On cubics: A survey

Abstract

Standard cubics, i.e., nonrational cubics, are the simplest twisted curves—hence their considerable importance for CAGD. Rational cubics allow modification of their fullness even when the end tangents are kept fixed; this is the reason why they are occasionally preferred to standard cubics in CAGD. We shall point out connections between the various representations and their underlying geometric properties. This should serve as an easy and intuitive introduction and help the potential user choose a suitable representation. (This survey was initiated by Forrest's article “The twisted cubic curve” [14] and the subsequent correspondence with A. Ball [4] on some minor misunderstandings.) Some new algorithms can be obtained in a straightforward way.