Bézier curve

English right|thumb|240px|Animation of the construction of a quadratic Bézier curve. The parameter t ∈ [0,1] defines by linear proportion the positions of the endpoints of the green line along the lines P0P1 and P1P2 as well as the point along the green line that is on the Bézier curve. right|thumb|A Bézier curve with four control points (a cubic Bézier curve) Etymology Probably a en fr courbe de Bézier 1. en Renault engineer Pierre Bézier = who popularized the curves developed earlier by Paul de Casteljau and used them to design curves for the bodywork of cars. Pronunciation en /ˈbɛz.i.eɪ/ + en curve Noun en mathematics computer graphics A type of parametric curve defined by a set of control points, two of which are the ends of the curve, while the others determine its shape. From 1960s. en bezier en Bézier curves are applied, for example in computer graphics and animation, to produce smooth, infinitely scalable curves. en 2003 Samuel Buss Samuel R. Buss 3D Computer Graphics: A Mathematical Introduction with OpenGL https://books.google.com.au/books?id=LBm1khtUX0cC&pg=PA163&dq=%22B%C3%A9zier+curve%22%7C%22B%C3%A9zier+curves%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwiPvu-Y3fPyAhUWq5UCHW6WDBAQ6AF6BAghEAI#v=onepage&q=%22B%C3%A9zier%20curve%22%7C%22B%C3%A9zier%20curves%22&f=false 163 w:Cambridge University Press So, instead, it is often better to combine multiple Bézier curves to form a longer, more complicated curve called a piecewise Bézier curve.This section discusses how to join Bézier curves together – especially how to join them so as to preserve continuity and smoothness (i.e., continuity of the first derivative). en 2008 T. Theoharis; G. Papaioannou; N. Platis; N. M. Patrikalakis Graphics & Visualization: Principles and Algorithms Taylor & Francis (A K Peters / CRC Press) https://books.google.com.au/books?id=9UwhIdwmS9cC&pg=PA193&dq=%22B%C3%A9zier+curve%22%7C%22B%C3%A9zier+curves%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwiPvu-Y3fPyAhUWq5UCHW6WDBAQ6AF6BAhTEAI#v=onepage&q=%22B%C3%A9zier%20curve%22%7C%22B%C3%A9zier%20curves%22&f=false 193 The initial points p0, p1, and p2 are called control points of the Bézier curve.The process outlined above for the generation of a quadratic Bézier curve from its three control points can be generalized for more control points in a straightforward manner. en 2012 Frank Klawonn Introduction to Computer Graphics: Using Java 2D and 3D 2nd https://books.google.com.au/books?id=Kw4rxkDvDsIC&pg=PA151&dq=%22B%C3%A9zier+curve%22%7C%22B%C3%A9zier+curves%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwiPvu-Y3fPyAhUWq5UCHW6WDBAQ6AF6BAhZEAI#v=onepage&q=%22B%C3%A9zier%20curve%22%7C%22B%C3%A9zier%20curves%22&f=false 151 Springer When an affine transformation is applied to the control points, the resulting Bézier curve with respect to the new control points coincides with the transformed Bézier curve. Therefore, Bézier curves are invariant under affine transformations like rotation, translation or scaling. Bézier curves are also symmetric with respect to their control points. Related terms en composite Bézier curve bezier Bézier coefficient Bézier spline Bézier surface Bézier triangle Translations kind of parametric curve Chinese: Mandarin: cmn 貝茲曲線 Bèizī qǔxiàn Czech: cs Bézierova křivka f Finnish: fi Bézier-käyrä French: fr courbe de Bézier f German: de Bézierkurve f Italian: it curva di Bézier f Japanese: ja ベジェ曲線 Beje kyokusen Russian: ru крива́я Безье́ f Spanish: es curva de Bézier f Swedish: sv Bézier-kurva c See also en Bernstein form en control point en spline Further reading Bernstein polynomial De Casteljau's algorithm Bézier curve on Encyclopedia of Mathematics