Question
Denote tkematrinIzla,6c) Ila&c) J(,6,c) Juc(a.&,c) Ile64) fv(4,6,4) Jzr(a,&,c) Iale.&.c) f(a.b,c)_ AFdl only Mf tke agcbra) Dfle.6) dirertions From the theory quadratie fOn (lineI three principle mIfIOl? vositi, that Jry(a,6,c)] (LZl d(I(a,6,e)) 6.c)Jlo,ke)] and fa(a, c) fola.kc) ine(a.6c) I (a,6.c) fsle,&c) f(a.6,c) Ja(a.6,c) Js(4,6,c) Jle6.c) puint c). Tbe condlitions For 4 ECal minimum & f(5.9 =) at # crical the scond Jenuln Ma Kich -I In this Til_ "Thts Fuximur
Denote tke matrin Izla,6c) Ila&c) J(,6,c) Juc(a.&,c) Ile64) fv(4,6,4) Jzr(a,&,c) Iale.&.c) f(a.b,c)_ AFdl only Mf tke agcbra) Dfle.6) dirertions From the theory quadratie fOn (lineI three principle mIfIOl? vositi, that Jry(a,6,c)] (LZl d(I(a,6,e)) 6.c)Jlo,ke)] and fa(a, c) fola.kc) ine(a.6c) I (a,6.c) fsle,&c) f(a.6,c) Ja(a.6,c) Js(4,6,c) Jle6.c) puint c). Tbe condlitions For 4 ECal minimum & f(5.9 =) at # crical the scond Jenuln Ma Kich -I In this Til_ "Thts Fuximur follov fromn this ftult JGDcr flV:) loc Inilon IULt utcratc xign, that nqquireanent thut tbt Trat principle Ja(a.6,c) Jey (4,6.e) (Ila,k,c)) ~ad] fa(a.6.c) Ju(a.6,e) _ [fa(a,6,c) Jole.6.c) Ila,b.c) fu(e.6,c) Ix(a.6.c) Jala.&,c) Jae(a-6.c) fa(a,6.c) Jala.6,c)_ Juvalleetc ec Elogiag funxseeh Exc bcsl uarinumn Jocal minimum points 01 56695?+0+s 039 f(s,6,*) ~ { _ 4 +X<%0 8+847 3 * IS 38 < *-" '04 Sis.4* 8)
Answers
Given control points $\mathbf{p}_{0}, \mathbf{p}_{1}, \mathbf{p}_{2},$ and $\mathbf{p}_{3}$ in $\mathbb{R}^{n},$ let $\mathbf{g}_{1}(t)$ for $0 \leq t \leq 1$ be the quadratic Bézier curve from Exercise 23 determined by $\mathbf{p}_{0}, \mathbf{p}_{1},$ and $\mathbf{p}_{2},$ and let $\mathbf{g}_{2}(t)$ be defined similarly for $\mathbf{p}_{1}, \mathbf{p}_{2},$ and $\mathbf{p}_{3} .$ For $0 \leq t \leq 1,$ define $\mathbf{h}(t)=(1-t) \mathbf{g}_{1}(t)+t \mathbf{g}_{2}(t) .$ Show that the graph of $\mathbf{h}(t)$ 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.
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