A round pickup should not become a square just because a box test was convenient.
Use it in a project
All coordinates and RADIUS are unsigned bytes in the same non-wrapping coordinate space. Return absolute DX/DY and HIT=1 when DX²+DY²≤RADIUS², including the boundary. Radius zero accepts only the centre. Inputs are preserved.
Include component.inc once (MOS callers include the shared 6502 implementation through it). Reserve the public fields in this exact order: HIT, DX, DY, POINT_X, POINT_Y, CENTRE_X, CENTRE_Y, RADIUS. They occupy 8 contiguous bytes. Also reserve the private workspace: T0, T1, T2, T3, T4, T5, T6, T7, T8, T9 (10 bytes). Array names in the explanation refer to the numbered consecutive bindings, such as C0, C1 and so on.
Call point_circle_init once at the intended reset boundary. It clears inputs as well as outputs and workspace. Set your starting state and configuration afterwards, then call point_circle_step whenever the described event occurs. The helper reads no device, waits for no frame and installs no interrupt handler.
| Ownership | Fields |
|---|---|
| Retain between calls | none |
| Supply before service | POINT_X, POINT_Y, CENTRE_X, CENTRE_Y and RADIUS |
| Read after service | HIT, DX and DY |
Reject a point outside the radius on either axis first. For a surviving point, compare the second squared displacement with the squared-radius budget left after the first. No square root is needed.
Reusable initialisation and service
; SPDX-License-Identifier: MIT
; Test a Point Against a Circle. See README.md for the byte interface and ownership.
; init clears the public fields and private workspace; set inputs afterwards.
point_circle_init:
lda #0
sta HIT
sta DX
sta DY
sta POINT_X
sta POINT_Y
sta CENTRE_X
sta CENTRE_Y
sta RADIUS
sta T0
sta T1
sta T2
sta T3
sta T4
sta T5
sta T6
sta T7
sta T8
sta T9
rts
point_circle_step:
; HIT = 0
lda #0
sta HIT
; if POINT_X >= CENTRE_X:
lda POINT_X
cmp CENTRE_X
bcs point_circle_L4
point_circle_L5:
jmp point_circle_L2
point_circle_L4:
jmp point_circle_L1
point_circle_L1:
; DX = POINT_X - CENTRE_X
lda POINT_X
sec
sbc CENTRE_X
sta DX
jmp point_circle_L3
point_circle_L2:
; DX = CENTRE_X - POINT_X
lda CENTRE_X
sec
sbc POINT_X
sta DX
point_circle_L3:
; if POINT_Y >= CENTRE_Y:
lda POINT_Y
cmp CENTRE_Y
bcs point_circle_L9
point_circle_L10:
jmp point_circle_L7
point_circle_L9:
jmp point_circle_L6
point_circle_L6:
; DY = POINT_Y - CENTRE_Y
lda POINT_Y
sec
sbc CENTRE_Y
sta DY
jmp point_circle_L8
point_circle_L7:
; DY = CENTRE_Y - POINT_Y
lda CENTRE_Y
sec
sbc POINT_Y
sta DY
point_circle_L8:
; if DX > RADIUS:
lda DX
cmp RADIUS
beq point_circle_L15
bcs point_circle_L14
point_circle_L15:
jmp point_circle_L12
point_circle_L14:
jmp point_circle_L11
point_circle_L11:
; return
rts
jmp point_circle_L13
point_circle_L12:
point_circle_L13:
; if DY > RADIUS:
lda DY
cmp RADIUS
beq point_circle_L20
bcs point_circle_L19
