Rings

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Sections: Geometry
Kinematics
Statics
Dynamics

Geometry

Definition Ring.D1:

Let R1 and R2 be two externally connected, thickly connected regions, whose outsides meet in two disconnected surface patches. Then R1 union R2 is a ring , also called a thick ring If the common boundary of the two contains a surface patch and a disconnected point set, then R1 union R2 is a thin ring. ("Thin ring" is a more general category than "ring".)

Declaration:
make_thin_ring(R1,R2,R : region; S1 : surface; S2 : pointset) -- Predicate
ring(R : region) --- Predicate.
thin_ring(R: region) --- Predicate.
ring_slice(RS, R : region) --- Predicate.

Formalism:
make_thin_ring(R1,R2,R : region; S1 : surface; S2 : pointset) < = >
rcc_EC(R1,R2) ^ R = R1 union R2 ^ thickly_connected(R1) ^ thickly_connected(R2) ^
R1 intersect R2 = S1 union S2 ^ connected(S1) ^ connected(S2) ^ PP(S1,outside(R1)) ^ PP(S2,outside(R2)) ^ DC(S1,S2).

ring(R) < = >
exists(R1,R2 : region; S1, S2 : surface) make_thin_ring(R1,R2,R,S1,S2).

thin_ring(R) < = >
exists(R1,R2 : region; S1 : surface; S2 : pointset) make_thin_ring(R1,R2,R,S1,S2).

ring_slice(RS,R) < = >
exists(R2) make_thin_ring(RS,R2,R).

Formalism:
thin_ring(R) < = >
exists(R1,R2 : region; S1,S2 : point_set) rcc_EC(R1,R2) ^ connected_component(S1,R1 cup R2) ^ connected_component(S2,R1 cup R2) ^ disjoint(S1,S2).

Definition Ring.D2:

RH is a ring hole through ring R if RH is a hole through R relative to some filling-in RF and all of the outlets of RH meet the outside of RF.

Formalism
ring_hole(RH,R : region) --- Predicate.

Definition: ring_hole(RH,R) < = >
exists(RF : region; RHS : set[region]) holes(RHS,R,RF) ^ RH in RHS ^
[forall((S : surface) in outlets(RH,R)) subset(S,outside(RF))]

Fact Ring.F1:

R is a thin ring if and only if there is a hole through R.

Characteristics : Topology.

Formalism:
thin_ring(R) < = > exists(RH : region) ring_hole(RH,R)

Fact Ring.F2:

If R1 and R2 combine to make a ring, then either R1 or R2 is not convex.

Characteristics : Geometry.

Formalism:
make_thin_ring(R1,R2,R,S1,S2) => ~convex(R1) V ~convex(R2).

Fact Ring.F3:

A ring is not a blob.

Characteristics : Topology

Formalism:
ring(R) => ~blob(R).

Definition Ring.D3:

Let R1 and R2 be disjoint thin rings. R1 interlocks with R2 iff every blob that contains R1 overlaps R2.

Declaration:
interlock(R1,R2 : region) --- Predicate.

Formalism:
interlock(R1,R2) < = >
thin_ring(R1) ^ thin_ring(R2) ^ DS(R1,R2) ^
forall(RS : region) [blob(RS) ^ PP(R1,RS)] => rcc_O(RS,R2).

Fact Ring.F4

If R1 interlocks R2 then R2 interlocks R1

Characteristics : Topology.

Formalism:
interlocks(R1,R2) < = > interlocks(R2,R1)

Fact Ring.F5

Let R1 be a ring and R2 be a region outside R1. Then there exists a blob handle RH such that R2 union RH interlocks with R1, but RH by itself does not interlock with R1.

Characteristics : Geometry.

Formalism:
thin_ring(R1) ^ PP(R2,outside(R1)) =>
exists(RH : region) rcc_EC(R2,RH) ^ blob(RH) ^ interlocks(R1,R2 union RH). ^ ~interlocks(R1,RH).

Fact Ring.F6

Let R1 and R2 be interlocking rings. Let R1A be a connected superset of R1 and let R2A be a connected superset of R2. If R1A and R2A are disjoint, then they are interlocking rings.

Characteristics : Geometry.

Formalism
interlocks(R1,R2) ^ PP(R1,R1A) ^ PP(R2,R2A) ^ rcc_DS(R1A,R2A) ^ connected(R1A) ^ connected(R2A) =>
interlocks(R1A,R2A).

Fact Ring.F7

If R1 is a ring, R2 is a connected subset of R1, and R2 is not a ring, then R1-R2 contains a slice through R.

Characteristics : Topology.

