Doors

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Definition Dd.1: Let OF, OD, OM be objects : frame, door, and mover respectively. Let CA, CB be configurations of CM. Let CO, CC be the scenarios over OD and OF (open and closed) Then OD is a door through OF for OM from RA to RB with opening motion PCC if the following three conditions are met:

• 1. It is possible to open the door while keeping the door frame fixed
• 2. Both the starting and ending positions of OM are out of the way of the door.
• 3. There is a way for OM to get from RA to RB which is feasible if the door is open but not when the door is closed. There is no way for OM to get from RA to RB which is feasible when the door is closed but not when the door is open.

• 1. There exists a feasible path PCC over OD,OF from CO to CC such that OF is fixed under PCC.
• 2. For every configuration C in PCC, neither place(OF,C) nor place(OD,C) overlaps place(OM,CA) or place(OM,CB).
• 3. Let PPMO [repectively PPMC] be the spaces of configuration space paths PM for OF,OD,OM such that
  • For all configurations CP in PM, position(OD,CP) = position(OD,CO) [respectively, position(OD,CC)] and position(OF,CP) = position(OF,CO) = position(OF,CC).
  • position(CO,start(PM)) = CA; position(CO,end(PM)) = CB;
  • PM is kinematically feasible
  Then
  • 3.a. There is a connected component of PPMO that is disjoint from PPMC.
  • 3.b. There is no connected component of PPMC that is disjoint from PPMO.

Definition Dd.2: A door is unavoidable if PPMC is empty; that is, there is no way for OM to get from RA to RB if the door remains closed.

D.1: Let OD be an unavoidable door through OF for OM from RA to RB with open position CO and closed position CC. If OF, OD is in CC from time T1 to time T2 and OM is inside RA at T1 then OM is not inside RB at time T2. Equivalently, if OF is fixed at position(OF,CC) throughout [T1,T2], OM is inside RA at T1 and inside RB at T2, then OD is not in position(OD,CC) throughout [T1,T2].
Kinematics; deterministic; no gravity; no external force; prediction/interpolation.

Definition Dd.3: Object OD is a standard door with frame OF, open position CO, closed position CC, gap LG, and conduit RO (a region)

• The shape of OD is the union of a rectangular box of dimensions denoted ``body(OD)'' with smaller regions (e.g. handle; hinge).
• OD and OF are hinged with vertical axis L.
• OF is physically fixed.
• We define a global coordinate system COOR and the lengths LDW, LDT, LDH, such that, in closed position CC, the door OD occupies the box [0,LDW] x [0,LDT] x [0,LDH]; the nearest vertical edge to the axis of the hinge is the line X=0,Y=0; and the door opens with negative rotation in the $X-Y$ plane. (Note that this may require COOR to be a left-handed coordinate system.) Let LAX be the distance from axis L to the origin.
• The region place(OF,CC) contains the four rectangles that frame the closed door at distance LG away: namely, [0,LDW] x [0,LDT] x [-LG,-LG]; [-LG,-LG] x [0,LDT] x [0,LDH]; [0,LDW] x [0,LDT] x [LDH+LG,LDH+LG]; and [LDW+LG,LDW+LG] x [0,LDT] x [0,LDH]. (This is stronger than is physically necessary, but it's hard to formulate in adequate, reasonable, weaker condition.)
• Conduit RO is disjoint from place(OF,CO) and from place(OD,CO).
• Conduit RO is equal to the rectangular box [LX,LX+LW] x [-LY1, LY2] x [LZ,LZ+LH] in COOR for some LX,LZ,LW,LZ such that
  • a. LX > = -LG;
  • c. LX+LW < = LDW+LG;
  • LY1 > LAX+LDW
  • LY2 > LDT
  • LZ > = -LG;
  • LZ+LH < = LDH+LG;
  • LW > 2 LG; LZ > 2 LG
• place(OF,CC) does not overlap the swath of OD in rotating around L from CC to CO.

D.2: Let OD be a standard door with frame OF, open position CO, closed position CC, gap LG, and conduit RO (a region). Let COOR be the standard coordinate system of the door in CC, and let LDW be the width of the door. Let RA, RB be any subregions of RO such that the y-coordinate of all of RA is less than -(LAX+LDW) and the y-coordinate of all of RB is greater than LDT. Let OM be an object such that shape(OM) fits in RA and fits in RB and contains a sphere of radius LG. Then OD is a door through OF for OM from RA to RB with closed position CC and open position CO.
Deterministic, no gravity, no external force, system identification.

D.3: One-way door:
D.3.a: Can be pushed open.
D.3.b: Can be pulled open with a handle
D.3.c: Can perhaps be pulled open by wedging an object between frame and door and dragging.
D.3.d: Can perhaps be banged open

D.4: Swinging door: Can be pushed open from either side.

D.5: Boulder in front of doorframe.

D.6: Sliding door.

D.7: Cat door: Hinged on horizontal line, swings down to one-way closing. Can be pushed from one side, pulled and held from the other. Self-closing.

D.8: Swinging cat door: Hinged on horizontal line, swings through. Can be pushed from either side.

D.9: Upside down cat door. Hinged on horizontal line below opening. Self-opening. Must be held or pinned shut.

D.10: Trap door.
D.10.a: Lying loose over hole.
D.10.a.i: Can/cannot fall through hole.
D.10.b: Hinged.
D.10.c: Sliding.

D.11: Trap door underneath opening. Hinged. Must be pinned to stay closed.

D.12: Misintalled vertical door, with hinges on a non-vertical line. Either self-closing or self-opening.

D.13: Minsintalled hinged door, with non-collinear hinges. Jammed.

D.14: Cylinder with hole through it in cylindrical frame. Rotate into place.

D.15: Door with hinges along center line. Rotates about center.
D.15.a. One way. D.15.b: Two way.

D.16: Failed door, with door too large to clear frame.
D.16.a: Jammed shut frictively.
D.16.b: Closes but not cleanly.

D.17: Sequential double doors.
D.17a: Unopenable sequential double doors.

D.18: Side by side double doors.

D.19: Frictive sliding double windows.

D.20: Door inside a door: (Cat door 2).