Message-Id: <199702041536.KAA03008@delorie.com>
From: MTEEL DOT DTECHS AT dtechs DOT com (Mark Teel)
Date: Tue, 04 Feb 1997 09:32 EST
To: djgpp AT delorie DOT com
Cc: djgpp AT delorie DOT com
Subject: fwd: Re: ellipses at an angle

======== Original Message ========


On Mon, 3 Feb 1997, Benjamin D Chambers wrote:

> (Might of been Newton, I was sleeping through that part... the general
> equations and such get a lot more fun than a bunch of history :)

check out arthur koestler's "sleepwalkers" - history of man's conception 
of the universe. you might begin to like history.

> The point is, shapes aren't mathematical equations - they're shapes. 
> However, they CAN be modelled with math (by the way, it's easier {IMHO}
> to use
[snipped equation]

> The point is, computers don't run on equations - they run on algorithms. 

how true! there is a way however to get "shapes" out of equations: its 
called the "parametric equation". the advantage of these parametric 
equation is that the solution is built into the expressions. (sorry to 
bore people who know..) an example is a circle:

x^2 + y^2 == a^2,

to plot that, use some optimised version of the two expressions:

x = a * cos(t)
y = a * sin(t)

with these expressions (not equations, mind! note the == in the equation 
as opposed to = in the expression) the equation for the circle 
comes "presolved". one can get a parametric expression for the ellipse 
too and i had posted it a few days ago (with a mistake, later corrected). 
in any case here it is again:

x = r1 * cos(t)
y = r2 * sin(t)

these expressions are designed to "presolve" the equation for the ellipse 
for any value of t (the angle actually). rotation expressions were given 
in my earlier mail which i cannot locate now. but if you are interested, 
i would be happy to send them to you.

with warmest regards
======== Fwd by: Mark Teel ========
I'm sorry you all have such disdain for equations - I rather like them!
BTW, an algorithm can be nothing more than a language specific
representation of
an equation, in the rare cases where an equation will suffice.  And you are
over
simplifying the parametric "expressions" (which are also equations, BTW). 
They
are derived from the polar coordinate system treatment of these "shapes". 
An
ellipse is a graphical representation of the solution set of the
aforementioned
equation, not the other way around.  If you want to draw something "kinda"
like
an ellipse, then use any algorithm you like.

MST