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:
ld a,0
ld (HIT),a
ld (DX),a
ld (DY),a
ld (POINT_X),a
ld (POINT_Y),a
ld (CENTRE_X),a
ld (CENTRE_Y),a
ld (RADIUS),a
ld (T0),a
ld (T1),a
ld (T2),a
ld (T3),a
ld (T4),a
ld (T5),a
ld (T6),a
ld (T7),a
ld (T8),a
ld (T9),a
ret
point_circle_step:
; HIT = 0
ld a,0
ld (HIT),a
; if POINT_X >= CENTRE_X:
ld a,(POINT_X)
ld b,a
ld a,(CENTRE_X)
ld c,a
ld a,b
cp c
jp nc,point_circle_L1
jp point_circle_L2
point_circle_L1:
; DX = POINT_X - CENTRE_X
ld a,(POINT_X)
ld b,a
ld a,(CENTRE_X)
ld c,a
ld a,b
sub c
ld (DX),a
jp point_circle_L3
point_circle_L2:
; DX = CENTRE_X - POINT_X
ld a,(CENTRE_X)
ld b,a
ld a,(POINT_X)
ld c,a
ld a,b
sub c
ld (DX),a
point_circle_L3:
; if POINT_Y >= CENTRE_Y:
ld a,(POINT_Y)
ld b,a
ld a,(CENTRE_Y)
ld c,a
ld a,b
cp c
jp nc,point_circle_L4
jp point_circle_L5
point_circle_L4:
; DY = POINT_Y - CENTRE_Y
ld a,(POINT_Y)
ld b,a
ld a,(CENTRE_Y)
ld c,a
ld a,b
sub c
ld (DY),a
jp point_circle_L6
point_circle_L5:
; DY = CENTRE_Y - POINT_Y
ld a,(CENTRE_Y)
ld b,a
ld a,(POINT_Y)
ld c,a
ld a,b
sub c
ld (DY),a
point_circle_L6:
; if DX > RADIUS:
ld a,(DX)
ld b,a
ld a,(RADIUS)
ld c,a
ld a,b
cp c
jp z,point_circle_L8
jp nc,point_circle_L7
jp point_circle_L8
point_circle_L7:
; return
ret
jp point_circle_L9
point_circle_L8:
point_circle_L9:
; if DY > RADIUS:
ld a,(DY)
ld b,a
ld a,(RADIUS)
ld c,a
ld a,b
cp c
jp z,point_circle_L11
jp nc,point_circle_L10
jp point_circle_L11
point_circle_L10:
; return
ret
jp point_circle_L12
point_circle_L11:
point_circle_L12:
; T4 = RADIUS
ld a,(RADIUS)
ld (T4),a
; T5 = 0
ld a,0
ld (T5),a
; T1 = RADIUS
ld a,(RADIUS)
ld (T1),a
; T6 = 0
ld a,0
ld (T6),a
; T7 = 0
ld a,0
ld (T7),a
; T0 = 8
ld a,8
ld (T0),a
; while T0 != 0:
point_circle_L13:
ld a,(T0)
cp 0
jp nz,point_circle_L14
jp point_circle_L15
point_circle_L14:
; T9 = T1 & 1
ld a,(T1)
and 1
ld (T9),a
; if T9 != 0:
ld a,(T9)
cp 0
jp nz,point_circle_L16
jp point_circle_L17
point_circle_L16:
; T8 = T6
ld a,(T6)
ld (T8),a
; T6 = T6 + T4
ld a,(T6)
ld b,a
ld a,(T4)
ld c,a
ld a,b
add a,c
ld (T6),a
; T9 = 0
ld a,0
ld (T9),a
; if T6 < T8:
ld a,(T6)
ld b,a
ld a,(T8)
ld c,a
ld a,b
cp c
jp c,point_circle_L19
jp point_circle_L20
point_circle_L19:
; T9 = 1
ld a,1
ld (T9),a
jp point_circle_L21
point_circle_L20:
point_circle_L21:
