This routine will take an 8 bit integer that corresponds to the numerator of a fraction whose denominator is 256 and find its arctangent. So the input ranges from 0 to 255 which corresponds to 0 to 255/256 = 0.996 . The output for an arctangent routine that returns a floating point number would be from 0 (atan(0)) to 0.783 (atan(255/256)) radians; or if you prefer, 0 to 44.89 degrees. However, this routine scales the output so that pi/4 radians (or 45 degrees) corresponds to 256. So for the input range of 0 to 255 you get an output of 0 to 255 ( atan(255/256) * 256 / (pi/4) is about 255). It's probably a little more interesting to see an intermediate data point or two:
| CODE |
Intger Float x | atan(x) | x | atan(x) ------+----------+-------+--------- 0x4a | 0x5b | .289 | .281 0x62 | 0x77 | .383 | .366 0x6f | 0x84 | .434 | .409 0xa6 | 0xbb | .648 | .575 0xdb | 0xe6 | .855 | .707 |
The only thing that's left is combining the fractional division and the swapping of the x and y values if y is greater than x (and then subtracting the result from pi/2 or actually 512 in this case).
| CODE |
;---------------------------------------------------------- ; ;arctan (as adapted from the similar arcsin function) ; ; The purpose of this routine is to take the arctan of an ;8-bit number that ranges from 0 < x < 255/256. In other ;words, the input, x, is an unsigned fraction whose implicit ;divisor is 256. ; The output is in a conveniently awkward format of binary ;radians (brads?). The output corresponds to the range of zero ;to pi/4 for the normal arctan function. Specifically, this ;algorithm computes: ; ; arctan(x) = real_arctan(x/256) * 256 / (pi/4) ; for 0 <= x <= 255 ; ; where, real_arctan returns the real arctan of its argument ;in radians. ; ; The algorithm is a table look-up algorithm plus first order ;linear interpolation. The psuedo code is: ; ;unsigned char arctan(unsigned char x) ;{ ; unsigned char i; ; ; i = x >> 4; ; return(arctan[i] + ((arctan[i+1] - arctan[i]) * (x & 0xf))/16); ;} ; ; arctan SWAPF x,W ANDLW 0xf ADDLW 1 MOVWF temp ;Temporarily store the index CALL arc_tan_table ;Get a2=atan( (x>>4) + 1) MOVWF result ;Store temporarily in result DECF temp,W ;Get the saved index CALL arc_tan_table ;Get a1=atan( (x>>4) ) SUBWF result,W ;W=a2-a1, This is always positive. SUBWF result,F ;a1 = a1 - (a1-W) = W CLRF temp ;Clear the product CLRC BTFSC x,0 ADDWF temp,F RRF temp,F CLRC BTFSC x,1 ADDWF temp,F RRF temp,F CLRC BTFSC x,2 ADDWF temp,F RRF temp,F CLRC BTFSC x,3 ADDWF temp,F RRF temp,W ADDWF result,F RETURN arc_tan_table ADDWF PCL,F RETLW 0 RETLW 20 ;atan(1/16) = 3.576deg * 256/45 RETLW 41 RETLW 60 RETLW 80 RETLW 99 RETLW 117 RETLW 134 RETLW 151 RETLW 167 RETLW 182 RETLW 196 RETLW 210 RETLW 222 RETLW 234 RETLW 245 RETLW 0 ;atan(32/32) = 45deg * 256/45 |
The other part of the problem is implementing the division
| CODE |
FRAC_DIV: ;------------------- ;Fractional division ; ; Given x,y this routine finds: ; a = 256 * y / x ; movlw 8 ;number of bits in the result movwf cnt clrf a ; the result movf x,w L1: clrc rlf y,f ;if msb of y is set we know x<y rlf a,f ;and that the lsb of 'a' should be set subwf y,f ;But we still need to subtract the ;divisor from the dividend just in ;case y is less than 256. skpnc ;If y>x, but y<256 bsf a,0 ; we still need to set a:0 btfss a,0 ;If y<x then we shouldn't have addwf y,f ;done the subtraction decfsz cnt,f goto L1 return |
It's easy enough to combine these two routines to obtain a 4-quadrant arctan(y/x) routine. However, you do need to keed in mind that the arctangent routine posted above is only valid over 1/8th of the unit circle. To obtain the other 7/8th's you'll need to apply the appropriate trig identities.
| CODE |
// pic routines: extern int arctan(int x); extern int frac_div(int y, int x); // Untested c code that implements a 4-quadrant arctan function int arctan(int x, int y) { int f, swapped; int reference_angle; swapped = 0; if(x < 0) { if(y < 0) reference_angle = 256 * 2; // pi else reference_angle = 256; // pi/2 } else { if( y < 0) reference_angle = 256 * 3; // 3*pi/2 else reference_angle = 0; } if (x<=y) { f = y; y = x; x = f; swapped = 1; } f = frac_div(y,x); f = arctan(f); if (swapped) f = 256 - f; return f + reference_angle; } |