Formalism:
thin_ring(R1) ^ PP(R2,R1) ^ subset(R2,R1) ^ ~thin_ring(R2) =>
exists(R3 : region) ring_slice(R3,R1).

Fact Ring.F8

If R is a ring and R1 is a connected subset of R and not a ring, then R-R1 contains a slice of R.

Characteristics : Topology.

Formalism:
ring(R) ^ PP(R1,R) ^ ~ring(R1) => exists(R2) P(R2,R--R1) ^ ring_slice(R2,R).

Definition Ring.D4

A ring R is simple if it is composed of two slices that are not, themselves, rings.

Declaration: simple_ring(R : region)

Formalism:
simple_ring(R) < = >
exists(R1,R2,S1,S2) ~ring(R1) ^ ~ring(R2) ^ make_ring(R1,R2,R,S1,S2)

Fact Ring.F9

If R is a simple ring, and RS is a slice of R and not a ring, then R-RS is not a ring.

Characteristics : Topology.

Formalism:
simple_ring(R) ^ slice(R1,R) ^ ~ring(R1) => ~ring(R--R1).

Fact Ring.F10

If R is a simple ring and RS1 and RS2 are disconnected slices of R then R--RS1--RS2 is not connected.

Characteristics : Topology.

Formalism:
thin_ring(R1) ^ slice(RS1,R) ^ slice(RS2,R) ^ DC(RS1,RS2) => ~connected(R--R1--R2).

Fact Ring.F11

A ring can be divided into two rings.

Characteristics : Topology.

Formalism:
ring(R) => exists(R1,R2) ring(R1) ^ ring(R2) ^ rcc_EC(R1,R2) ^ R = R1 intersect R.

Fact Ring.F12

If R1 and R2 are interlocking solid rings, then there is no plane that separates R1 and R2.

Characteristics: Geometry

Formalism:
interlock(R1,R2) => ~[exists(SP) plane(SP) ^ plane_separates(SP,R1,R2)].

Kinematics

Fact Ring.F13

If O1 is a ring, O2 is a [thin] ring, and O1 and O2 are interlocking, then they have always been interlocking and will always be interlocking.

Characteristics: Kinematics

Formalism:
ring(O1 : object) ^ thin_ring(O2 : object) ^ holds(S1,interlock(O1,O2)) ^ S1 in H ^ S2 in H ^ kinematic(H) =>
holds(S2,interlock(O1,O2))

B.10: Let O1 and O2 be interlocking solid rings and let P be a point. Then not every rotation of O1 around P is feasible. (Note that this is not true for an object inside a box.)

B.11: Let R1 and R2 be interlocking solid rings. Suppose that at time T1, place(O1) is a subset of R1 and place(O2)=R2, and at time T2, O1 and O2 are separated by a plane. Then there is a slice RS through R1 such that place(O1) subset R1-RS and O2 goes through RS.

If two rings R1 and R2 are interlocked, and R1 undergoes a large enough rotation not around the Z-axis, then R2 must ultimately undergo a similar rotation.

If a ring is on a pole, and the pole is blocked at both ends with a cap that cannot go through the ring, then the ring will always be on the pole.

If a ring is on a pole and the pole is blocked at one end with a cap that cannot go through the ring, then the ring can only come off the pole if and only if the pole comes out through it at the open end.

If a broken ring is on a pole that is capped at both ends with a cap that cannot go through the ring, then the ring will come off the pole if and only if the pole slides through the gap in the ring.

If a ring is on a pole, then there is a maximal difference between the angle of the axes of the ring and the pole. Therefore, if the ring is rotated far enough around the X or Y axes, and the ring remains on the pole, then the pole must undergo a similar rotation, and vice versa.

MULTIPLE RINGS / POLES THROUGH RING.

Rings in order on pole / in cyclic order on ring

Rings that can pass through one another on pole/ring

CHAIN OF RINGS.

Non-simple chains: Cyclic. Double. Altenate double/single etc.

Statics

Definition Bd.4: Let C1 be a strictly convex planar curve and let A2 be a strictly convex planar region. Let P be a point inside A2. Let R be the region generated by sweeping A2 normal to C1, holding P on A2. Suppose that the inner boundary of the cross section of R through C1 is convex. Then R is called an ordinary ring.

