Which of the following equations is/are incorrect for the given grid conversion formulas?

• A. i, ii, iii, vi
• B. i, ii, iv only
• C. iii, iv only
• D. ii, iv only

Answer: (B)

Grid coordinate conversion uses a 2D similarity transformation: rotate the coordinates by a specific angle to align the axes, apply a uniform scale to account for differences in unit length, and then translate to the target grid origin. The rotation must be applied with the same scale factor to both axes, and the translation should occur after the rotation and scaling, so the final coordinates reflect the intended grid alignment.

If an equation leaves out the rotation terms or uses the sine and cosine with the wrong signs, the axes won’t line up as intended and the mapping will be off. If the scale is not applied uniformly to both coordinates or is incorporated in an inconsistent way, distances and angles won’t be preserved properly. If translation is misapplied—taken before rotation or placed inside the rotation inappropriately—the final coordinates will end up in the wrong location on the grid.

Thus, only the form that adheres to the proper rotation plus uniform scale about the correct origin is correct; the others fail for these reasons, which is why that set of equations is identified as incorrect.