RE: [Algorithms] portal engines in outdoor environments

[Tom F. <to...@mu...>]

You can find it perfectly of course, but if you can't be bothered, or you're
doing lots of these (a common case), then a good conservative approximation
of the real size is to find the radius of a circle that is r units closer to
the viewer than S, i.e. just move P closer by r. It's virtually the same
thing for sensible-distance-away spheres, it's only as the sphere gets
really very close that it starts to give poor results, and this is usually
acceptable in things like hierachial culling.

Tom Forsyth - Muckyfoot bloke.
Whizzing and pasting and pooting through the day.

> -----Original Message-----
> From: Jamie Fowlston [mailto:j.f...@re...]
> Sent: 21 August 2000 12:05
> To: gda...@li...
> Subject: Re: [Algorithms] portal engines in outdoor environments
> 
> 
> > Spheres are particularly nice because the sphere radius is 
> the same as the
> > circles radius after projection
> 
> Careful. There's a common misconception here (which you may 
> or may not have made
> :).
> 
> Let the sphere have radius r.
> Let S be the centre of the sphere.
> Let V be some vector perpendicular to the view vector of length r.
> 
> Let P = S + V
> 
> Some people claim that projecting point P gives you a point 
> on the edge of the
> circle which is the rasterisation of the sphere. This is not true.
> 
> Demonstration that this is so (in 2D, so hopefully it's clearer :) :
> 
> Take a circle with centre C.
> Place an arbitrary point P outside the circle. The closer it 
> is to the circle,
> the clearer my point (unintentional... sorry:) should be.
> Let the 2 tangents to the circle passing through P be T1, T2.
> Let P1 be the point of intersection between T1 and C. Define 
> P2 similarly.
> 
> It should be clear that the projections of P1 and P2 are 
> equivalent to points on
> the edge of the rasterisation.
> But (P - C) is not perpendicular to (T1 - C) or (T2 - C).
> 
> Although as | P - C | approaches infinity, they approach 
> perpendicular. If you
> can be sure you'll never be close enough to appreciate the 
> error, then you'll be
> fine :)
> 
> 
> Back to the sphere: this means that the true rasterisation of 
> the sphere is
> larger than the circle calculated by projecting P.
> 
> I'll expand more if anybody needs it... or gives a monkey's :)
> 
> Jamie
> 
> 
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