point_circle_L20:
jmp point_circle_L17
point_circle_L19:
jmp point_circle_L16
point_circle_L16:
; return
rts
jmp point_circle_L18
point_circle_L17:
point_circle_L18:
; T4 = RADIUS
lda RADIUS
sta T4
; T5 = 0
lda #0
sta T5
; T1 = RADIUS
lda RADIUS
sta T1
; T6 = 0
lda #0
sta T6
; T7 = 0
lda #0
sta T7
; T0 = 8
lda #8
sta T0
; while T0 != 0:
point_circle_L21:
lda T0
cmp #0
bne point_circle_L24
point_circle_L25:
jmp point_circle_L23
point_circle_L24:
jmp point_circle_L22
point_circle_L22:
; T9 = T1 & 1
lda T1
and #1
sta T9
; if T9 != 0:
lda T9
cmp #0
bne point_circle_L29
point_circle_L30:
jmp point_circle_L27
point_circle_L29:
jmp point_circle_L26
point_circle_L26:
; T8 = T6
lda T6
sta T8
; T6 = T6 + T4
lda T6
clc
adc T4
sta T6
; T9 = 0
lda #0
sta T9
; if T6 < T8:
lda T6
cmp T8
bcc point_circle_L34
point_circle_L35:
jmp point_circle_L32
point_circle_L34:
jmp point_circle_L31
point_circle_L31:
; T9 = 1
lda #1
sta T9
jmp point_circle_L33
point_circle_L32:
point_circle_L33:
; T7 = T7 + T5
lda T7
clc
adc T5
sta T7
; T7 = T7 + T9
lda T7
clc
adc T9
sta T7
jmp point_circle_L28
point_circle_L27:
point_circle_L28:
; T9 = T4 >> 7
lda T4
lsr
lsr
lsr
lsr
lsr
lsr
lsr
sta T9
; T4 = T4 << 1
lda T4
asl
sta T4
; T5 = T5 << 1
lda T5
asl
sta T5
; T5 = T5 | T9
lda T5
ora T9
sta T5
; T1 = T1 >> 1
lda T1
lsr
sta T1
; T0 = T0 - 1
lda T0
sec
sbc #1
sta T0
jmp point_circle_L21
point_circle_L23:
; T2 = T6
lda T6
sta T2
; T3 = T7
lda T7
sta T3
; T4 = DX
lda DX
sta T4
; T5 = 0
lda #0
sta T5
; T1 = DX
lda DX
sta T1
; T6 = 0
lda #0
sta T6
; T7 = 0
lda #0
sta T7
; T0 = 8
lda #8
sta T0
; while T0 != 0:
point_circle_L36:
lda T0
cmp #0
bne point_circle_L39
point_circle_L40:
jmp point_circle_L38
point_circle_L39:
jmp point_circle_L37
point_circle_L37:
; T9 = T1 & 1
lda T1
and #1
sta T9
; if T9 != 0:
lda T9
cmp #0
bne point_circle_L44
point_circle_L45:
jmp point_circle_L42
point_circle_L44:
jmp point_circle_L41
point_circle_L41:
; T8 = T6
lda T6
sta T8
; T6 = T6 + T4
lda T6
clc
adc T4
sta T6
; T9 = 0
lda #0
sta T9
; if T6 < T8:
lda T6
cmp T8
bcc point_circle_L49
point_circle_L50:
jmp point_circle_L47
point_circle_L49:
jmp point_circle_L46
point_circle_L46:
; T9 = 1
lda #1
sta T9
jmp point_circle_L48
point_circle_L47:
point_circle_L48:
; T7 = T7 + T5
lda T7
clc
adc T5
sta T7
; T7 = T7 + T9
lda T7
clc
adc T9
sta T7
jmp point_circle_L43
point_circle_L42:
point_circle_L43:
; T9 = T4 >> 7
lda T4
lsr
lsr
lsr
lsr
lsr
lsr
lsr
sta T9
; T4 = T4 << 1
lda T4
asl
sta T4
; T5 = T5 << 1
lda T5
asl
sta T5
; T5 = T5 | T9
lda T5
ora T9
sta T5
; T1 = T1 >> 1
lda T1
lsr
sta T1
; T0 = T0 - 1
lda T0
sec
sbc #1
sta T0
jmp point_circle_L36
point_circle_L38:
; T8 = 0
lda #0
sta T8
; if T2 < T6:
lda T2
cmp T6
bcc point_circle_L54
point_circle_L55:
jmp point_circle_L52
point_circle_L54:
jmp point_circle_L51
point_circle_L51:
; T8 = 1
lda #1
sta T8
jmp point_circle_L53
point_circle_L52:
point_circle_L53:
