Two ways to construct a smooth piecewise rational curve
The subject of this paper are two ways of constructing a smooth piecewise rational curve. The first is called the analytic and the second the geometric approach. In Section 2 the two approaches are described for a continuous piecewise cubic polynomial \(S:\mathbb{R}\to\mathbb{R}^ s\), \(s\geq 2\). The particular interest is in curves which look smooth where the smoothness is defined as geometrical continuity of order 2 (in short: \(G^ 2\)), if the curve has a continuous unit tangent and a continuous curvature at all points where \(S'\) is non zero. The analytic approach for cubic polynomials then looks as follows: \(S\) is assumed to have knots at the integers, i.e., for each \(i\in\mathbb{Z}\), \(S\mid [i,i+1]\) is a cubic polynomial. For \(S\) to be in \(G^ 2\) at \(i\in\mathbb{Z}\) it can be shown to be necessary and sufficient that \[ (S'(i^ +),S''(i^ +))^ T = A_ i(S'(i^ -),S''(i^ -))^ T \] where \(A_ i=\bigl( {{\beta_ i} \atop {\gamma_ i}} {0 \atop {\beta^ 2_ i}}\bigr)\), for some \(\beta_ i>0\) and some \(\gamma_ i\in\mathbb{R}\), is the so called connection matrix. If \(\gamma_ i\geq 0\) for all \(i\in\mathbb{Z}\), it has been shown that \(S\) can be written uniquely in the form \(S(t)=\sum^ \infty_{i=-\infty} P^ i N_ i(t)\) where \(P^ i\in\mathbb{R}^ s\) and \(N_ i: \mathbb{R}\to\mathbb{R}\) has support on \([i,i+4]\) with \(\sum^ \infty_{i=-\infty} N_ i=1\). In addition to this further properties of piecewise cubic polynomial curves are discussed in Section 2. In the geometric approach the curve is assumed to be of the form \(S(t)=p_ i(t-1)\), \(i\leq t\leq i+1\), \(i\in\mathbb{Z}\) where \[ p_ i(t)=b^ o_ 0(1-t)^ 3+ b^ i_ 1 3t(1- t)^ 2+ b^ i_ 2 3t^ 2(1-t)+ b^ i_ 3 t^ 3, \qquad 0\leq t\leq 1, \] \(i\in \mathbb{Z}\), for points \(b^ i_ 0, b^ i_ 1, b^ i_ 2, b^ i_ 3\in \mathbb{R}^ s\) which are called Bézier points. Starting with arbitrary points \(P^ i\in\mathbb{R}^ s\) and numbers \(\beta_ i, \gamma_ i, \mu_ i>0\), \(i\in\mathbb{Z}\) with \(\lambda_ i\mu_ i=\beta^ 2_ i\) it is shown how the Bézier points \(b^ i_ 0, b^ i_ 1, b^ i_ 2, b^ i_ 3\in\mathbb{R}^ s\), \(i\in\mathbb{Z}\), can be constructed such that \(S: \mathbb{R}\to \mathbb{R}^ s\) can also be represented in the form \(S(t)=\sum^ \infty _{i=-\infty} P^ i N_ i(t)\) where \(N_ i: \mathbb{R}\to\mathbb{R}\) is positive with support on \([i,i+4]\) and \(\sum^ \infty_{i=-\infty} N_ i=1\). It is also shown that \[ (S'(i^ +),S''(i^ +))^ T= A_{i+2}(S'(i^ -),S''(i^ -))^ T \] where \(A_ i=\bigl( {{\beta_ i} \atop {\gamma_ i}} {0 \atop {\delta_ i}}\bigr)\) for \(\beta_ i>0\), \(\delta_ i>0\) and \(\gamma_ i>- 2(\beta_ i+\delta_ i)\). To ensure \(S\) to be in \(G^ 2\) one can take \(\delta_ i=\beta^ 2_ i\) and \(\gamma_ i>-2(\beta_ i+\delta_ i)\) which is more general than in the analytic approach. The two approaches can then be extended to piecewise cubic rationals and piecewise general rationals.
scientific article; zbMATH DE number 222117