Name
Abstract | ||
|---|---|---|
A polynomial Pythagorean-hodograph (PH) curve r(t)=(x"1(t),...,x"n(t)) in R^n is characterized by the property that its derivative components satisfy the Pythagorean condition x"1^'^2(t)+...+x"n^'^2(t)=@s^2(t) for some polynomial @s(t), ensuring that the arc length s(t)=@!@s(t)dt is simply a polynomial in the curve parameter t. PH curves have thus far been extensively studied in R^2 and R^3, by means of the complex-number and the quaternion or Hopf map representations, and the basic theory and algorithms for their practical construction and analysis are currently well-developed. However, the case of PH curves in R^n for n3 remains largely unexplored, due to difficulties with the characterization of Pythagorean (n+1)-tuples when n3. Invoking recent results from number theory, we characterize the structure of PH curves in dimensions n=5 and n=9, and investigate some of their properties. |
Year | DOI | Venue |
|---|---|---|
2012 | 10.1016/j.cam.2012.04.002 | J. Computational Applied Mathematics |
Keywords | Field | DocType |
derivative component,pythagorean condition,arc length,pythagorean-hodograph curve,euclidean space,dimensions n,basic theory,curve parameter,hopf map representation,number theory,polynomial pythagorean-hodograph,ph curve,quaternions,hopf map,complex numbers | Mathematical optimization,Complex number,Polynomial,Mathematical analysis,Quaternion,Hopf fibration,Arc length,Euclidean geometry,Pythagorean theorem,Number theory,Mathematics | Journal |
Volume | Issue | ISSN |
236 | 17 | 0377-0427 |
Citations | PageRank | References |
5 | 0.48 | 10 |
Authors | ||
2 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Takis Sakkalis | 1 | 347 | 34.52 |
| Rida T. Farouki | 2 | 1396 | 137.40 |