; T7 = T7 + T5
ld a,(T7)
ld b,a
ld a,(T5)
ld c,a
ld a,b
add a,c
ld (T7),a
; T7 = T7 + T9
ld a,(T7)
ld b,a
ld a,(T9)
ld c,a
ld a,b
add a,c
ld (T7),a
jp point_circle_L18
point_circle_L17:
point_circle_L18:
; T9 = T4 >> 7
ld a,(T4)
srl a
srl a
srl a
srl a
srl a
srl a
srl a
ld (T9),a
; T4 = T4 << 1
ld a,(T4)
sla a
ld (T4),a
; T5 = T5 << 1
ld a,(T5)
sla a
ld (T5),a
; T5 = T5 | T9
ld a,(T5)
ld b,a
ld a,(T9)
ld c,a
ld a,b
or c
ld (T5),a
; T1 = T1 >> 1
ld a,(T1)
srl a
ld (T1),a
; T0 = T0 - 1
ld a,(T0)
sub 1
ld (T0),a
jp point_circle_L13
point_circle_L15:
; T2 = T6
ld a,(T6)
ld (T2),a
; T3 = T7
ld a,(T7)
ld (T3),a
; T4 = DX
ld a,(DX)
ld (T4),a
; T5 = 0
ld a,0
ld (T5),a
; T1 = DX
ld a,(DX)
ld (T1),a
; T6 = 0
ld a,0
ld (T6),a
; T7 = 0
ld a,0
ld (T7),a
; T0 = 8
ld a,8
ld (T0),a
; while T0 != 0:
point_circle_L22:
ld a,(T0)
cp 0
jp nz,point_circle_L23
jp point_circle_L24
point_circle_L23:
; T9 = T1 & 1
ld a,(T1)
and 1
ld (T9),a
; if T9 != 0:
ld a,(T9)
cp 0
jp nz,point_circle_L25
jp point_circle_L26
point_circle_L25:
; T8 = T6
ld a,(T6)
ld (T8),a
; T6 = T6 + T4
ld a,(T6)
ld b,a
ld a,(T4)
ld c,a
ld a,b
add a,c
ld (T6),a
; T9 = 0
ld a,0
ld (T9),a
; if T6 < T8:
ld a,(T6)
ld b,a
ld a,(T8)
ld c,a
ld a,b
cp c
jp c,point_circle_L28
jp point_circle_L29
point_circle_L28:
; T9 = 1
ld a,1
ld (T9),a
jp point_circle_L30
point_circle_L29:
point_circle_L30:
; T7 = T7 + T5
ld a,(T7)
ld b,a
ld a,(T5)
ld c,a
ld a,b
add a,c
ld (T7),a
; T7 = T7 + T9
ld a,(T7)
ld b,a
ld a,(T9)
ld c,a
ld a,b
add a,c
ld (T7),a
jp point_circle_L27
point_circle_L26:
point_circle_L27:
; T9 = T4 >> 7
ld a,(T4)
srl a
srl a
srl a
srl a
srl a
srl a
srl a
ld (T9),a
; T4 = T4 << 1
ld a,(T4)
sla a
ld (T4),a
; T5 = T5 << 1
ld a,(T5)
sla a
ld (T5),a
; T5 = T5 | T9
ld a,(T5)
ld b,a
ld a,(T9)
ld c,a
ld a,b
or c
ld (T5),a
; T1 = T1 >> 1
ld a,(T1)
srl a
ld (T1),a
; T0 = T0 - 1
ld a,(T0)
sub 1
ld (T0),a
jp point_circle_L22
point_circle_L24:
; T8 = 0
ld a,0
ld (T8),a
; if T2 < T6:
ld a,(T2)
ld b,a
ld a,(T6)
ld c,a
ld a,b
cp c
jp c,point_circle_L31
jp point_circle_L32
point_circle_L31:
; T8 = 1
ld a,1
ld (T8),a
jp point_circle_L33
point_circle_L32:
point_circle_L33:
; T2 = T2 - T6
ld a,(T2)
ld b,a
ld a,(T6)
ld c,a
ld a,b
sub c
ld (T2),a
; T3 = T3 - T7
ld a,(T3)
ld b,a
ld a,(T7)
ld c,a
ld a,b
sub c
ld (T3),a
