One of the simplest shape properties is monotonicity. The sequence of control points (Po, . . . ,Pn) and the corresponding points of the curve j(t), t E [a, 6], are projected on a line /. Let us denote by Pi the projection of Pi, i = 0 , . . . , n, and by 7 (t) the projection of 7 (t) on /. Then we want the following property to be satisfied for any control polygon and any line Z: if the projected points P0, . . . , Pn are ordered, that is,
Pi = P0 + 0 < o^x < - - < ttn,
and v is a directional vector of the line /, then the projected points 7 (t), t E [a, 6], are also ordered, that is, 7 (2) = P0 + a(t)v, where a(t) is an increasing function. This is equivalent to the following property:
(•t) is an increasing function (1)
for each P0, • • •, Pn E IRfc, v E Hfe. Now, let (t/o,...,wn) be a system of blending functions, and let 7 be
the curve defined by 7 (t) = Y^=oui(t)Pi‘ Then we may write vTj(t) = S IL o(vTPi)ui(t)- Thus the property suggested by (1) motivates the following definition. Definition 1.1. A system of blending functions (uq , . . . , un) is monotonicity preserving if for any increasing vector A = (Ao,. . . , An)T, (A0 < • • • < \ n), the function u(t) = X^=o A,-ut*(t) is an increasing function.