Chapter 5.2

Notes

Linguistic Mapping Exercises

Knowledge Checks

  1. Take \(p_1=(1,1),p_2=(4,2),p_3=(2,3)\). Use the shoelace formula to determine the area of the triangle \(\Delta p_1p_2p_3\) and to determine whether or not the triangle is positively or negatively oriented.

  2. Take \(p_1=(1,1),p_2=(4,2),p_3=(2,3)\). Use Heron’s formula to determine the area of the triangle \(\Delta p_1p_2p_3\) by first finding the length of each of its sides.

  3. Take \(\square(1,1)(4,1)(5,2)(2,2)\) to be the parallelogram with the given ordered vertex set. Calculate \(\mathcal{A}(\square (1,1)(4,1)(5,2)(2,2)).\)

  4. Take \(((-1,1),(1,-2),(3,-1),(4,3))\) to be the ordered vertex set of a polygonal path with epoch \((0,2,3,6,10)\). Identify the given path with a piecewise linear function, \(c\). Simulate a particle whose position at time \(t\) is \(c(t)\).

  5. Take \(((-1,1),(1,-2),(3,-1),(4,3))\) to be the ordered vertex set of a polygon \(P\). Identify a positively oriented triangulation for \(P\) and identify all of the triangles in this triangulation. Calculate the area of your triangulation.

  6. Take \(((-1,1),(1,-2),(3,-1),(4,3))\) to be the ordered vertex set of a polygon \(P\). Identify a negatively oriented triangulation for \(P\) and identify all of the triangles in this triangulation.

  7. Take \(((-1,1),(1,-2),(3,-1),(4,3))\) to be the ordered vertex set of a polygon \(P\). Use the shoelace formula to determine the area of \(P\) and the orientation of \(\partial_o P\).

Practice Problems

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