====== Generating Sine Tables from Parabolas ======
by White Flame

It's been a long-standing tradition in games & demos that sine waves can be approximated by parabolas (see the graph at the bottom).  They're a little boxier, and deviate to an error of about 6%, but generally work for doing quick and dirty trig.

{{:base:sine-parabolas2.png|}}

Parabolas are easy to generate, as they can represent a value under constant acceleration, which is discretely defined as
<code c>
loop
{
  x += dx
  dx += Constant
}
</code>

Here's a simple implementation ripped from from [[http://noname.c64.org/csdb/release/?id=11730|Too(C)o(M)p(L)ex]] by Cruzer/Camelot, and adjusted a bit for clarity.  The original source is in the download from the CSDb page, and uses self-modifying code to hold the value and delta.

<code asm>
; ca65 syntax

initSineTable:

 ldy #$3f
 ldx #$00

; Accumulate the delta (normal 16-bit addition)
: lda value
  clc
  adc delta
  sta value
  lda value+1
  adc delta+1
  sta value+1

; Reflect the value around for a sine wave
  sta sine+$c0,x
  sta sine+$80,y
  eor #$ff
  sta sine+$40,x
  sta sine+$00,y

; Increase the delta, which creates the "acceleration" for a parabola
  lda delta
  adc #$10   ; this value adds up to the proper amplitude
  sta delta
  bcc :+
   inc delta+1
:

; Loop
  inx
  dey
 bpl :--

 rts

value: .word 0
delta: .word 0

sine: .res 256
</code>


===== Notes on the code =====

==== Precision ====

The accelerating value we calculate is held in a 16-bit number, the high byte of which we will use to fill in the values in the 0-255 sine table.  This is required, as when the curve is on its more "flat" regions, the delta is much less than 1/256th of the amplitude (what a single byte can hold).

==== A piece at a time ====

The outer loop only spans 1/4th of the period (ie, 0-1 from the graph), as each quarter can be reflected onto the other.  As the value accelerates from 0-127, it's stored mirrored around $c0 (x=3 on the graph), while the inverted value is mirrored around $40 (x=1 on the graph).  It uses an incrementing .X and decrementing .Y to accomplish the mirroring.

==== Output values ====

The final table follows the same shape as the graph, going through this progression:
  * table + $00 = ~$80
  * table + $40 = $ff
  * table + $80 = ~$80
  * table + $c0 = $00
  * table + $ff = ~$80
Thus, the sine values are unsigned with a DC offset of $80.

===== Modifications =====

==== Cosine instead of Sine ====

Simply change how the output is written:

<code asm>
; Reflect the value around for a cosine wave
  sta sine+$80,x
  sta sine+$40,y
  eor #$ff
  sta sine+$00,x
  sta sine+$c0,y
</code>

==== Amplitude ====

If you want a range of $00-$7f instead of $00-$ff (as is in the original demo):
  * change the EOR value from #$ff to #$7f
  * change the delta acceleration from #$10 to #$08
Any base-2 amplitude should be likewise possible.