Queries On PH Quintic Splines - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T13:12:17Z http://mathoverflow.net/feeds/question/41555 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/41555/queries-on-ph-quintic-splines Queries On PH Quintic Splines Ganesh 2010-10-09T01:09:49Z 2010-10-09T01:09:49Z <p>This is with respect to the Pythagorean Hodograph Splines of degree 5, developed here: <a href="http://www.jstor.org/stable/2153373?seq=3" rel="nofollow">link text</a>. </p> <p>I'm trying to code these up and can't really get clear about a couple of points:</p> <p>(1) Are these Splines approximation or interpolation splines, i.e. do they pass close to their control points, or do they pass exactly through them ? (Tentative answer: They are approximation splines, due to their Bernstein-Bezier form , as the latter are themselves approximative).</p> <p>(2) Consider the representation $\textbf{r}(t)= \displaystyle \sum_{0\leq k\leq n} \binom{n}{k}\textbf{p}_k \times t^{k} (1-t)^{n-k}$ given in page 2 of the above paper. I understand this to mean that points $\textbf{r}(t)=(x(t),y(t))$ obey the said parametric equation. For this curve to be a hodograph,the $x(t)$ and $y(t)$ need to have $x'(t)=w(u^2-v^2), y'(t)=2wuv$ as given in Eq. (7) of the paper, where $w,u,v$ are polynomial functions in $t$. If $\textbf{r}(t)$ obeys the parametric form above,does it follow that the components $x'$ and $y'$ automatically obey the Hodograph form?</p>