower tilted dashed edge. If we set y = 0 in the equation of y' we get y' = A * x. That's the equation of the bottom tilted dashed line.

I have no idea from where you have got your formulas. But what I wrote is the mapping and it will not be changed to your version if one turns upside down the direction of the Y-axis like programs Photoshop, Illustrator etc. do.

The equation of the mapped line y = 0 can be reversed to equivalent form x = (1/A) * y' but that's neither your version.

Shearing horizontally also doesn't give your version. In horizontally sheared image line Y=0 stays intact. But horizontal shearing gives something that resembles a part of your question:

Point (x,y) in the blue rectangle should be mapped to (x',y')

where y' = y and x' = x + (tan(S) * y)

A guess based on zero programming knowledge: the dot in x′=x+shx.y is not multiplying but a name separator in a programming language. shx is something which returns a numerical value for math operations. y could be the name of the input for shx or the name of a numerical property of shx

• No it is a multiplication allright. K *x is a wedge. Shx is just the amount of linear shear at y=1 (trigonometric functions are computationally heavy so you dont want to do that for each element if you can avoid it so you convert it to a linear operation). But essentially this is a matrix opearation written out. So linear algebra. Anyway you dont actually need to do more than one shear you can transform it to any other shear with T^-1 . Shear . T where . Is matrix multiplication. Hence the notation above. Commented Oct 26, 2021 at 13:37