An inverse transformation undoes a transformation: apply the original mapping to a point, then apply its inverse to recover the starting point. For geometric work, this is how transformed coordinates can be brought back to the original shape. section 4.3.2
Projective transformations make this especially practical. Homogeneous coordinates express translation, rotation, scaling, shear, and their compositions as matrix multiplications. Because a projective point may be represented by scaled coordinate tuples, one transformation has a continuum of equivalent matrices; that freedom can make an inverse transformation cheaper than computing an ordinary inverse matrix. section 4.0 section 4.7
A useful workflow is:
- Compose the required geometric steps into one transformation matrix, rather than repeatedly transforming every point. section 4.3.2
- Transform the points of the shape with the composed matrix. section 4.3.2
- Compute an inverse matrix, or an equivalent projective inverse representation, and apply it to the transformed points to recover their original coordinates. section 4.3.2 section 4.7
Rotating a rectangle and reversing it
A rectangle centered near (1, 1) is translated to the origin, rotated 90 degrees clockwise, and translated back. The three 3×3 matrices are multiplied into one matrix c. In SymPy, c.inv() produces c_inv; multiplying c_inv by each transformed corner restores the four original corners, including (0.75, 0.5) and (1.25, 1.5). section 4.3.2 section 4.3.2
pt1 = c * p1 pt2 = c * p2 pt3 = c * p3 pt4 = c * p4 ... print(c_inv * pt1) print(c_inv * pt2) print(c_inv * pt3) print(c_inv * pt4)
This SymPy listing applies a composed transformation and then its inverse to each rectangle corner, making the recovery check explicit. section 4.3.2
Not every transformation has a usable one-to-one inverse. Biquadratic and polynomial transformations can map different input points to the same output point, so reversing them may be ambiguous rather than a function. section 10.5
