Rotating a Search Pattern to Match the Drone's Heading
A search pattern for a small uncrewed aerial vehicle is stored as offsets in the aircraft's own frame, written as (forward, left) in metres from the launch point: W0 (0, 0); W1 (0, 6); W2 (8, 6); W3 (8, 0). The aircraft flies the waypoints in order and returns to W0, sweeping the rectangle they enclose. The launch point is the origin of a map grid in which +x is east and +y is north. The heading is fixed before launch and does not change in flight. The camera footprint is small compared with the pattern, so coverage equals the swept rectangle. A person of interest is reported at map coordinate (5, 4). The operator must choose the launch heading on the ground, before take-off, and cannot change it later. Wind is negligible and the pattern is flown at constant altitude.
1. Convert all four waypoints to map coordinates for a launch on heading 090 (east), and state whether the rectangle covers (5, 4).
Read the worked answer
On heading 090 the aircraft's forward axis points east and its left axis points north, so the transform is the identity: (forward, left) → (x, y).
- W1 (0, 6) → (0, 6)
- W2 (8, 6) → (8, 6)
- W3 (8, 0) → (8, 0)
- W0 returns to the origin.
Coverage is the rectangle x ∈ [0, 8], y ∈ [0, 6].
The target (5, 4) satisfies 0 ≤ 5 ≤ 8 and 0 ≤ 4 ≤ 6, so yes, heading 090 covers it. Note the pattern is anchored at a corner (the launch point), not at its centre - that is why heading choice matters so much.
2. Give the transformed W2 for headings 000 and 270, and say whether either covers the target.
Read the worked answer
Heading 000 (north): forward is +y, left is −x, so (f, l) → (−l, f). W2 (8, 6) → (−6, 8). Corners: (0,0), (−6,0), (−6,8), (0,8). Coverage x ∈ [−6, 0], y ∈ [0, 8]. Target (5, 4) has x = 5 outside the range → not covered.
Heading 270 (west): forward is −x, left is −y, so (f, l) → (−f, −l). W2 (8, 6) → (−8, −6). Corners: (0,0), (0,−6), (−8,−6), (−8,0). Coverage x ∈ [−8, 0], y ∈ [−6, 0]. Target x = 5 outside → not covered.
Only heading 090 puts the footprint in the north-east quadrant where the target sits.
3. The report is corrected: the target is at (5, −4). Which heading now works, and what does the answer reveal about the pattern's symmetry?
Read the worked answer
Heading 180 (south): forward is −y, left is +x, so (f, l) → (l, −f).
- W1 (0, 6) → (6, 0)
- W2 (8, 6) → (6, −8)
- W3 (8, 0) → (0, −8)
Coverage x ∈ [0, 6], y ∈ [−8, 0]. Target (5, −4) is inside → heading 180 works.
What it reveals: the footprint under heading 180 is exactly the heading-090 footprint rotated 90° clockwise about the origin, which is consistent with a south-east target. The pattern's own internal symmetry - an 8 × 6 rectangle is symmetric under 180° rotation about its centre - is irrelevant, because the aircraft rotates about the launch point at a corner. Do not reason from the shape's symmetry; reason from the transform of the corner.