An ordinary ring OR that has only one point of contact P with another object OS is stably supported if and only if the following conditions are met:

1. OR is in vertical position; that is, C1 is in a vertical plane. Let us align the coordinate system so that C1 is in the XZ plane, and the point of contact is on the Z axis.
2. The normal to OR at P is vertically down; the normal to OS at P is vertically up. (See ??)
3.
- 3.A: In the cross-section of OS in the YZ plane, P is a non-strict local minimum in Z.
- 3.B: In the cross-section of OS in the XZ plane, P is a strict local maximum in Z.
- 3.C: In the cross-section of OR in the YZ plane, P is a strict local minimum in Z.
- 3.D: In the cross-section of OR in the XZ plane, P is a strict local maximum in Z
- 3.E: In the cross-section of OR in the XZ plane, P is a non-strict local maximum of curvature.

Under these circumstances, the following motions of OR are easy:

OR may swing in the XZ plane quite freely.
OR may swing in the YZ plane quite freely.
OR may spin around the Z axis through a quite small angle.
OR may slip in a rotation around an axis parallel to the Y-axis.
OR may slip along the Y axis.

However, any such motion, if small enough, will return to a stable support at a single point. If (3.E) above is strict, then the point in OR will be the same as at the start. If (3.A) above is strict, the the point in OS will be the same as at the start.

There are six main ways in which an ordinary ring can be stably supported.

1. It is kinematically fixed. See ??.
2. It is supported from below by points on one side of the object's XZ plane. The contact points points must encompass the center in the XY projection.
3. It is supported on the top inside surface. This approximates the condition in ?? --- the closer all the support points are together, the better the approximation. The conclusions in ?? therefore approximately apply.
4. It is kinematically fixed in the object's X-Y plane, except possibly for rotation about the Z axis. Small motions along the Z axis maintain the kinematic fixture in the X-Y plane. There are substantial friction against motion along the Z-axis.
5. It fits into a slot, and it is supported on the lower outside surface. The ring has to slide a distance into the slot that is substantial as compared to the diameter of C2 (NOT QUITE THE RIGHT MEASURE) and there must be contact points on both sides of C1 and on both sides of the Z'-axis. The angle into the slot can be anything from vertically down to horizontal.
6. It fits into a slot along the object's Z' axis. Motion in the X'Y' plane maintains contact with both sides of the slot, and engenders substantial frictive resistance.

There are probably other possibilities as well, but these would be more specialized.

Let OR1 and OR2 be ordinary rings of similar size. OR2 can stably support OR1 in either of two ways:
1. Both rings are horizontal, and OR1 lies directly above OR2. This can be strongly or weakly stable.
2. OR1 and OR2 are interlocked, and the bottom inside surface of OR2 is in contact with the upper inside surface of OR1. OR1 may be able to swing in the plane of C1, to swing in the plane of C2, or to slide along C1, but this configuration is ultimately stable, in the sense that it will always ultimately return back to some such position.

An ordinary ring in vertical position cannot be supported from the inside bottom or the outside top or both. It can be supported stably from the outside bottom only if it is slotted in; that is, it has contacts on either side of the plane of C1 and on either side (in the perpendicular vertical plane) of the lowest point of C1. Since the ring OR must fit despite these contacts, there must be a slot for the ring in its support(s).

Rings on vertical pole with base.

Rings on ring with gap on top.

Rings piled neatly on stick

Dynamics

If a ring is flat on the ground and a vertical pole is loose inside the ring, then lifting the pole vertically will not move the ring.

If a vertical pole goes through a ring, and the pole has a cap at the bottom that the ring cannot go through, then if the pole is moved smoothly and kept vertical, the ring will remain on the pole. If the ring is initially partially supported by the cap at the bottom, then it will remain supported by the cap at the bottom. If the ring can fit off the top of the pole and the pole is moved violently and/or moved far from vertical, then the ring may come off the top of the pole; this becomes more likely as the pole becomes shorter, the motion becomes more violent, and the departure from the vertical further. The same applies to a broken ring if the gap is too small for the pole to slide through. If the pole is turned upside down and held there, the ring will come off.

If a ring is on a horizontal pole and the hole in the ring is large enough to slide off either end, and the pole is moved, then it is likely that the ring will move with the pole for a while and then slide off. For the ring to stay on, the pole must be moved very carefully. If the pole curves up at both ends, then less care must be taken; the larger the curve, both in terms of vertical distance and in terms of vertical angle, the less the care required. If the pole curves down at both ends, then it may be difficult or impossible to keep the ring on the pole. If the pole is capped at one end or curves up at one end, then the pole should be tilted so that that end is lower and the other end is higher.

If a pole is vertical and has a cap on the top and there is a ring around it, and the ring is moved smoothly and upright, the pole will stay inside the ring. If the ring is moved violently or tipped sharply from the vertical, then the pole may come out; the likelihood goes up as the pole becomes shorter, the motion becomes more violent, and the departure from the vertical becomse larger.