From - Wed Feb 2 19:29:31 2005 Return-Path: Received: from coxeter.math.toronto.edu (coxeter.math.toronto.edu [128.100.68.3]) by nef2.ens.fr (8.12.11/1.01.28121999) with ESMTP id j12ITRGn019062 for ; Wed, 2 Feb 2005 19:29:28 +0100 (CET) Received: (from laurent@localhost) by coxeter.math.toronto.edu (AIX5.2/8.11.6p2/8.11.0/UTMath 1.0) id j12IT2V136794; Wed, 2 Feb 2005 13:29:02 -0500 Date: Wed, 2 Feb 2005 13:29:02 -0500 From: Larry Siebenmann Message-Id: <200502021829.j12IT2V136794@coxeter.math.toronto.edu> To: lfinsto1@gwdg.de, metafont@ens.fr, metapost@tug.org Subject: Re: Intersections of NURBs X-Greylist: Recipient e-mail whitelisted, not delayed by milter-greylist-1.5.10 (nef2.ens.fr [129.199.96.32]); Wed, 02 Feb 2005 19:29:28 +0100 (CET) Hi Laurence F, LF> In case anyone's wondering, the reason why points with the same > z-coordinate as Focus::position can't be projected is because this > would involve division by 0. Objects consisting of points whose > z-coordinates are greater than 0 could be projected, but are generally > culled, because people don't see that way. Projective (homogeneous) coordinates permit division by 0. Projective transformations are bijective and everywhere defined. The projective plane RP^2 was created to make that true!! Curiously in certain circumstances we *can* see images that are located behind us. This happens in Newtonian lens systems; the observer does not realize the image is located behind her until she notices that paralax has switched sign. This is a real memory; she was an A-level highscool physics student in England writing up her optics experiment. Naturally her report was culled. Naturally Galileo recanted. Cheers LS From - Thu Feb 3 21:41:35 2005 Return-Path: Received: from coxeter.math.toronto.edu (coxeter.math.toronto.edu [128.100.68.3]) by nef2.ens.fr (8.12.11/1.01.28121999) with ESMTP id j13Kdv40073031 for ; Thu, 3 Feb 2005 21:41:30 +0100 (CET) Received: (from laurent@localhost) by coxeter.math.toronto.edu (AIX5.2/8.11.6p2/8.11.0/UTMath 1.0) id j13Kao5154692; Thu, 3 Feb 2005 15:36:50 -0500 Date: Thu, 3 Feb 2005 15:36:50 -0500 From: Larry Siebenmann Message-Id: <200502032036.j13Kao5154692@coxeter.math.toronto.edu> To: lfinsto1@gwdg.de, metafont@ens.fr, metapost@tug.org Subject: Re: Intersections of NURBs X-Greylist: Recipient e-mail whitelisted, not delayed by milter-greylist-1.5.10 (nef2.ens.fr [129.199.96.32]); Thu, 03 Feb 2005 21:41:33 +0100 (CET) Hi Laurence F. I carelessly wrote on Sun, 30 Jan 2005 22:02:27 -0500: > The notion of convex hull in RP^n seems well defined for > objects that miss some hyperplane. I should have written | The notion of convex hull in n-dimensional real | projective space RP^n seems well defined for | objects that miss some hyperplane and are affine convex | in the the complement of that hyperplane. OR (consequently) | The notion of convex hull in RP^n seems well defined for | all small objects. I had just carefully given a counterexample to the quoted statement!! -- namely, my four point set {A',B',C',D'}. The modified statements seem to get around the difficulty I used {A',B',C',D'} to illustrate to Laurence F. concerning generalization of the de Casteljau convex hull property for projectively transformed bezier curves. LF> At any rate, my sources on 3D graphics all indicate that > NURBs are the de facto standard for 3D applications. So > far, I've had to take this on faith. I lack faith. NURBSs cannot be the choice to end all choices