Bézier curves of positive curvature
Let \(n+1\) distinct points \(P_ i\), \(i=0,1,...,n\), be given and let \(v_ i\), \(i=0,1,...,n\), be the vector from \(P_ i\) to \(P_{i+1}\). Let \(\vartheta_{k,j}\), \(k=0,1,...,n-1\), \(j=0,1,...,n-1\), be the counter- clockwise angle from \(v_ k\) to \(v_ j\). The data \(P_ i\) have positive discrete curvature if \(0<\vartheta_{i-1,i}<\pi\) for \(i=1,2,...,n-1\). Let \(B_ n(t)=(\dot x(t),y(t))=\sum^{n}_{k=0}P_ k\binom{n}{k}t^ k(1-t)^{n-k}\) be the nth degree Bézier curve for the control points \(P_ i=(x_ i,y_ i)\). The positive discrete curvature of the data points \(P_ i\) is not sufficient to insure the positivity of the curvature of the resulting curve \(B_ n(t)\). The author obtains sufficient conditions on the control points in the form of easy to check algorithms for positive curvature of \(B_ n(t)\). Many interesting examples are also given.
scientific article; zbMATH DE number 4058718
scientific article