Spectrum ROM: Routine at 09090

The address of this routine is found in the parameter table.

This routine is entered with the co-ordinates of a point X0, Y0, say, in COORDS. If only two parameters X, Y are given with the DRAW command, it draws an approximation to a straight line from the point X0, Y0 to X0+X, Y0+Y. If a third parameter G is given, it draws an approximation to a circular arc from X0, Y0 to X0+X, Y0+Y turning anti-clockwise through an angle G radians.

The routine has four parts:

i. Just draws a line if only 2 parameters are given or if the diameter of the implied circle is less than 1.
ii. Calls CD_PRMS1 to set the first parameters.
iii. Sets up the remaining parameters, including the initial displacements for the first arc.
iv. Enters the arc-drawing loop and draws the arc as a series of smaller arcs approximated by straight lines, calling the line-drawing subroutine at DRAW_LINE as necessary.

Two subroutines, CD_PRMS1 and DRAW_LINE, follow the main routine. The above 4 parts of the main routine will now be treated in turn.

i. If there are only 2 parameters, a jump is made to LINE_DRAW. A line is also drawn if the quantity Z=(ABS X+ABS Y)/ABS SIN(G/2) is less than 1. Z lies between 1 and 1.5 times the diameter of the implied circle. In this section mem-0 is set to SIN (G/2), mem-1 to Y, and mem-5 to G.

ii. Just calls CD_PRMS1. This subroutine saves in the B register the number of shorter arcs required for the complete arc, viz. A=4*INT (G'*SQR Z/8)+4 (where G'=ABS G), or 252 if this expression exceeds 252 (as can happen with a large chord and a small angle). So A is a multiple of 4 from 4 to 252. The subroutine also stores in mem-0 to mem-4 the quantities G/A, SIN (G/2*A), 0, COS (G/A), SIN (G/A).

iii. Sets up the rest of the parameters as follow. The stack will hold these 4 items, reading up to the top: X0+X and Y0+Y as end of last arc; then X0 and Y0 as beginning of first arc. Mem-0 will hold X0 and mem-5 Y0. Mem-1 and mem-2 will hold the initial displacements for the first arc, U and V; and mem-3 and mem-4 will hold COS (G/A) and SIN (G/A) for use in the arc-drawing loop.

The formulae for U and V can be explained as follows. Instead of stepping along the final chord, of length L, say, with displacements X and Y, we want to step along an initial chord (which may be longer) of length L*W, where W=SIN (G/2*A)/SIN (G/2), with displacements X*W and Y*W, but turned through an angle (G/2-G/2*A), hence with true displacements:

U=Y*W*SIN (G/2-G/2*A)+X*W*COS (G/2-G/2*A)
Y=Y*W*COS (G/2-G/2*A)-X*W*SIN (G/2-G/2*A)

These formulae can be checked from a diagram, using the normal expansion of COS (P-Q) and SIN (P-Q), where Q=G/2-G/2*A.

This entry point is used by the routine at CIRCLE.

iv. The arc-drawing loop. This is entered at ARC_START with the co-ordinates of the starting point on top of the stack, and the initial displacements for the first arc in mem-1 and mem-2. It uses simple trigonometry to ensure that all subsequent arcs will be drawn to points that lie on the same circle as the first two, subtending the same angle at the centre. It can be shown that if 2 points X1, Y1 and X2, Y2 lie on a circle and subtend an angle N at the centre, which is also the origin of co-ordinates, then X2=X1*COS N-Y1*SIN N, and Y2=X1*SIN N+Y1*COS N. But because the origin is here at the increments, say Un=Xn+1-Xn and Vn=Yn+1-Yn, thus achieving the desired result. The stack is shown below on the (n+1)th pass through the loop, as Xn and Yn are incremented by Un and Vn, after these are obtained from Un-1 and Vn-1. The 4 values on the top of the stack at ARC_LOOP are, in DRAW, reading upwards, X0+X, Y0+Y, Xn and Yn but to save space these are not shown until ARC_START. For the initial values in CIRCLE, see the end of CIRCLE, above. In CIRCLE too, the angle G must be taken to be 2π.