; T2 = T2 - T6
lda T2
sec
sbc T6
sta T2
; T3 = T3 - T7
lda T3
sec
sbc T7
sta T3
; T3 = T3 - T8
lda T3
sec
sbc T8
sta T3
; T4 = DY
lda DY
sta T4
; T5 = 0
lda #0
sta T5
; T1 = DY
lda DY
sta T1
; T6 = 0
lda #0
sta T6
; T7 = 0
lda #0
sta T7
; T0 = 8
lda #8
sta T0
; while T0 != 0:
point_circle_L56:
lda T0
cmp #0
bne point_circle_L59
point_circle_L60:
jmp point_circle_L58
point_circle_L59:
jmp point_circle_L57
point_circle_L57:
; T9 = T1 & 1
lda T1
and #1
sta T9
; if T9 != 0:
lda T9
cmp #0
bne point_circle_L64
point_circle_L65:
jmp point_circle_L62
point_circle_L64:
jmp point_circle_L61
point_circle_L61:
; T8 = T6
lda T6
sta T8
; T6 = T6 + T4
lda T6
clc
adc T4
sta T6
; T9 = 0
lda #0
sta T9
; if T6 < T8:
lda T6
cmp T8
bcc point_circle_L69
point_circle_L70:
jmp point_circle_L67
point_circle_L69:
jmp point_circle_L66
point_circle_L66:
; T9 = 1
lda #1
sta T9
jmp point_circle_L68
point_circle_L67:
point_circle_L68:
; T7 = T7 + T5
lda T7
clc
adc T5
sta T7
; T7 = T7 + T9
lda T7
clc
adc T9
sta T7
jmp point_circle_L63
point_circle_L62:
point_circle_L63:
; T9 = T4 >> 7
lda T4
lsr
lsr
lsr
lsr
lsr
lsr
lsr
sta T9
; T4 = T4 << 1
lda T4
asl
sta T4
; T5 = T5 << 1
lda T5
asl
sta T5
; T5 = T5 | T9
lda T5
ora T9
sta T5
; T1 = T1 >> 1
lda T1
lsr
sta T1
; T0 = T0 - 1
lda T0
sec
sbc #1
sta T0
jmp point_circle_L56
point_circle_L58:
; if T7 > T3:
lda T7
cmp T3
beq point_circle_L75
bcs point_circle_L74
point_circle_L75:
jmp point_circle_L72
point_circle_L74:
jmp point_circle_L71
point_circle_L71:
; return
rts
jmp point_circle_L73
point_circle_L72:
point_circle_L73:
; if T7 == T3:
lda T7
cmp T3
beq point_circle_L79
point_circle_L80:
jmp point_circle_L77
point_circle_L79:
jmp point_circle_L76
point_circle_L76:
; if T6 > T2:
lda T6
cmp T2
beq point_circle_L85
bcs point_circle_L84
point_circle_L85:
jmp point_circle_L82
point_circle_L84:
jmp point_circle_L81
point_circle_L81:
; return
rts
jmp point_circle_L83
point_circle_L82:
point_circle_L83:
jmp point_circle_L78
point_circle_L77:
point_circle_L78:
; HIT = 1
lda #1
sta HIT
rts
component_end:Machine contract
A/X/status may change; Y and decimal mode are preserved. Use CLD for binary arithmetic. A direct JSR needs two return-address stack bytes. The helper makes no nested calls and adds no other stack use. Its named scratch is private workspace, not a pointer, a ROM variable or shared zero-page space that the game may borrow concurrently. The samples use fixed symbol bindings. Scope or rename every label and binding, including the component_end measurement marker, and allocate a separate block for each independent instance. Different helpers can also use common field names, so combining them needs separate namespaces or explicitly renamed bindings. Give each block one mutation owner.
The diagnostic PRG owns screen, font and colour RAM and disables interrupts; it does not return to BASIC. The reusable helper does none of those things.
