🚀 go-pugleaf

dandb+@cs.cmu.edu (Dean Rubine) writes:

>In article <1750@tekig7.MAP.TEK.COM> gaulandm@tekig7.MAP.TEK.COM (Mike Gauland) writes:
>>I'm trying to implement a sine-wave generator using a DSP.  Anyone have
>>an especially quick algorithm they'd like to share?


>   I seem to recall from a standard graphics text that this method is
>slightly unstable: when used to draw circles it produces spirals.
>I'm not sure about this, but I seem to remember there was a simple fix,
>which unfortunately involved another multiply.

A standard CG trick to generate circle points is the "register saving" technique
originating from a programmer's mistake.

Rotating a point around the origin can be done by multiplying by a
rotation matrix:

                /  cos(phi)  -sin(phi)  \
[x' y'] = [x y] |                       |
		\  sin(phi)   cos(phi)  /

As an approximation for small angles you can use the matrix

    / 1       -phi \
    |              |
    \ phi       1  /

A program rotating the point:

   tmp = x + y * phi
   y = y - x * phi
   x = tmp

The mistake was:

   x = x + y * phi
   y = y - x * phi  (saving one scratch register)

This corresponds to the matrix

    / 1       -phi \
    |              |
    \ phi  1-phi^2 /

which has a determinant equal to 1. This algorithm draws an ellipse very close
to a circle. For further information see a CG book like
Graphics Gems, by A. Glassner.

I don't know whether this algorithm is stable, but you could restart it
after one full circle and use the X component as output.

Ge' Weijers
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