since the envelope of a circular penstroke along a bezier cubic is not a NURBS; it is in general an *non*-rational algebraic curve; hence not a NURBS! Curve nirvana has not yet been found. Cheers Laurent S. From - Thu Feb 10 13:36:25 2005 Return-Path: Received: from mailer.gwdg.de (mailer.gwdg.de [134.76.10.26]) by nef2.ens.fr (8.12.11/1.01.28121999) with ESMTP id j1ACaITa069402 for ; Thu, 10 Feb 2005 13:36:22 +0100 (CET) Received: from gwdu71.gwdg.de ([134.76.8.21]) by mailer.gwdg.de with esmtp (Exim 4.42) id 1CzDXq-0006KM-Fg; Thu, 10 Feb 2005 13:35:52 +0100 Received: from localhost by gwdu71.gwdg.de (8.11.1/1.1.29.3/16Jun03-0924AM) id j1ACZ8e0000241153; Thu, 10 Feb 2005 13:35:08 +0100 (MET) X-Authentication-Warning: gwdu71.gwdg.de: lfinsto1 owned process doing -bs Date: Thu, 10 Feb 2005 13:35:08 +0100 (MET) From: Laurence Finston To: Larry Siebenmann cc: liste metafont , metapost@tug.org Subject: Re: [metapost] Re: Some observations on turning and self intersection In-Reply-To: <200502082124.j18LOxE32357@f7.net> Message-ID: References: <200502082124.j18LOxE32357@f7.net> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Spam-Level: / X-Spam-Report: Content analysis: 0.0 points, 6.0 required X-Virus-Scanned: (clean) by exiscan+sophie X-Greylist: Recipient e-mail whitelisted, not delayed by milter-greylist-1.5.10 (nef2.ens.fr [129.199.96.32]); Thu, 10 Feb 2005 13:36:22 +0100 (CET) On Tue, 8 Feb 2005, Larry Siebenmann wrote: > There are now dozens of books on computer graphics and any > one of them could/should include the classification. But our > math library has few -- and *none* of the volumes recommended > by/to Laurence F. I think Salomon and Jones are serious omissions. The latter has been the most useful book on computer graphics to me by far. As far as the others are concerned, I've found books on recreational mathematics, and particularly the reprints from Dover Books, to be a very fruitful source of ideas. A book I find particularly interesting is _Fast platonische K{\"o}rper_ (Nearly Platonic Figures) by Christoph P{\"o}ppe. It's published in the "Papiermechanik" (paper mechanics) series of "Spektrum der Wissenschaft" (Spectrum of the Sciences), the German version of Scientific American. Perhaps such topics might be of interest to the Megapost developers, if they decide to leave "Flatland". I think it's a shame if a math library ignores recreational mathematics, but I suppose any library has to decide how to spend a limited budget. Laurence From - Mon Feb 21 05:28:01 2005 Return-Path: Received: from cogs.susx.ac.uk (as07081.sogang.ac.kr [163.239.22.81]) by nef2.ens.fr (8.13.2/1.01.28121999) with ESMTP id j1L4RuMM037269 for ; Mon, 21 Feb 2005 05:27:57 +0100 (CET) Message-Id: <200502210427.j1L4RuMM037269@nef2.ens.fr> From: alanje@cogs.susx.ac.uk To: metafont@ens.fr Subject: Returned mail: see transcript for details Date: Mon, 21 Feb 2005 13:27:57 +0900 MIME-Version: 1.0 Content-Type: multipart/mixed; boundary="----=_NextPart_000_0003_167C5EA3.D34A2A3E" X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2600.0000 X-MIMEOLE: Produced By Microsoft MimeOLE V6.00.2600.0000 X-Greylist: Recipient e-mail whitelisted, not delayed by milter-greylist-1.5.10 (nef2.ens.fr [129.199.96.32]); Mon, 21 Feb 2005 05:27:59 +0100 (CET) This is a multi-part message in MIME format. ------=_NextPart_000_0003_167C5EA3.D34A2A3E Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit 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