# Queries On PH Quintic Splines

This is with respect to the Pythagorean Hodograph Splines of degree 5, developed here: link text.

I'm trying to code these up and can't really get clear about a couple of points:

(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).

(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?

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Freely accessible version of the linked paper: ams.org/journals/mcom/1995-64-212/S0025-5718-1995-1308452-6/… –  J. M. Oct 9 '10 at 1:53
For (1), if you read the paper, you see that the PH curve has two endpoints (with the derivatives at those points specified, thus a Hermite interpolation), and that the other control points specify the remaining degrees of freedom. –  J. M. Oct 9 '10 at 2:04