Question

A grid PQRS is distorted such that its edges get stretched or compressed UNIFORMLY to form P’Q’R’S’ as shown on the left. Identify the correct distorted grid from the options.

• A. Fig A
• B. Fig B
• C. Fig C
• D. Fig D

Hint:

Linear stretching of boundaries always preserves the straightness of grid lines. If the problem states "uniform stretch," you can immediately eliminate any option that contains curved lines.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This question belongs to the topic of

Projective Transformations and Grid Distortions.
We are given a square grid PQRS that undergoes a uniform boundary stretch and compression to form a new quadrilateral P'Q'R'S'. We need to identify which of the options correctly represents this uniformly distorted grid.

Step 2: Key Formula or Approach:

• "Uniformly stretched or compressed" means that the relative spacing between adjacent grid line endpoints along any given boundary edge remains equal (constant spacing along the edge).

• In a 2D uniform linear transformation, straight grid lines connecting opposite boundaries must remain perfectly straight lines; they do not bend or curve.

• Therefore, any option containing curved grid lines (like B or C) is mathematically incorrect.


Step 3: Detailed Explanation:

• Let us analyze the boundary edges of P'Q'R'S':
- The boundary lines of P'Q'R'S' are straight lines.
- If we divide each of these four straight edges into 10 equal segments, and connect the corresponding division points on opposite sides with straight lines, we generate a bilinear patch.
- Because the division points along the boundaries are uniformly spaced, the internal grid lines must also be straight and show a linear gradient of spacing from one side to the other.

• Let us evaluate the options:
- Option A: Shows straight grid lines with uniform spacing along each boundary edge. This is the mathematically correct bilinear grid.
- Option B and C: Show curved lines inside, which would only happen if the stretching was non-uniform or if there was a warp applied.
- Option D: Shows straight lines, but the spacing along the edges is compressed towards the center, which violates the "uniform" stretching condition.


Step 4: Final Answer:
Option (A) is the correct distorted grid.