A convex hull can bridge a concavity. Its enclosed area can include locations outside the original region.
Retain the enclosure definition
A convex hull encloses a selected set of points with a convex boundary. Area computed from that hull belongs to the enclosure. It need not equal the area of an original concave region whose outline supplied the points, because an indentation can be bridged by the enclosing geometry. Primary reference: MathWorks convex hull and hull area.
For an original worksheet, keep the ordered original boundary and the hull boundary separately. Identify whether the inputs are point locations or pixel-cell support. Do not silently substitute an enclosing descriptor for the original occupied region when the question concerns missing or empty space.
Compare a fictional concave polygon
Use the invented polygon (0,0), (4,0), (4,1), (1,1), (1,4), (0,4). It forms two unit-wide arms with original area 4 + 4 − 1 = 7. Its convex hull bridges the indentation from (4,1) to (1,4), enclosing area 11.5.
The hull therefore includes an additional 4.5 square coordinate units outside the original polygon. These are continuous polygon areas, not a claim about a particular pixel-raster hull convention. The teaching shape does not establish physical material area or a defect measurement.
Inspect the bridged region
Prepare an original overlay coloring the original region and the added enclosure separately. If a hull-based descriptor changes, inspect the input boundary and the bridged space before explaining the result. Retain the coordinate definition so continuous polygon area is not mixed with foreground sample counts.
This guide gives no universal concavity threshold and no HM inspection capability. Its reader decision is whether an enclosing hull has been identified as such. Empty space admitted by the enclosure should remain visible rather than being described as originally occupied foreground.
Customer Questions
What is the original fictional polygon area?
Seven square coordinate units.
What is its hull area?
Eleven point five square coordinate units.
Is all hull area originally occupied?
No. The enclosure bridges empty concave space.
Primary References
These references support the technical principles discussed in this guide. The worked examples and review questions are educational.
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