The address of this routine is found in the parameter table. This routine is entered with the co-ordinates of a point X0, Y0, say, in COORDS. If only two parameters X, Y are given with the DRAW command, it draws an approximation to a straight line from the point X0, Y0 to X0+X, Y0+Y. If a third parameter G is given, it draws an approximation to a circular arc from X0, Y0 to X0+X, Y0+Y turning anti-clockwise through an angle G radians. The routine has four parts: i. Just draws a line if only 2 parameters are given or if the diameter of the implied circle is less than 1. ii. Calls CD_PRMS1 to set the first parameters. iii. Sets up the remaining parameters, including the initial displacements for the first arc. iv. Enters the arc-drawing loop and draws the arc as a series of smaller arcs approximated by straight lines, calling the line-drawing subroutine at DRAW_LINE as necessary. Two subroutines, CD_PRMS1 and DRAW_LINE, follow the main routine. The above 4 parts of the main routine will now be treated in turn. i. If there are only 2 parameters, a jump is made to LINE_DRAW. A line is also drawn if the quantity Z=(ABS X+ABS Y)/ABS SIN(G/2) is less than 1. Z lies between 1 and 1.5 times the diameter of the implied circle. In this section mem-0 is set to SIN (G/2), mem-1 to Y, and mem-5 to G.
DRAW	09090	RST 24	Get the current character.
	09091	CP ","	If it is a comma, then jump.
	09093	JR Z,DR_3_PRMS	If it is a comma, then jump.
	09095	CALL CHECK_END	Move on to next statement if checking syntax.
	09098	JP LINE_DRAW	Jump to just draw the line.
DR_3_PRMS	09101	RST 32	Get next character (the angle).
	09102	CALL CLASS_06	Angle to calculator stack.
	09105	CALL CHECK_END	Move on to next statement if checking syntax.
	09108	RST 40	X, Y, G are on the stack.
	09109	DEFB 197	st_mem_5: (G is copied to mem-5)
	09110	DEFB 162	stk_half: X, Y, G, 0.5
	09111	DEFB 4	multiply: X, Y, G/2
	09112	DEFB 31	sin: X, Y, SIN (G/2)
	09113	DEFB 49	duplicate: X, Y, SIN (G/2), SIN (G/2)
	09114	DEFB 48	f_not: X, Y, SIN (G/2), (0/1)
	09115	DEFB 48	f_not: X, Y, SIN (G/2), (1/0)
	09116	DEFB 0	jump_true: X, Y, SIN (G/2)
	09117	DEFB 6	to DR_SIN_NZ (if SIN (G/2)=0 i.e. G=2πN just draw a straight line).
	09118	DEFB 2	delete: X, Y
	09119	DEFB 56	end_calc
	09120	JP LINE_DRAW	Line X0, Y0 to X0+X, Y0+Y.
DR_SIN_NZ	09123	DEFB 192	st_mem_0: (SIN (G/2) is copied to mem-0)
	09124	DEFB 2	delete: X, Y are now on the stack.
	09125	DEFB 193	st_mem_1: (Y is copied to mem-1).
	09126	DEFB 2	delete: X
	09127	DEFB 49	duplicate: X, X
	09128	DEFB 42	abs: X, X' (X'=ABS X)
	09129	DEFB 225	get_mem_1: X, X', Y
	09130	DEFB 1	exchange: X, Y, X'
	09131	DEFB 225	get_mem_1: X, Y, X', Y
	09132	DEFB 42	abs: X, Y, X', Y' (Y'=ABS Y)
	09133	DEFB 15	addition: X, Y, X'+Y'
	09134	DEFB 224	get_mem_0: X, Y, X'+Y', SIN (G/2)
	09135	DEFB 5	division: X, Y, (X'+Y')/SIN (G/2)=Z', say
	09136	DEFB 42	abs: X, Y, Z (Z=ABS Z')
	09137	DEFB 224	get_mem_0: X, Y, Z, SIN (G/2)
	09138	DEFB 1	exchange: X, Y, SIN (G/2), Z
	09139	DEFB 61	re_stack: (Z is re-stacked to make sure that its exponent is available).
	09140	DEFB 56	end_calc
	09141	LD A,(HL)	Get exponent of Z.
	09142	CP 129	If Z is greater than or equal to 1, jump.
