I have yet to obtain a copy of Norman's book, but I have taken a quick look at the introductory chapter on the web. It appears that Norman specifies a plane rotation in terms of two ratios. These ratios are not necessarily rational numbers, they could be real numbers. So Norman's "rational trigonometry" is called "rational" because it deals with ratios, not because it deals with rational numbers. That said, any angle can be approximated arbitrarily closely by rational numbers.
I need to study the book in more detail to discover what the metrical relationship is between standard measures of angle and Norman's ratios. For what it is worth on a skimpy reading, Norman's approach looks better than my approach to rational trigonometry. If it is better, I will adopt it and abandon my own approach. But, regardless of who has the better approach, there are some useful properties of any rational-numbered approach to trigonometry. I have tried to rank these from most important to least important. 1) Visual calculations can be carried out to arbitrary precision, not at a precision fixed by floating-point arithmetic. Think about the leverage between the retina and the lens of your eye, roughly an inch, and an object a mile, or ten miles away. A small angular error at the retina implies a huge linear error in the position of the object. This makes all manner of visual algorithms ill-conditioned, but computing to arbitrary, rational precision means that algorithms do not terminate on round-off errors - as they would with floating-point arithmetic. In many cases they can plough through the ill-conditioning and arrive at a good solution. Of course, the computational cost is still a function of the ill-conditioning so this is not a panacea - just a great help in many practical cases. It means that we can all hack out effective vision algorithms without learning a whole lot of Numerical Analysis. More to the point, it means that an AGI can develop its own vision algorithms, via genetic algorithms or whatever, without having to represent Numerical Analysis. 2) Vectors can be made exact so that there are no gaps between facets describing a surface. This means we can hack out geometrical solutions to visual problems without having to think about topology. And so can an AGI ... 3) Bottom-up rational solutions can be approximated by top-down floating point ones, so if you cannot afford the computational cost of rational numbers then give your AGI expectations about the world and let it confirm these in a global-to-fine strategy. Sadly this is a whole lot easier to say than to do! And that's all I have time for right now. Cheers, James -----Original Message----- From: Robert Stewart [mailto:[EMAIL PROTECTED] Sent: 01 November 2005 13:11 To: [email protected] Subject: [agi] New Book- Divine Proportions: Rational Trigonometry to Universal Geometry by N J Wildberger I have just come across this book and think that it may be relevant to wireframe modelling (and many other things besides). Details can be found here: http://web.maths.unsw.edu.au.nyud.net:8090/~norman/book.htm I wonder if Dr Anderson could comment on how this material relates to his own work on rational geometry and the perspex machine. Best, Rob __________________________________ Yahoo! Mail - PC Magazine Editors' Choice 2005 http://mail.yahoo.com ------- To unsubscribe, change your address, or temporarily deactivate your subscription, please go to http://v2.listbox.com/member/[EMAIL PROTECTED] ------- To unsubscribe, change your address, or temporarily deactivate your subscription, please go to http://v2.listbox.com/member/[EMAIL PROTECTED]