Adapt the caller
demo.asm supplies the bindings and include order. exercise.inc first runs table cases, then a separate client retaining state between calls. A game can call the public step entry directly and remove the diagnostic packet and display. Do not copy the diagnostic’s hardware takeover into an existing game loop.
Build with make ASM198X=/path/to/asm198x. The Makefile produces normal, custom, deliberately wrong and repaired programs. Tools must already be installed; the sample downloads nothing.
Bindings and include order
HIT=$80
DX=$81
DY=$82
POINT_X=$83
POINT_Y=$84
CENTRE_X=$85
CENTRE_Y=$86
RADIUS=$87
T0=$88
T1=$89
T2=$8a
T3=$8b
T4=$8c
T5=$8d
T6=$8e
T7=$8f
T8=$90
T9=$91
STATE=$80
input_ptr=$70
cases_left=$72
ptr=$74
saved_y=$76
OUTPUT=$c000
preview=OUTPUT+1400
ready=OUTPUT+1408
CUSTOM=0
BROKEN=0
COYOTE_STEPS=4
BUFFER_STEPS=4
TEXT_CAP=8
PROPERTY_COUNT=16
STREAM_BYTES=13
CRC_POLY=7
!source "caller.inc"
!source "exercise.inc"
component_start:
!source "component.inc"
!source "data.inc"
!source "cases.inc"
hex_font:
!byte 0,0,0,0,0,0,0,0,0,56,40,40,40,56,0,0,0,16,48,16,16,56,0,0,0,56,8,56,32,56,0,0,0,56,8,56,8,56,0,0,0,40,40,56,8,8,0,0,0,56,32,56,8,56,0,0,0,56,32,56,40,56,0,0,0,56,8,8,8,8,0,0,0,56,40,56,40,56,0,0,0,56,40,56,8,56,0,0,0,16,40,56,40,40,0,0,0,48,40,48,40,48,0,0,0,56,32,32,32,56,0,0,0,48,40,40,40,48,0,0,0,56,32,48,32,56,0,0,0,56,32,48,32,32,0,0
code_end:A client retaining state between calls
preview_rules:
!if CUSTOM {
; POINT_X = 3
lda #3
sta POINT_X
; POINT_Y = 4
lda #4
sta POINT_Y
; RADIUS = 5
lda #5
sta RADIUS
!if BROKEN {
; POINT_Y = CENTRE_Y
lda CENTRE_Y
sta POINT_Y
}
jsr checked_step
jsr emit_state
rts
} else {
; POINT_X = 4
lda #4
sta POINT_X
; POINT_Y = 4
lda #4
sta POINT_Y
; RADIUS = 5
lda #5
sta RADIUS
!if BROKEN {
; POINT_Y = CENTRE_Y
lda CENTRE_Y
sta POINT_Y
}
jsr checked_step
jsr emit_state
}
rtsCustomise it
Move onto the familiar 3–4–5 boundary. The maintained custom caller changes its inputs or client sequence rather than hiding a different implementation behind the same label. Compare every state of the two clients, not only their final display.
Custom client selection
HIT=$80
DX=$81
DY=$82
POINT_X=$83
POINT_Y=$84
CENTRE_X=$85
CENTRE_Y=$86
RADIUS=$87
T0=$88
T1=$89
T2=$8a
T3=$8b
T4=$8c
T5=$8d
T6=$8e
T7=$8f
T8=$90
T9=$91
STATE=$80
input_ptr=$70
cases_left=$72
ptr=$74
saved_y=$76
OUTPUT=$c000
preview=OUTPUT+1400
ready=OUTPUT+1408
CUSTOM=1
BROKEN=0
COYOTE_STEPS=2
BUFFER_STEPS=2
TEXT_CAP=4
PROPERTY_COUNT=16
STREAM_BYTES=13
CRC_POLY=49
!source "caller.inc"
!source "exercise.inc"
component_start:
!source "component.inc"
!source "data.inc"
!source "cases.inc"
hex_font:
!byte 0,0,0,0,0,0,0,0,0,56,40,40,40,56,0,0,0,16,48,16,16,56,0,0,0,56,8,56,32,56,0,0,0,56,8,56,8,56,0,0,0,40,40,56,8,8,0,0,0,56,32,56,8,56,0,0,0,56,32,56,40,56,0,0,0,56,8,8,8,8,0,0,0,56,40,56,40,56,0,0,0,56,40,56,8,56,0,0,0,16,40,56,40,40,0,0,0,48,40,48,40,48,0,0,0,56,32,32,32,56,0,0,0,48,40,40,40,48,0,0,0,56,32,48,32,56,0,0,0,56,32,48,32,32,0,0
code_end:The custom diagnostic row is 01 03 04 03 04 00 00 05. The full client trace in the source explains how it gets there.