	09144	JR NC,DR_PRMS	If Z is greater than or equal to 1, jump.
	09146	RST 40	X, Y, SIN (G/2), Z
	09147	DEFB 2	delete: X, Y, SIN (G/2)
	09148	DEFB 2	delete: X, Y
	09149	DEFB 56	end_calc
	09150	JP LINE_DRAW	Just draw the line from X0, Y0 to X0+X, Y0+Y.
ii. Just calls CD_PRMS1. This subroutine saves in the B register the number of shorter arcs required for the complete arc, viz. A=4INT (G'SQR Z/8)+4 (where G'=ABS G), or 252 if this expression exceeds 252 (as can happen with a large chord and a small angle). So A is a multiple of 4 from 4 to 252. The subroutine also stores in mem-0 to mem-4 the quantities G/A, SIN (G/2*A), 0, COS (G/A), SIN (G/A).
DR_PRMS	09153	CALL CD_PRMS1	The subroutine is called.
iii. Sets up the rest of the parameters as follow. The stack will hold these 4 items, reading up to the top: X0+X and Y0+Y as end of last arc; then X0 and Y0 as beginning of first arc. Mem-0 will hold X0 and mem-5 Y0. Mem-1 and mem-2 will hold the initial displacements for the first arc, U and V; and mem-3 and mem-4 will hold COS (G/A) and SIN (G/A) for use in the arc-drawing loop. The formulae for U and V can be explained as follows. Instead of stepping along the final chord, of length L, say, with displacements X and Y, we want to step along an initial chord (which may be longer) of length LW, where W=SIN (G/2A)/SIN (G/2), with displacements XW and YW, but turned through an angle (G/2-G/2A), hence with true displacements: U=YWSIN (G/2-G/2A)+XWCOS (G/2-G/2A) Y=YWCOS (G/2-G/2A)-XWSIN (G/2-G/2A) These formulae can be checked from a diagram, using the normal expansion of COS (P-Q) and SIN (P-Q), where Q=G/2-G/2A.
	09156	PUSH BC	Save the arc-counter in B.
	09157	RST 40	X, Y, SIN(G/2), Z
	09158	DEFB 2	delete: X, Y, SIN(G/2)
	09159	DEFB 225	get_mem_1: X, Y, SIN(G/2), SIN(G/2*A)
	09160	DEFB 1	exchange: X, Y, SIN(G/2*A), SIN(G/2)
	09161	DEFB 5	division: X, Y, SIN(G/2*A)/SIN(G/2)=W
	09162	DEFB 193	st_mem_1: (W is copied to mem-1).
	09163	DEFB 2	delete: X, Y
	09164	DEFB 1	exchange: Y, X
	09165	DEFB 49	duplicate: Y, X, X
	09166	DEFB 225	get_mem_1: Y, X, X, W
	09167	DEFB 4	multiply: Y, X, X*W
	09168	DEFB 194	st_mem_2: (X*W is copied to mem-2).
	09169	DEFB 2	delete: Y, X
	09170	DEFB 1	exchange: X, Y
	09171	DEFB 49	duplicate: X, Y, Y
	09172	DEFB 225	get_mem_1: X, Y, Y, W
	09173	DEFB 4	multiply: X, Y, Y*W
	09174	DEFB 226	get_mem_2: X, Y, YW, XW
	09175	DEFB 229	get_mem_5: X, Y, YW, XW,G
	09176	DEFB 224	get_mem_0: X, Y, YW, XW, G, G/A
	09177	DEFB 3	subtract: X, Y, YW, XW, G-G/A
	09178	DEFB 162	stk_half: X, Y, YW, XW, G-G/A, 1/2
	09179	DEFB 4	multiply: X, Y, YW, XW, G/2-G/2*A=F
	09180	DEFB 49	duplicate: X, Y, YW, XW, F, F
	09181	DEFB 31	sin: X, Y, YW, XW, F, SIN F
	09182	DEFB 197	st_mem_5: (SIN F is copied to mem-5).
	09183	DEFB 2	delete: X, Y, YW, XW,F
	09184	DEFB 32	cos: X, Y, YW, XW, COS F
	09185	DEFB 192	st_mem_0: (COS F is copied to mem-0).
	09186	DEFB 2	delete: X, Y, YW, XW
	09187	DEFB 194	st_mem_2: (X*W is copied to mem-2).
	09188	DEFB 2	delete: X, Y, Y*W
	09189	DEFB 193	st_mem_1: (Y*W is copied to mem-1).
	09190	DEFB 229	get_mem_5: X, Y, Y*W, SIN F
	09191	DEFB 4	multiply: X, Y, YWSIN F
	09192	DEFB 224	get_mem_0: X, Y, YWSIN F, X*W
	09193	DEFB 226	get_mem_2: X, Y, YWSIN F, X*W, COS F
