12 multiple-choice questions, progressively harder.
The directed edge goes from P(2,3)P(2,3)P(2,3) to Q(5,7)Q(5,7)Q(5,7). What is its shoelace product difference?
Solution
Correct answer: B
Multiply the first vertex's horizontal coordinate by the second vertex's vertical coordinate. Subtract the product of the other two coordinates.
An edge runs from (−4,3)(-4,3)(−4,3) to (−4,8)(-4,8)(−4,8) above a horizontal reference line. What is its signed trapezoid contribution?
Correct answer: D
The horizontal coordinate stays at −4-4−4, so the signed width is zero. Multiplying by any average height gives zero.
Triangle ABCABCABC has boundary vertices A(1,1)A(1,1)A(1,1), B(5,2)B(5,2)B(5,2), and C(2,5)C(2,5)C(2,5). What is its ordinary area?
Correct answer: A
The differences for A→BA\to BA→B, B→CB\to CB→C, and C→AC\to AC→A are −3-3−3, 212121, and −3-3−3.
The signed area is 15/215/215/2, so the ordinary area is 15/215/215/2 square units.
In boundary order, the first shoelace product total of a simple polygon is 272727. The second is 363636. What is the signed area?
Correct answer: C
Subtract the second total from the first. Divide by two and keep the sign because the question asks for signed area.
An ordered vertex list starts at P(2,4)P(2,4)P(2,4) and its final listed vertex is T(−1,3)T(-1,3)T(−1,3). What is the product difference for the closing edge?
The closing edge is T→PT\to PT→P, so the coordinates of TTT come first in the two products.
The arrows give a boundary walk. Which vertex should come immediately after LLL if you start the walk at LLL?
Read the outgoing arrow at LLL. Changing the starting point leaves that arrow in place.
A simple polygon is listed clockwise. Which statement describes its signed area?
Clockwise order reverses each edge from its counterclockwise direction. For positive ordinary area KKK, the signed area SSS is its negative.
The four product differences for a simple polygon are 777, −3-3−3, 999, and −1-1−1. What is its ordinary area?
Add the differences with their signs before halving and taking the absolute value.
The ordinary area is 6 square units.
An edge runs from (3,1)(3,1)(3,1) to (−1,4)(-1,4)(−1,4). Use y=−2y=-2y=−2 as the reference line. What is the signed trapezoid contribution?
The edge moves 4 units left. Its endpoint heights are 3 and 6 units above the reference line.
In a vertical strip, one piece of a counterclockwise polygon lies between an upper edge and a lower edge. Which trapezoid is subtracted?
The interior lies above the lower edge. To keep the interior on the left, the lower edge travels rightward. Its signed trapezoid contribution is negative, so subtract the lower trapezoid.
The signed area of a simple polygon in the listed order is 131313. Which direction does the list follow?
The signed area is positive.
With the usual coordinate axes, this sign means the boundary is counterclockwise.
Which description is enough to show that a simple polygon is a lattice polygon?
A lattice polygon has all its vertices at integer grid points. For each vertex PiP_iPi, both coordinates must be integers.
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