Question

Given control points \({{\rm{p}}_{\rm{o}}}\) , \({{\rm{p}}_{\rm{1}}}\) , \({{\rm{p}}_{\rm{2}}}\) and \({{\rm{p}}_{\rm{3}}}\) in \({\mathbb{R}^n}\) , let \({g_1}\left( t \right)\)for \(0 \le t \le 1\) be the quadratic Bézier curve from Exercise 23 determined by \({{\rm{p}}_{\rm{o}}}\) , \({{\rm{p}}_{\rm{1}}}\) , and \({{\rm{p}}_{\rm{2}}}\), and let \({g_2}\left( t \right)\)be defined similarly for \({{\rm{p}}_{\rm{1}}}\) , \({{\rm{p}}_{\rm{2}}}\) and \({{\rm{p}}_{\rm{3}}}\). For \(0 \le t \le 1\), define \(h\left( t \right) = \left( {1 - t} \right){g_1}\left( t \right) + t{g_2}\left( t \right)\). Show that the graph of \(h\left( t \right)\)lies in the convex hull of the four control points. This curve is called a cubic Bézier curve, and its definition here is one step in an algorithm for constructing Bézier curves (discussed later in Section 8.6). A Bézier curve of degree \(k\) is determined by \(k + 1\) control points, and its graph lies in the convex hull of these control points.

Short answer

\({\bf{h}}\left( t \right) = \left( {1 - 3t + 3{t^2} - {t^3}} \right){{\bf{p}}_{\bf{o}}} + \left( {3t - 6{t^2} + 3{t^3}} \right){{\bf{p}}_{\bf{1}}} + \left( {3{t^2} - 3{t^3}} \right){{\bf{p}}_{\bf{2}}} + {t^3}{{\bf{p}}_{\bf{3}}}\)

It is shown that \({\bf{h}}\left( t \right)\)is the convex combination of \(\left\{ {{{\bf{p}}_{\bf{o}}}{\bf{,}}\,{{\bf{p}}_{\bf{1}}}{\bf{,}}\,{{\bf{p}}_{\bf{2}}}{\bf{,}}{{\bf{p}}_{\bf{3}}}} \right\}\).

Step-by-step solution

Use the definition of affine hull

Assume that \({{\bf{p}}_{\bf{o}}}{\bf{,}}\,{{\bf{p}}_{\bf{1}}}{\bf{,}}{{\bf{p}}_{\bf{2}}} \in {{\bf{g}}_{\bf{o}}}\left( t \right)\) and \({{\bf{p}}_{\bf{1}}}{\bf{,}}\,{{\bf{p}}_{\bf{2}}}{\bf{,}}{{\bf{p}}_{\bf{3}}} \in {{\bf{g}}_{\bf{1}}}\left( t \right)\).

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 \({{\bf{g}}_{\bf{o}}}\left( t \right)\)and \({{\bf{g}}_{\bf{1}}}\left( t \right)\)is \({\bf{h}}\left( t \right) = \left( {1 - t} \right){{\bf{g}}_{\bf{o}}}\left( t \right) + t{{\bf{g}}_{\bf{1}}}\left( t \right)\).

 Apply the affine hull for \({g_o}\left( t \right)\), and\({g_1}\left( t \right)\)

\(\begin{aligned}{}{\bf{h}}\left( t \right) = \left( {1 - t} \right)\left( {{{\left( {1 - t} \right)}^2}{{\bf{p}}_{\bf{o}}} + 2t\left( {1 - t} \right){p_1} + {t^2}{p_2}} \right) + t\left( {{{\left( {1 - t} \right)}^2}{{\bf{p}}_{\bf{1}}} + 2t\left( {1 - t} \right){{\bf{p}}_{\bf{2}}} + {t^2}{{\bf{p}}_{\bf{3}}}} \right)\\ = {\left( {1 - t} \right)^3}{{\bf{p}}_{\bf{o}}} + 2t\left( {1 - 2t + {t^2}} \right){{\bf{p}}_{\bf{1}}} + \left( {{t^2} - {t^3}} \right){{\bf{p}}_{\bf{2}}} + t\left( {1 - 2t + {t^2}} \right){{\bf{p}}_{\bf{1}}} + 2{t^2}\left( {1 - t} \right){{\bf{p}}_{\bf{2}}} + {t^3}{{\bf{p}}_{\bf{3}}}\\ = \left( {1 - 3t + 3{t^2} - {t^3}} \right){{\bf{p}}_{\bf{o}}} + \left( {2t - 4{t^2} + 2{t^3}} \right){{\bf{p}}_{\bf{1}}} + \left( {{t^2} - {t^3}} \right){{\bf{p}}_{\bf{2}}} + \left( {t - 2{t^2} + {t^3}} \right){{\bf{p}}_{\bf{1}}}\\ + \left( {2{t^2} - 2{t^3}} \right){{\bf{p}}_{\bf{2}}} + {t^3}{{\bf{p}}_{\bf{3}}}\\ = \left( {1 - 3t + 3{t^2} - {t^3}} \right){{\bf{p}}_{\bf{o}}} + \left( {3t - 6{t^2} + 3{t^3}} \right){{\bf{p}}_{\bf{1}}} + \left( {3{t^2} - 3{t^3}} \right){p_2} + {t^3}{{\bf{p}}_{\bf{3}}}\end{aligned}\)

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 \(h\left( t \right)\) should be 1; that is, \(\left( {1 - 3t + 3{t^2} - {t^3}} \right) + \left( {3t - 6{t^2} + 3{t^3}} \right) + \left( {3{t^2} - 3{t^3}} \right) + {t^3} = 1\).

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

The weight sum is \(\left( {1 - 3t + 3{t^2} - {t^3}} \right) + \left( {3t - 6{t^2} + 3{t^3}} \right) + \left( {3{t^2} - 3{t^3}} \right) + {t^3} = 1\). This sum also varies between 0 and 1 if \(0 \le t \le 1\).

This implies, it is convex combination of \(\left\{ {{{\bf{p}}_{\bf{o}}}{\bf{,}}\,{{\bf{p}}_{\bf{1}}}{\bf{,}}\,{{\bf{p}}_{\bf{2}}}{\bf{,}}{{\bf{p}}_{\bf{3}}}} \right\}\).