Tony Moss wrote: > Patrick Kessler wrote: > > >Can anyone recommend an essay on steriographic projection? In particular I > >am searching for a proof that circles on the sphere are mapped onto the > >equatorial plane as circles. > > > > As ever my response to this query is via graphical rather than numeric
> methods using computer scale drawing at high magnification. (3200%) > > As the necessary extreme example I took a small circle of about 1/4 globe > diameter just north of the equator. > > Two diameters of the circle projected onto the equatorial plane were > measured. The first on a radius from the polar axis and the second at > right angles to it. This second diameter appears to be about 1.95% > longer. Now comes the hard part - proving it! > ><snip> > Tony Moss Tony's measurement of the projected lengths of the diameters of a circle shows them to be different. From this one might be led to doubt that the projection is a circle. But the projections of diameters of circles are not, in general, diameters of the projection. One way to see this is to consider a circle centred on the equator, extending 30 degrees north, south, east and west of the centre. On an astrolabe with equator radius r, the circle's north-south diameter projects to a radial line from radius r/sqrt(3) to r*sqrt(3), so its length is about r*1.1547. Its east-west diameter projects to a portion of the equator spanning 60 degrees, so has length r. That's more than 15% difference! But the east-west diameter of the original circle is not a diameter of the projected figure. I have taken the liberty of attaching a ZIPped file containing a Microsoft Paint picture to show this effect graphically. While this doesn't prove that the projection is a true circle, it does explain Tony's measurement. Chris Lusby Taylor 51.3N 1.4W Attachment converted: Macintosh HD:astrolabe1.zip (pZIP/pZIP) (00012C67) Content-Type: text/x-vcard; charset=us-ascii; name="clusby.vcf" Content-Transfer-Encoding: 7bit Content-Description: Card for Chris Lusby Taylor Content-Disposition: attachment; filename="clusby.vcf" Attachment converted: Macintosh HD:clusby.vcf (TEXT/ttxt) (00012C68)
