Question

Let\(g\left( t \right)\) be defined as in Exercise 21. Its graph is called a quadratic Bézier curve, and it is used in some computer graphics designs. The points\({p_o}\),\({p_1}\), and\({p_2}\)are called the control points for the curve. Compute a formula for\(g\left( t \right)\)that involves only\({p_o}\),\({p_1}\), and\({p_2}\). Then show that\(g\left( t \right)\)is in\({\rm{conv }}\left\{ {{{\rm{p}}_{\rm{o}}}{\rm{,}}{{\rm{p}}_{\rm{1}}}{\rm{,}}{{\rm{p}}_{\rm{2}}}} \right\}\)for\(0 < t < 1\).

Short answer

\(g\left( t \right) = {\left( {1 - t} \right)^2}{{\bf{p}}_{\bf{o}}} + 2t\left( {1 - t} \right){{\bf{p}}_{\bf{1}}} + {t^2}{{\bf{p}}_{\bf{2}}}\)

It shows that the convex combination of \(\left\{ {{p_o},\,{p_1},\,{p_2}} \right\}\).

Step-by-step solution

Step 1:Use the definition of affine hull

Assume that \({{\bf{p}}_{\bf{o}}}{\bf{,}}\,{{\bf{p}}_{\bf{1}}} \in {f_o}\left( t \right)\) and \({{\bf{p}}_{\bf{1}}}{\bf{,}}\,{{\bf{p}}_{\bf{2}}} \in {f_1}\left( t \right)\).

The affine hull of distinct points \({v_1}\) and \({v_2}\) is \(y = \left( {1 - t} \right){v_1} + t{v_2}\).

Similarly, the affine hull of \({f_o}\left( t \right)\)and \({f_1}\left( t \right)\)is \(g\left( t \right) = \left( {1 - t} \right){f_o}\left( t \right) + t{f_1}\left( t \right)\).

Step 2:Apply the affine hull for \({f_o}\left( t \right)\), and\({f_1}\left( t \right)\)

\(\begin{aligned}{}g\left( t \right) &= \left( {1 - t} \right)\left( {\left( {1 - t} \right){{\bf{p}}_{\bf{o}}} + t{{\bf{p}}_{\bf{1}}}} \right) + t\left( {\left( {1 - t} \right){{\bf{p}}_{\bf{1}}} + t{{\bf{p}}_{\bf{2}}}} \right)\\ &= {\left( {1 - t} \right)^2}{{\bf{p}}_{\bf{o}}} + 2t\left( {1 - t} \right){{\bf{p}}_{\bf{1}}} + {t^2}{{\bf{p}}_{\bf{2}}}\end{aligned}\)

Use the concept that weights in linear combination sum to 1

A point \(y\) in\({\mathbb{R}^n}\) is an affine combination of \({v_1},.......,{v_p}\)in \({\mathbb{R}^n}\) if \(\overline y = {c_1}{\overline v _1} + ...... + {c_p}{\overline v _p}\) such that \({c_1} + ...... + {c_p} = 1\). So, the sum of the weight of \(g\left( t \right)\) should be 1; that is, \({\left( {1 - t} \right)^2} + 2t\left( {1 - t} \right) + {t^2} = 1\), which is simplified to \(\left( {1 - 2t + {t^2}} \right) + \left( {2t - 2{t^2}} \right) + {t^2} = 1\).

Find the range of weight when \(0 \le t \le 1\)

The weighted sum is \(\left( {1 - 2t + {t^2}} \right) + \left( {2t - 2{t^2}} \right) + {t^2} = 1\). This sum also varies between 0 and 1 if \(0 \le t \le 1\) . This implies that a convex combination of \(\left\{ {{{\bf{p}}_{\bf{o}}}{\bf{,}}\,{{\bf{p}}_{\bf{1}}}{\bf{,}}\,{{\bf{p}}_{\bf{2}}}} \right\}\).