	09194	DEFB 4	multiply: X, Y, YWSIN F, XWCOS F
	09195	DEFB 15	addition: X, Y, YWSIN F+XWCOS F=U
	09196	DEFB 225	get_mem_1: X, Y, U, Y*W
	09197	DEFB 1	exchange: X, Y, Y*W, U
	09198	DEFB 193	st_mem_1: (U is copied to mem-1)
	09199	DEFB 2	delete: X, Y, Y*W
	09200	DEFB 224	get_mem_0: X, Y, Y*W, COS F
	09201	DEFB 4	multiply: X, Y, YWCOS F
	09202	DEFB 226	get_mem_2: X, Y, YWCOS F, X*W
	09203	DEFB 229	get_mem_5: X, Y, YWCOS F, X*W, SIN F
	09204	DEFB 4	multiply: X, Y, YWCOS F, XWSIN F
	09205	DEFB 3	subtract: X, Y, YWCOS F-XWSIN F=V
	09206	DEFB 194	st_mem_2: (V is copied to mem-2).
	09207	DEFB 42	abs: X, Y, V' (V'=ABS V)
	09208	DEFB 225	get_mem_1: X, Y, V', U
	09209	DEFB 42	abs: X, Y, V', U' (U'=ABS U)
	09210	DEFB 15	addition: X, Y, U'+V'
	09211	DEFB 2	delete: X, Y
	09212	DEFB 56	end_calc: (DE now points to U'+V').
	09213	LD A,(DE)	Get exponent of U'+V'.
	09214	CP 129	If U'+V' is less than 1, just tidy the stack and draw the line from X0, Y0 to X0+X, Y0+Y.
	09216	POP BC
	09217	JP C,LINE_DRAW
	09220	PUSH BC	Otherwise, continue with the parameters: X, Y, on the stack.
	09221	RST 40
	09222	DEFB 1	exchange: Y, X
	09223	DEFB 56	end_calc
	09224	LD A,(23677)	Get X0 from COORDS into A and so on to the stack.
	09227	CALL STACK_A	Get X0 from COORDS into A and so on to the stack.
	09230	RST 40	Y, X, X0
	09231	DEFB 192	st_mem_0: (X0 is copied to mem-0).
	09232	DEFB 15	addition: Y, X0+X
	09233	DEFB 1	exchange: X0+X, Y
	09234	DEFB 56	end_calc
	09235	LD A,(23678)	Get Y0 from COORDS into A and so on to the stack.
	09238	CALL STACK_A	Get Y0 from COORDS into A and so on to the stack.
	09241	RST 40	X0+X, Y, Y0
	09242	DEFB 197	st_mem_5: (Y0 is copied to mem-5).
	09243	DEFB 15	addition: X0+X, Y0+Y
	09244	DEFB 224	get_mem_0: X0+X, Y0+Y, X0
	09245	DEFB 229	get_mem_5: X0+X, Y0+Y, X0, Y0
	09246	DEFB 56	end_calc
	09247	POP BC	Restore the arc-counter in B.
This entry point is used by the routine at CIRCLE. iv. The arc-drawing loop. This is entered at ARC_START with the co-ordinates of the starting point on top of the stack, and the initial displacements for the first arc in mem-1 and mem-2. It uses simple trigonometry to ensure that all subsequent arcs will be drawn to points that lie on the same circle as the first two, subtending the same angle at the centre. It can be shown that if 2 points X1, Y1 and X2, Y2 lie on a circle and subtend an angle N at the centre, which is also the origin of co-ordinates, then X2=X1COS N-Y1SIN N, and Y2=X1SIN N+Y1COS N. But because the origin is here at the increments, say Un=Xn+1-Xn and Vn=Yn+1-Yn, thus achieving the desired result. The stack is shown below on the (n+1)th pass through the loop, as Xn and Yn are incremented by Un and Vn, after these are obtained from Un-1 and Vn-1. The 4 values on the top of the stack at ARC_LOOP are, in DRAW, reading upwards, X0+X, Y0+Y, Xn and Yn but to save space these are not shown until ARC_START. For the initial values in CIRCLE, see the end of CIRCLE, above. In CIRCLE too, the angle G must be taken to be 2π.
DRW_STEPS	09248	DEC B	B counts the passes through the loop.
	09249	JR Z,ARC_END	Jump when B has reached zero.
	09251	JR ARC_START	Jump into the loop to start.
