From: robjohn9@idt.net (Rob Johnson) Subject: Re: Please help with conic and nurbs Date: 6 Feb 2000 13:34:26 GMT Newsgroups: sci.math Summary: [missing] In article <389cad1e_3@news5.newsfeeds.com>, "Kevin N." wrote: >Can anybody show a mathematical proof that any >degree 2 NURBS traces out a conic or portion of >a conic section? Let us represent a degree 2 NURB by a_1 p_1 (1-t)^2 + a_2 p_2 t(1-t) + a_3 p_3 t^2 p(t) = ---------------------------------------------- [1] a_1 (1-t)^2 + a_2 t(1-t) + a_3 t^2 where the p_i are points and the a_i are positive weights. Let p_i have coordinates (x_i,y_i), and p(t) have coordinates x(t) and y(t). Then we have the following matrix equation from [1]: +- -+ +- -+ +- -+ | x_1 x_2 x_3 | | a_1 (1-t)^2 | | x(t) w(t) | | y_1 y_2 y_3 | | a_2 t(1-t) | = | y(t) w(t) | [2] | 1 1 1 | | a_3 t^2 | | w(t) | +- -+ +- -+ +- -+ Let P be the inverse of the leftmost matrix in [2]. Then we have +- -+ +- -+ | a_1 (1-t)^2 | | x(t) w(t) | | a_2 t(1-t) | = P | y(t) w(t) | [3] | a_3 t^2 | | w(t) | +- -+ +- -+ Let b = -2 a_1 a_3 / a_2^2, then for all t, +- -+T +- -+ +- -+ | a_1 (1-t)^2 | | 0 0 1 | | a_1 (1-t)^2 | | a_2 t(1-t) | | 0 b 0 | | a_2 t(1-t) | = 0 [4] | a_3 t^2 | | 1 0 0 | | a_3 t^2 | +- -+ +- -+ +- -+ Combining [3] and [4], +- -+T +- -+ +- -+ | x(t) w(t) | T | 0 0 1 | | x(t) w(t) | | y(t) w(t) | P | 0 b 0 | P | y(t) w(t) | = 0 [5] | w(t) | | 1 0 0 | | w(t) | +- -+ +- -+ +- -+ Dividing the leftmost and rightmost matrices in [5] by w(t), we get +- -+T +- -+ +- -+ | x(t) | T | 0 0 1 | | x(t) | | y(t) | P | 0 b 0 | P | y(t) | = 0 [6] | 1 | | 1 0 0 | | 1 | +- -+ +- -+ +- -+ The product of the three matrices in the middle of [6] is obviously symmetric. Thus, [6] is of the form ax^2 + bxy + cy^2 + dx + ey + f = 0 [7] which is the equation for a conic section. Rob Johnson robjohn9@idt.net ============================================================================== From: "Thomas J Scally" Subject: Re: Please help with conic and nurbs Date: Sun, 6 Feb 2000 23:38:33 -0500 Newsgroups: sci.math Hop David asked: >What is a nurb? NURBS are a kind of spline used in 3d modelling. acronym = non-uniform rational B-spline, as i recall (btw, thanks for that Escher post, I enjoyed it) -Tom (remove obvious spamblock to reply) ============================================================================== From: "Tony Richards, Optical Systems Group" Subject: Re: Please help with conic and nurbs Date: Mon, 7 Feb 2000 11:34:38 -0000 Newsgroups: sci.math Hop David wrote in message news:389D30FE.146FAC0C@tabletoptelephone.com... > What is a nurb A 'NURBS' is a Non-Uniform, Rational B-Spline A spline curve is a parametric curve made up of sections each of which is expressable as polynomial functions of a parameter and which obey some one or more continuity equations at the points where the sections join. The points where the sections join are called the knots. Uniform knot spacings imply equal ranges of the parameter between successive knots. Non-uniform knot spacings imply un-equal ranges of the parameter between successive knots. The 'B' comes from Bezier, I think, a particular type of curve based on particular cubic basis functions (e.g. t^3, t^2*(1-t), t*(1-t)^2, (1-t)^3 , t= the parameter). 'Rational' means that the curves are expressable as ratios of polynomial functions. (Unity is a perfectly acceptable polynomial to have as the denominator in such a ratio, so that ALL spline curves can be called RATIONAL splines, even though the denominator in many cases may be the constant 1 )