Piecewise Curves and Surfaces
After an introductory section, in Section 9.2 we will consider Bézier curves. Then we will discuss the important notions of parametric C k - continuity and geometric C k -continuity of piecewise curves. Section 9.3 is devoted to (parametrically C k -conti
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Piecewise Curves and Surfaces
After an introductory section, in Section 9.2 we will consider B´ezier curves. Then we will discuss the important notions of parametric Ck -continuity and geometric Ck continuity of piecewise curves. Section 9.3 is devoted to (parametrically Ck -continuous) Hermite interpolation and its applications to spline curves. In Section 9.4 we will study β -splines, which are geometrically Ck -continuous. We will also briefly survey their particular case, B-splines, which are parametrically Ck -continuous. In Sections 9.5 and 9.6 we will study similar problems for piecewise surfaces. Along the way we will create and apply several M-files that are given in Section A.4.
9.1 Introduction The main problem discussed in this chapter is this: given points (control vertices) P = {P1 , P2 , . . . , Pn } arbitrarily placed in the plane or in space, construct a smooth curve passing near or through these points and satisfying some additional conditions. A polygon that joins neighboring points from P is called a control polygon; P1 and Pn are called boundary points, and P2 , . . . , Pn−1 are called inner points. The equation of the curve will be written in the form of weighted sum r(t) = ∑ni=1 ai (t)Pi (t ∈ [a, b]), where the functions ai (t) must be derived. Spline curves are obtained by using a similar scheme to that in Section 1.6. We fix the net a = t1 < t2 < · · · < tm = b and build the vector function r(t) with the following two conditions: (i) r restricted to [ti ,ti+1 ] is a polynomial of degree at most d, (ii) r ∈ Cd−1 [a, b]. We call r a spline of degree d ≥ 1 with respect to the {ti }, the knots. For interpolating splines the nodes {r(ti )} coincide with the control vertices. For smoothing splines the R V. Rovenski, Modeling of Curves and Surfaces with MATLAB , Springer Undergraduate Texts in Mathematics and Technology 7, DOI 10.1007/978-0-387-71278-9 9, c Springer Science+Business Media, LLC 2010
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9 Piecewise Curves and Surfaces
nodes need not coincide with the control vertices. Each control vertex participates in at most d + 1 curve segments, so the curve features local control: moving one control vertex affects only those segments and not the entire curve. A spline may also be defined on an infinite interval. From the above definition, we see that a linear spline r(t) (with d = 1) is just a continuous polygonal arc. The most common case is that of cubic splines (d = 3). Similarly to curves, a piecewise (spline) surface can be obtained by constructing a number of patches and connecting them. The method used to construct the patch should allow for smooth connection of the patches. A surface patch can be displayed either as a wire frame or as a solid surface. Recall that a regular curve γ can be generated by a regular parameterization r(t); i.e., r (t) = 0 for all t ∈ I. A curve γ is called parametrically Ck -continuous if it can be generated by a regular parameterization r(t) of class Ck (i.e., r(k) (t) is a continuous vector function). One may consider a Ck -regular change of
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