Question

Let \(S = \left\{ {{{\bf{v}}_1},\,{{\bf{v}}_2},\,{{\bf{v}}_3},\,{{\bf{v}}_4}} \right\}\) be an affinely independent set. Consider the points \({{\bf{p}}_{\bf{1}}},.....,{{\bf{p}}_{\bf{5}}}\) whose barycentric coordinates with respect to S are given by \(\left( {{\bf{2}},{\bf{0}},{\bf{0}}, - {\bf{1}}} \right)\), \(\left( {{\bf{0}},\frac{{\bf{1}}}{{\bf{2}}},\frac{{\bf{1}}}{{\bf{4}}},\frac{{\bf{1}}}{{\bf{4}}}} \right)\), \(\left( {\frac{{\bf{1}}}{{\bf{2}}},{\bf{0}},\frac{{\bf{3}}}{{\bf{2}}}, - {\bf{1}}} \right)\), \(\left( {\frac{{\bf{1}}}{{\bf{3}}},\frac{{\bf{1}}}{{\bf{4}}},\frac{{\bf{1}}}{{\bf{4}}},\frac{{\bf{1}}}{{\bf{6}}}} \right)\), and \(\left( {\frac{{\bf{1}}}{{\bf{3}}},{\bf{0}},\frac{{\bf{2}}}{{\bf{3}}},{\bf{0}}} \right)\), respectively. Determine whether each of \({{\bf{p}}_{\bf{1}}},.....,{{\bf{p}}_{\bf{5}}}\) is inside,outside, or on the surface of conv S, a tetrahedron. Are any of these points on an edge of conv S?

Short answer

\({{\bf{p}}_1}\) and \({{\bf{p}}_3}\) are outside the tetrahedron \({\rm{conv}}\,\,S\). \({{\bf{p}}_2}\) is on the face with vertices \({{\bf{v}}_2}\), \({{\bf{v}}_3}\), and \({{\bf{v}}_4}\). \({{\bf{p}}_4}\) is inside \({\rm{conv}}\,S\). \({{\bf{p}}_5}\) is on the edge between \({{\bf{v}}_1}\) and \({{\bf{v}}_3}\).

Step-by-step solution

Check for the first coordinate

The barycentric coordinatesof \({{\bf{p}}_1}\) and \({{\bf{p}}_3}\)has negative numbers, so \({{\bf{p}}_1}\) and \({{\bf{p}}_3}\) are outside the tetrahedron convex S.

Check for the second coordinate

For point \({{\bf{p}}_2}\), the first number is zero. Hence, \({{\bf{p}}_2}\) is on the face containing \({{\bf{v}}_2}\), \({{\bf{v}}_3}\), and \({{\bf{v}}_4}\).

Check for the third coordinate

The barycentric coordinates of the point \({{\bf{p}}_4}\) are positive, hence \({{\bf{p}}_4}\) is inside convex S.

Check for the fourth coordinate

Two barycentric coordinates in \({{\bf{p}}_5}\) are positive, and two of them are zero. Hence, \({{\bf{p}}_5}\) is on the edge of \({{\bf{v}}_1}\) and \({{\bf{v}}_3}\).