Question 5 of 12: 2-D Transformation for Digitized Map Corners

Question 5: 2-D Transformation for Digitized Map Corners (5 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

The appropriate transformation is a 2-D affine (six-parameter) transformation. When a map sheet is registered on a digitizer (or scanned and registered on screen) by its four corners, the raw table/pixel coordinates must be converted to the map's real-world coordinate system. The affine transformation, $X = a_0 + a_1 x + a_2 y$ and $Y = b_0 + b_1 x + b_2 y$, models the four effects that actually occur in this situation: translation (the sheet's origin is offset), rotation (the sheet is not aligned with the table axes), differential scale in $x$ and $y$ (paper and film shrink or stretch by different amounts along and across the sheet), and skew/non-orthogonality of the axes. It has six parameters, so it needs a minimum of three control points; digitizing all four corners gives four points — two redundant observations — which lets the transformation be solved by least squares and yields residuals that flag a mis-digitized corner. A four-parameter similarity (conformal) transformation would be wrong here because it forces a single uniform scale and cannot absorb the differential shrinkage of the medium. If the four corners must be honoured exactly (zero residuals) and the sheet has trapezoidal distortion, a projective or bilinear transformation using all four points is the alternative; but for routine map-sheet registration the least-squares affine is the standard, robust choice.