; T3 = T3 - T8
ld a,(T3)
ld b,a
ld a,(T8)
ld c,a
ld a,b
sub c
ld (T3),a
; T4 = DY
ld a,(DY)
ld (T4),a
; T5 = 0
ld a,0
ld (T5),a
; T1 = DY
ld a,(DY)
ld (T1),a
; T6 = 0
ld a,0
ld (T6),a
; T7 = 0
ld a,0
ld (T7),a
; T0 = 8
ld a,8
ld (T0),a
; while T0 != 0:
point_circle_L34:
ld a,(T0)
cp 0
jp nz,point_circle_L35
jp point_circle_L36
point_circle_L35:
; T9 = T1 & 1
ld a,(T1)
and 1
ld (T9),a
; if T9 != 0:
ld a,(T9)
cp 0
jp nz,point_circle_L37
jp point_circle_L38
point_circle_L37:
; T8 = T6
ld a,(T6)
ld (T8),a
; T6 = T6 + T4
ld a,(T6)
ld b,a
ld a,(T4)
ld c,a
ld a,b
add a,c
ld (T6),a
; T9 = 0
ld a,0
ld (T9),a
; if T6 < T8:
ld a,(T6)
ld b,a
ld a,(T8)
ld c,a
ld a,b
cp c
jp c,point_circle_L40
jp point_circle_L41
point_circle_L40:
; T9 = 1
ld a,1
ld (T9),a
jp point_circle_L42
point_circle_L41:
point_circle_L42:
; T7 = T7 + T5
ld a,(T7)
ld b,a
ld a,(T5)
ld c,a
ld a,b
add a,c
ld (T7),a
; T7 = T7 + T9
ld a,(T7)
ld b,a
ld a,(T9)
ld c,a
ld a,b
add a,c
ld (T7),a
jp point_circle_L39
point_circle_L38:
point_circle_L39:
; T9 = T4 >> 7
ld a,(T4)
srl a
srl a
srl a
srl a
srl a
srl a
srl a
ld (T9),a
; T4 = T4 << 1
ld a,(T4)
sla a
ld (T4),a
; T5 = T5 << 1
ld a,(T5)
sla a
ld (T5),a
; T5 = T5 | T9
ld a,(T5)
ld b,a
ld a,(T9)
ld c,a
ld a,b
or c
ld (T5),a
; T1 = T1 >> 1
ld a,(T1)
srl a
ld (T1),a
; T0 = T0 - 1
ld a,(T0)
sub 1
ld (T0),a
jp point_circle_L34
point_circle_L36:
; if T7 > T3:
ld a,(T7)
ld b,a
ld a,(T3)
ld c,a
ld a,b
cp c
jp z,point_circle_L44
jp nc,point_circle_L43
jp point_circle_L44
point_circle_L43:
; return
ret
jp point_circle_L45
point_circle_L44:
point_circle_L45:
; if T7 == T3:
ld a,(T7)
ld b,a
ld a,(T3)
ld c,a
ld a,b
cp c
jp z,point_circle_L46
jp point_circle_L47
point_circle_L46:
; if T6 > T2:
ld a,(T6)
ld b,a
ld a,(T2)
ld c,a
ld a,b
cp c
jp z,point_circle_L50
jp nc,point_circle_L49
jp point_circle_L50
point_circle_L49:
; return
ret
jp point_circle_L51
point_circle_L50:
point_circle_L51:
jp point_circle_L48
point_circle_L47:
point_circle_L48:
; HIT = 1
ld a,1
ld (HIT),a
ret
component_end:Machine contract
AF/BC/DE/HL may change; IX/IY are preserved. A direct CALL 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 48K diagnostic boots through ROM tape loading and USR, disables interrupts and owns the screen. The helper performs no port or screen writes.
