Maths (Geometry): ARCTAN

       Name: ARCTAN                                                  [Show more]
       Type: Subroutine
   Category: Maths (Geometry)
    Summary: Calculate A = arctan(P / Q)
  Deep dive: The sine, cosine and arctan tables
    Context: See this subroutine in context in the source code
 References: This subroutine is called as follows:
             * PLS4 calls ARCTAN


 Calculate the following:

   A = arctan(P / Q)

 In other words, this finds the angle in the right-angled triangle where the
 opposite side to angle A is length P and the adjacent side to angle A has
 length Q, so:

   tan(A) = P / Q

 The result in A is an integer representing the angle in radians. The routine
 returns values in the range 0 to 128, which covers 0 to 180 degrees (or 0 to
 PI radians).


.ARCTAN

 LDA P                  ; Set T1 = P EOR Q, which will have the sign of P * Q
 EOR Q                  ;
;AND #%10000000         ; The AND is commented out in the original source
 STA T1

 LDA Q                  ; If Q = 0, jump to AR2 to return a right angle
 BEQ AR2

 ASL A                  ; Set Q = |Q| * 2 (this is a quick way of clearing the
 STA Q                  ; sign bit, and we don't need to shift right again as we
                        ; only ever use this value in the division with |P| * 2,
                        ; which we set next)

 LDA P                  ; Set A = |P| * 2
 ASL A

 CMP Q                  ; If A >= Q, i.e. |P| > |Q|, jump to AR1 to swap P
 BCS AR1                ; and Q around, so we can still use the lookup table

 JSR ARS1               ; Call ARS1 to set the following from the lookup table:
                        ;
                        ;   A = arctan(A / Q)
                        ;     = arctan(|P / Q|)

 SEC                    ; Set the C flag so the SBC instruction in AR3 will be
                        ; correct, should we jump there

.AR4

 LDX T1                 ; If T1 is negative, i.e. P and Q have different signs,
 BMI AR3                ; jump down to AR3 to return arctan(-|P / Q|)

 RTS                    ; Otherwise P and Q have the same sign, so our result is
                        ; correct and we can return from the subroutine

.AR1

                        ; We want to calculate arctan(t) where |t| > 1, so we
                        ; can use the calculation described in the documentation
                        ; for the ACT table, i.e. 64 - arctan(1 / t)

 LDX Q                  ; Swap the values in Q and P, using the fact that we
 STA Q                  ; called AR1 with A = P
 STX P                  ;
 TXA                    ; This also sets A = P (which now contains the original
                        ; argument |Q|)

 JSR ARS1               ; Call ARS1 to set the following from the lookup table:
                        ;
                        ;   A = arctan(A / Q)
                        ;     = arctan(|Q / P|)
                        ;     = arctan(1 / |P / Q|)

 STA T                  ; Set T = 64 - T
 LDA #64
 SBC T

 BCS AR4                ; Jump to AR4 to continue the calculation (this BCS is
                        ; effectively a JMP as the subtraction will never
                        ; underflow, as ARS1 returns values in the range 0-31)

.AR2

                        ; If we get here then Q = 0, so tan(A) = infinity and
                        ; A is a right angle, or 0.25 of a circle. We allocate
                        ; 255 to a full circle, so we should return 63 for a
                        ; right angle

 LDA #63                ; Set A to 63, to represent a right angle

 RTS                    ; Return from the subroutine

.AR3

                        ; A contains arctan(|P / Q|) but P and Q have different
                        ; signs, so we need to return arctan(-|P / Q|), using
                        ; the calculation described in the documentation for the
                        ; ACT table, i.e. 128 - A

 STA T                  ; Set A = 128 - A
 LDA #128               ;
;SEC                    ; The SEC instruction is commented out in the original
 SBC T                  ; source, and isn't required as we did a SEC before
                        ; calling AR3

 RTS                    ; Return from the subroutine

.ARS1

                        ; This routine fetches arctan(A / Q) from the ACT table,
                        ; so A will be set to an integer in the range 0 to 31
                        ; that represents an angle from 0 to 45 degrees (or 0 to
                        ; PI / 4 radians)

 JSR LL28               ; Call LL28 to calculate:
                        ;
                        ;   R = 256 * A / Q

 LDA R                  ; Set X = R / 8
 LSR A                  ;       = 32 * A / Q
 LSR A                  ;
 LSR A                  ; so X has the value t * 32 where t = A / Q, which is
 TAX                    ; what we need to look up values in the ACT table

 LDA ACT,X              ; Fetch ACT+X from the ACT table into A, so now:
                        ;
                        ;   A = value in ACT + X
                        ;     = value in ACT + (32 * A / Q)
                        ;     = arctan(A / Q)

 RTS                    ; Return from the subroutine