Let us consider that the original point is \((x,y)\) and new transformed point…
2016
Let us consider that the original point is \((x,y)\) and new transformed point is \((x’,y’)\). Further \(Sh_𝑥\) and \(Sh_𝑦\) are shearing factors in \(x\) and \(y\) directions. If we perform the \(y\) direction shear relative to \(x=s_{𝑟𝑒𝑓}\) then the transformed point is given by
- A.
\(x’=x+Sh_x.(y-y_{ref}); \\ y’=y\) - B.
\(x’=x; \\ y’=y.Sh_x\) - C.
\(x’=x; \\ y’=Sh_y(x-x_{ref})+y\) - D.
\(x’=Sh_y.y; \\ y’=y.(x-x_{ref})\)
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Correct answer: C
Solution: Derivation of y-direction shear relative to x_ref.
Step 1: Translate coordinates so that x_ref becomes the origin: (x - x_ref, y).
Step 2: Apply a y-direction shear. The shear matrix for y-shear is [[1, 0], [Sh_y, 1]], which yields:
x_translated = x - x_ref
y_translated = y + Sh_y*(x - x_ref)
Step 3: Translate back to the original coordinate frame. For x:
x' = x_ref + (x - x_ref) = x
and for y:
y' = y + Sh_y*(x - x_ref)
Final formula: x' = x; y' = y + Sh_y*(x - x_ref).
Check: If x = x_ref then y' = y, so points on the vertical line x = x_ref are unchanged in y as required.