Hi John, You are basically correct. We have all been using Bezier Curves without even knowing it. Any time you use a drawing program to draw a curve approximating a series of points, you are likely using Bezier Curves. Even the outline of curves in fonts are often defined by Bezier Curves.
Spline curves (cubic splines etc.) are slightly different. With them the curve goes through the points. With Bezier Curves the line is controlled by the points but does not necessarily go through them. A curve like the analemma would be drawn from a series of linked Bezier curves approximating the adjacent points. There is not one curve like a polynomial that would define the whole curve. Anselmo's question remains a good one. How many points are required to give a good Bezier approximation of the analemma. From what I have seen on Excel charts, 12 is not enough and 365 is more than enough. Do a web search on the topic. All sorts of good info like this comes up. http://www.math.ucla.edu/~baker/java/hoefer/Bezier.htm "Bezier curves are created by taking a time-varying linear combination of the control points. The Bernstein polynomials are used to calculate this linear combination given by the following equation where Pi is the ith control point: P(t) = (1-t)^3P0 + 3(1-t)^2tP1 + 3(1-t)t^2P2 + t^3P3 with t running from 0 to 1." Roger Bailey Walking Shadow Designs N 51 W115 -----Original Message----- From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] Behalf Of John Carmichael Sent: January 22, 2002 6:26 AM To: Dave Bell Cc: Sundial List Subject: Re: On Bezier Curves and analemmae I've never heard of a "Bezier Curve". Is this the type of curve that Delta Cad draws when you use the "Spline" feature to connect points? John John L. Carmichael Jr. Sundial Sculptures 925 E. Foothills Dr. Tucson Arizona 85718 USA Tel: 520-696-1709 Email: [EMAIL PROTECTED] Website: <http://www.sundialsculptures.com> ----- Original Message ----- From: "Dave Bell" <[EMAIL PROTECTED]> To: "Th. Taudin-Chabot" <[EMAIL PROTECTED]> Cc: <[email protected]> Sent: Monday, January 21, 2002 3:09 PM Subject: Re: On Bezier Curves and analemmae > On Mon, 21 Jan 2002, Th. Taudin-Chabot wrote: > > > But through three point s it is always possible to draw a circle, so three > > points seems not to be the solution. > > > > Thibaud Chabot > > More precisely, through three points you can draw any second-degree > curve. The form of the equation that defines the curve is different > for circle, ellipse, parabola, and hyperbola. Three points are > insufficient to determine *which* curve, but just right to completely > define a particular one... > > Dave >