Mistake and repair
The wrong caller replaces the point’s vertical coordinate with the centre’s. A point outside the round boundary is tested as though it lay on the horizontal axis. Keep both axes in the same coordinate space.
The BROKEN branch lives in the sequential client, not the reusable helper. wrong.asm selects it; repaired.asm restores the normal call sequence. The repaired executable matches the normal program (for Spectrum, compare loaded CODE rather than the tape filename).
Select the deliberately wrong client
HIT=$80
DX=$81
DY=$82
POINT_X=$83
POINT_Y=$84
CENTRE_X=$85
CENTRE_Y=$86
RADIUS=$87
T0=$88
T1=$89
T2=$8a
T3=$8b
T4=$8c
T5=$8d
T6=$8e
T7=$8f
T8=$90
T9=$91
STATE=$80
input_ptr=$70
cases_left=$72
ptr=$74
saved_y=$76
OUTPUT=$c000
preview=OUTPUT+1400
ready=OUTPUT+1408
CUSTOM=0
BROKEN=1
COYOTE_STEPS=4
BUFFER_STEPS=4
TEXT_CAP=8
PROPERTY_COUNT=16
STREAM_BYTES=13
CRC_POLY=7
!source "caller.inc"
!source "exercise.inc"
component_start:
!source "component.inc"
!source "data.inc"
!source "cases.inc"
hex_font:
!byte 0,0,0,0,0,0,0,0,0,56,40,40,40,56,0,0,0,16,48,16,16,56,0,0,0,56,8,56,32,56,0,0,0,56,8,56,8,56,0,0,0,40,40,56,8,8,0,0,0,56,32,56,8,56,0,0,0,56,32,56,40,56,0,0,0,56,8,8,8,8,0,0,0,56,40,56,40,56,0,0,0,56,40,56,8,56,0,0,0,16,40,56,40,40,0,0,0,48,40,48,40,48,0,0,0,56,32,32,32,56,0,0,0,48,40,40,40,48,0,0,0,56,32,48,32,56,0,0,0,56,32,48,32,32,0,0
code_end:
Cost and alternatives
The measured helper extent is 686 code bytes; public state is 8 bytes and private scratch is 10 bytes. There are no immutable helper tables. Caller data, the test cases, return-address stack, alignment and display buffers cost additional space. These are assembled sizes, not cycle measurements.
Cheap axis rejection can avoid all three products. Surviving points use three eight-stage byte products. A bounding box can precede the test; a pixel mask can follow it when the actual opaque outline matters.
This is a point/circle test, not circle/circle or pixel-mask collision. Two byte-sized squared axis distances can sum beyond 65,535; the helper avoids that overflow by subtracting DX² from the radius budget before comparing DY².
Execution evidence
Four variants run on PAL stock-ROM PRG, using Asm198x 0.0.72, Emu198x 0.44.2. Each executes 64 boundary/varied/error table cases, dirty initialisation and the separate retained-state client. An independent mathematical/container model checks every public byte after every call. This is bounded example coverage, not exhaustive correctness for every possible state.
The records retain loaded immutable code, completion, register sentinels, full packets, source/tool/firmware hashes and captures. The public gate validates these retained observations and rejects altered evidence; it does not run restricted firmware. No firmware is distributed. The eight displayed bytes are the first eight public fields, zero-padded when shorter; the packet includes every field.
Where it fits
Rectangle overlap: Use a broad-phase box to discard distant pairs before paying for a more precise shape test.