ARC_LOOP	09253	RST 40	(See text above for the stack).
	09254	DEFB 225	get_mem_1: Un-1
	09255	DEFB 49	duplicate: Un-1, Un-1
	09256	DEFB 227	get_mem_3: Un-1, Un-1, COS(G/A)
	09257	DEFB 4	multiply: Un-1, Un-1*COS(G/A)
	09258	DEFB 226	get_mem_2: Un-1, Un-1*COS(G/A), Vn-1
	09259	DEFB 228	get_mem_4: Un-1, Un-1*COS(G/A), Vn-1, SIN(G/A)
	09260	DEFB 4	multiply: Un-1, Un-1COS(G/A), Vn-1SIN(G/A)
	09261	DEFB 3	subtract: Un-1, Un-1COS(G/A)-Vn-1SIN(G/A)=Un
	09262	DEFB 193	st_mem_1: (Un is copied to mem-1).
	09263	DEFB 2	delete: Un-1
	09264	DEFB 228	get_mem_4: Un-1, SIN(G/A)
	09265	DEFB 4	multiply: Un-1*SIN(G/A)
	09266	DEFB 226	get_mem_2: Un-1*SIN(G/A), Vn-1
	09267	DEFB 227	get_mem_3: Un-1*SIN(G/A), Vn-1, COS(G/A)
	09268	DEFB 4	multiply: Un-1SIN(G/A), Vn-1COS(G/A)
	09269	DEFB 15	addition: Un-1SIN(G/A)+Vn-1COS(G/A)=Vn
	09270	DEFB 194	st_mem_2: (Vn is copied to mem-2).
	09271	DEFB 2	delete: (As noted in the text, the stack in fact holds X0+X, Y0+Y, Xn and Yn).
	09272	DEFB 56	end_calc
ARC_START	09273	PUSH BC	Save the arc-counter.
	09274	RST 40	X0+X, Y0+y, Xn, Yn
	09275	DEFB 192	st_mem_0: (Yn is copied to mem-0).
	09276	DEFB 2	delete: X0+X, Y0+Y, Xn
	09277	DEFB 225	get_mem_1: X0+X, Y0+Y, Xn, Un
	09278	DEFB 15	addition: X0+X, Y0+Y, Xn+Un=Xn+1
	09279	DEFB 49	duplicate: X0+X, Y0+Y, Xn+1, Xn+1
	09280	DEFB 56	end_calc
	09281	LD A,(23677)	Next Xn', the approximate value of Xn reached by the line-drawing subroutine is copied from COORDS to A and hence to the stack.
	09284	CALL STACK_A
	09287	RST 40	X0+X, Y0+Y, Xn+1, Xn'
	09288	DEFB 3	subtract: X0+X, Y0+Y, Xn+1, Xn+1, Xn'-Xn'=Un'
	09289	DEFB 224	get_mem_0: X0+X, Y0+Y, Xn+1, Un', Yn
	09290	DEFB 226	get_mem_2: X0+X, Y0+Y, Xn+1, Un', Yn, Vn
	09291	DEFB 15	addition: X0+X, Y0+Y, Xn+1, Un', Yn+Vn=Yn+1
	09292	DEFB 192	st_mem_0: (Yn+1 is copied to mem-0).
	09293	DEFB 1	exchange: X0+X, Y0+Y, Xn+1, Yn+1, Un'
	09294	DEFB 224	get_mem_0: X0+X, Y0+Y, Xn+1, Yn+1, Un', Yn+1
	09295	DEFB 56	end_calc
	09296	LD A,(23678)	Yn', approximate like Xn', is copied from COORDS to A and hence to the stack.
	09299	CALL STACK_A
	09302	RST 40	X0+X, Y0+Y, Xn+1, Yn+1, Un', Yn+1, Yn'
	09303	DEFB 3	subtract: X0+X, Y0+Y, Xn+1, Yn+1, Un', Vn'
	09304	DEFB 56	end_calc
	09305	CALL DRAW_LINE	The next 'arc' is drawn.
	09308	POP BC	The arc-counter is restored.
	09309	DJNZ ARC_LOOP	Jump if more arcs to draw.
ARC_END	09311	RST 40	Jump if more arcs to draw.
	09312	DEFB 2	delete: The co-ordinates of the end of the last arc that was drawn are now deleted from the stack.
	09313	DEFB 2
	09314	DEFB 1	exchange: Y0+Y, X0+X
	09315	DEFB 56	end_calc
	09316	LD A,(23677)	The X-co-ordinate of the end of the last arc that was drawn, say Xz', is copied from COORDS to the stack.
	09319	CALL STACK_A
	09322	RST 40
	09323	DEFB 3	subtract: Y0+Y, X0+X-Xz'
	09324	DEFB 1	exchange: X0+X-Xz', Y0+Y
	09325	DEFB 56	end_calc
	09326	LD A,(23678)	The Y-co-ordinate is obtained from COORDS and stacked.
	09329	CALL STACK_A	The Y-co-ordinate is obtained from COORDS and stacked.
	09332	RST 40	X0+X-Xz', Y0+Y, Yz'
	09333	DEFB 3	subtract: X0+X-Xz', Y0+Y-Yz'
	09334	DEFB 56	end_calc
LINE_DRAW	09335	CALL DRAW_LINE	The final arc is drawn to reach X0+X, Y0+Y (or close the circle).
	09338	JP TEMPS	Exit, setting temporary colours.