Let \({{\bf{p}}_o}\) , \({{\bf{p}}_1}\) and \({{\bf{p}}_2}\) be points in \({\mathbb{R}^n}\) , and define \({{\bf{f}}_{\bf{0}}}\left( t \right) = \left( {1 - t} \right){{\bf{p}}_{\bf{0}}} + t{{\bf{p}}_{\bf{1}}}\), \({{\bf{f}}_1}\left( t \right) = \left( {1 - t} \right){{\bf{p}}_1} + t{{\bf{p}}_2}\) and \({\bf{g}}\left( t \right) = \left( {1 - t} \right){{\bf{f}}_0}\left( t \right) + t{{\bf{f}}_1}\left( t \right)\)for \(0 \le t \le 1\). For the points as shown below, draw a picture that shows \({{\bf{f}}_0}\left( {\frac{1}{2}} \right)\), \({{\bf{f}}_1}\left( {\frac{1}{2}} \right)\), and \({\bf{g}}\left( {\frac{1}{2}} \right)\).

It is given that \({{\rm{p}}_0},{{\rm{p}}_1},{{\rm{p}}_2} \in {\mathbb{R}^n}\). Also,\({{\rm{f}}_0}\left( t \right) = \left( {1 - t} \right){{\rm{p}}_0} + t{{\rm{p}}_1}\), \({{\rm{f}}_1}\left( t \right) = \left( {1 - t} \right){{\rm{p}}_1} + t{{\rm{p}}_2}\)and \({\rm{g}}\left( t \right) = \left( {1 - t} \right){{\rm{f}}_0}\left( t \right) + t{{\rm{f}}_1}\left( t \right)\).The graph of\({{\rm{f}}_0}\left( {\frac{1}{2}} \right)\),\({{\rm{f}}_1}\left( {\frac{1}{2}} \right)\) and\({\rm{g}}\left( {\frac{1}{2}} \right)\) is to be drawn.

The values of \({{\rm{f}}_0}\left( {\frac{1}{2}} \right)\),\({{\rm{f}}_1}\left( {\frac{1}{2}} \right)\) and\({\rm{g}}\left( {\frac{1}{2}} \right)\)are calculated as shown below:

\(\begin{aligned}{}{{\rm{f}}_0}\left( {\frac{1}{2}} \right) &= \left( {1 - \frac{1}{2}} \right){{\rm{p}}_0} + \frac{1}{2}{{\rm{p}}_1}\\ &= \frac{1}{2}{{\rm{p}}_0} + \frac{1}{2}{{\rm{p}}_1}\\ &= \frac{1}{2}\left( {{{\rm{p}}_0} + {{\rm{p}}_1}} \right)\end{aligned}\)

It shows that\({{\rm{f}}_0}\left( {\frac{1}{2}} \right)\)is at the midline formed by the points\({{\rm{p}}_0}{\rm{, }}{{\rm{p}}_1}\).

\(\begin{aligned}{}{{\rm{f}}_1}\left( {\frac{1}{2}} \right) &= \left( {1 - \frac{1}{2}} \right){{\rm{p}}_1} + \frac{1}{2}{{\rm{p}}_2}\\ &= \frac{1}{2}{{\rm{p}}_1} + \frac{1}{2}{{\rm{p}}_2}\\ &= \frac{1}{2}\left( {{{\rm{p}}_1} + {{\rm{p}}_2}} \right)\end{aligned}\)

It shows that \({{\rm{f}}_1}\left( {\frac{1}{2}} \right)\)is at the midline formed by the points\({{\rm{p}}_1}{\rm{, }}{{\rm{p}}_2}\).

\(\begin{aligned}{}{\rm{g}}\left( {\frac{1}{2}} \right) &= \left( {1 - \frac{1}{2}} \right){{\rm{f}}_0}\left( {\frac{1}{2}} \right) + \frac{1}{2}{{\rm{f}}_1}\left( {\frac{1}{2}} \right)\\ &= \frac{1}{2}{{\rm{f}}_0}\left( {\frac{1}{2}} \right) + \frac{1}{2}{{\rm{f}}_1}\left( {\frac{1}{2}} \right)\\\frac{1}{2}\left( {{{\rm{f}}_0}\left( {\frac{1}{2}} \right) + {{\rm{f}}_1}\left( {\frac{1}{2}} \right)} \right)\end{aligned}\)

It shows that \({\rm{g}}\left( {\frac{1}{2}} \right)\) is at the midline formed by the points of \({{\rm{f}}_0}\left( {\frac{1}{2}} \right)\),\({{\rm{f}}_1}\left( {\frac{1}{2}} \right)\).

Draw lines joining\({{\rm{p}}_0}{\rm{, }}{{\rm{p}}_1}\), then\({{\rm{p}}_1}{\rm{, }}{{\rm{p}}_2}\),such that\({{\rm{f}}_0}\left( {\frac{1}{2}} \right)\),\({{\rm{f}}_1}\left( {\frac{1}{2}} \right)\)bisects them respectively. Afterward, the value of\({\rm{g}}\left( {\frac{1}{2}} \right)\)is at the mid of the line joining\({{\rm{f}}_0}\left( {\frac{1}{2}} \right)\),\({{\rm{f}}_1}\left( {\frac{1}{2}} \right)\). Thus, the diagram is shown below: