Chapter 5.1

Notes

Linguistic Mapping Exercises

Knowledge Checks

  1. Take \(D\) to be the disk of radius \(4\) with center at \((1,3)\). Find a path that parameterizes the boundary of \(D\) in the clockwise direction.
  2. Take \(R\) to be the rectangle with vertices \((4,3), (7,5), (3,11)\) and \((0,9)\). Find a piecewise linear function \(c\) that parameterizes the boundary of \(R\) and simulate the motion of a particle whose position at time \(t\) is \(c(t)\).
  3. Take \(R\) to be the rectangle with vertices \((4,3), (7,5), (3,11)\) and \((0,9)\). Describe \(R\) as the feasible set of inequalities.
  4. Take \(R\) to be the rectangle with ordered vertex set \((4,3), (7,5), (3,11)\) and \((0,9)\). Determine whether \(R\) is positively oriented or negatively oriented.
  5. Take \(R\) to be the rectangle with ordered vertex set \((4,3), (7,5), (3,11)\) and \((0,9)\). Find a translation and rotation so that \((4,3)\) is moved to \((0,0)\) and one of the edges of \(R\) is on the \(x\)-axis.
  6. Determine the circumcircle for the triangle \(\Delta\) with vertex set \(\{(1,3),(2,4),(1,5)\}.\)
  7. 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.
  8. 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.
  9. Take \(\square(1,1)(4,1)(5,2)(2,2)\) to be the parallelogram with the given ordered vertex set. Calculate \(\mathcal{A}(\square (1,1)(4,1)(5,2)(2,2)).\)
  10. 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. Compare that to the answer you would get if you use the shoelace formula to determine the area of \(P\).

Remember, do the knowledge checks before checking the solutions.

Practice Problems

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