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
org $8000
CUSTOM equ 0
BROKEN equ 0
COYOTE_STEPS equ 4
BUFFER_STEPS equ 4
TEXT_CAP equ 8
PROPERTY_COUNT equ 16
STREAM_BYTES equ 13
CRC_POLY equ 7
include "caller.inc"
include "exercise.inc"
component_start:
include "component.inc"
include "data.inc"
include "cases.inc"
include "../../physics/rectangle-overlap/display.inc"
code_end:
STATE:
HIT: defb 0
DX: defb 0
DY: defb 0
POINT_X: defb 0
POINT_Y: defb 0
CENTRE_X: defb 0
CENTRE_Y: defb 0
RADIUS: defb 0
T0: defb 0
T1: defb 0
T2: defb 0
T3: defb 0
T4: defb 0
T5: defb 0
T6: defb 0
T7: defb 0
T8: defb 0
T9: defb 0
input_ptr: defw 0
output_ptr: defw 0
cases_left: defb 0
ready: defb 0
preview: defs 8,0
OUTPUT: defs 1300,0
end startA client retaining state between calls
preview_rules:
if CUSTOM
; POINT_X = 3
ld a,3
ld (POINT_X),a
; POINT_Y = 4
ld a,4
ld (POINT_Y),a
; RADIUS = 5
ld a,5
ld (RADIUS),a
if BROKEN
; POINT_Y = CENTRE_Y
ld a,(CENTRE_Y)
ld (POINT_Y),a
endif
call checked_step
call emit_state
ret
else
; POINT_X = 4
ld a,4
ld (POINT_X),a
; POINT_Y = 4
ld a,4
ld (POINT_Y),a
; RADIUS = 5
ld a,5
ld (RADIUS),a
if BROKEN
; POINT_Y = CENTRE_Y
ld a,(CENTRE_Y)
ld (POINT_Y),a
endif
call checked_step
call emit_state
endif
retCustomise 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
org $8000
CUSTOM equ 1
BROKEN equ 0
COYOTE_STEPS equ 2
BUFFER_STEPS equ 2
TEXT_CAP equ 4
PROPERTY_COUNT equ 16
STREAM_BYTES equ 13
CRC_POLY equ 49
include "caller.inc"
include "exercise.inc"
component_start:
include "component.inc"
include "data.inc"
include "cases.inc"
include "../../physics/rectangle-overlap/display.inc"
code_end:
STATE:
HIT: defb 0
DX: defb 0
DY: defb 0
POINT_X: defb 0
POINT_Y: defb 0
CENTRE_X: defb 0
CENTRE_Y: defb 0
RADIUS: defb 0
T0: defb 0
T1: defb 0
T2: defb 0
T3: defb 0
T4: defb 0
T5: defb 0
T6: defb 0
T7: defb 0
T8: defb 0
T9: defb 0
input_ptr: defw 0
output_ptr: defw 0
cases_left: defb 0
ready: defb 0
preview: defs 8,0
OUTPUT: defs 1300,0
end startThe 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
org $8000
CUSTOM equ 0
BROKEN equ 1
COYOTE_STEPS equ 4
BUFFER_STEPS equ 4
TEXT_CAP equ 8
PROPERTY_COUNT equ 16
STREAM_BYTES equ 13
CRC_POLY equ 7
include "caller.inc"
include "exercise.inc"
component_start:
include "component.inc"
include "data.inc"
include "cases.inc"
include "../../physics/rectangle-overlap/display.inc"
code_end:
STATE:
HIT: defb 0
DX: defb 0
DY: defb 0
POINT_X: defb 0
POINT_Y: defb 0
CENTRE_X: defb 0
CENTRE_Y: defb 0
RADIUS: defb 0
T0: defb 0
T1: defb 0
T2: defb 0
T3: defb 0
T4: defb 0
T5: defb 0
T6: defb 0
T7: defb 0
T8: defb 0
T9: defb 0
input_ptr: defw 0
output_ptr: defw 0
cases_left: defb 0
ready: defb 0
preview: defs 8,0
OUTPUT: defs 1300,0
end start
Cost and alternatives
The measured helper extent is 970 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 48K PAL ROM tape load and USR, 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.