Message-ID: <32F7B70A.149B@pobox.oleane.com>
Date: Tue, 04 Feb 1997 23:24:10 +0100
From: Francois Charton <deef AT pobox DOT oleane DOT com>
Organization: CCMSA
MIME-Version: 1.0
To: Benjamin D Chambers <chambersb AT juno DOT com>
CC: djgpp AT delorie DOT com
Subject: Re: Ellipses (again)
References: <19970203 DOT 185314 DOT 4575 DOT 7 DOT chambersb AT juno DOT com>
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Benjamin D Chambers wrote:
> 
> 
> A(x+w)^2   B(y+h)^2
> -------- + -------- = 1
>   C^2        C^2
> 

Not quite : this is a general conic equation, which may repesent 
ellipses, as well as parabolae and hyperbolae... In the above, if A and B 
are both negative, the equation has no solution (so this is *nothing*), 
if A and B are of opposite sign you get a hyperbola, and if any of them 
is zero you get a parabola... 

Only when A and B are both positive do you get an ellipse...

Here is my try at general conic equations (for conics centered on O):

A * (ux+vy)^2 + B * (px+ry)^2 = 1

with ur-pv!=0 and A and B not both zero, and not both negative

If A*B=0  this is a parabola
if A*B < 0 this is a hyperbola
if A*B > 0 this is an ellipse

As regards drawing ellipses, using equation is (IMHO) a bad thing : there 
are fast scan lines algorithms for general ellipses : the description of 
one can be found in Foley/van Dam, Computer Graphics, Principles and 
Practice (2nd Ed.), in chapter 19.

Francois