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| === Linkability formulae === | | {{Obsolete|Max distance has been raised from 54m to 64m as of 2023/06/27. Update was announced here. [https://community.secondlife.com/blogs/entry/13626-coming-soon-server-release-202306/ Tech Updates Server Release 2023.06]}} |
| | {{Multi-lang}} |
| | = New Linkability Rule = |
| | '''The linkability rule for objects changed as of Dec 2010.''' The main motivation was to make the linkability check more efficient but a consequence is that they are also much simpler. The size-dependency of the prims has been discarded. The only thing that matters is the bounding sphere of the collection of prim centers. The linkability rule can be written as: |
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| Consider two primitives, A and B. Whether they can be linked or not is determined
| | LINKABLE = {{HoverText|D|diameter of the smallest bounding sphere of the collection of prim centers}} < 54 AND {{HoverText|N|number of prims in the collection}} ≤ 256 |
| by measuring the ''span'' from the edge of one object's ''bounding sphere'' to
| | where D = diameter of the smallest bounding sphere of the collection of prim centers |
| the far opposite edge of the other's bounding sphere and comparing that value to the
| | and N = number of prims in the collection |
| ''maximum linkability span'' which is a function of the radii of the two bounding spheres:
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| (1) max_link_span = minimum( 3 * (radius_A + radius_B) + SMALLEST_MAX, LARGEST_MAX )
| | [[User:Andrew Linden|Andrew Linden]] 21:47, 02 December 2010 (UTC) |
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| where:
| | If you are interested in how the old formula worked, see [[Linkability Rules/Havok 4]]. |
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| (3) SMALLEST_MAX = 1.0 meters
| | === Notes === |
| (4) LARGEST_MAX = 54.0 meters
| | * [2011-08-27-16:52] Andrew Linden: 54 was the minimum sphere that still contained all of the legacy linkable content. |
| (5) radius_X = radius of the primitive X's ''bounding sphere'' (Figure A)
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| (6) minimum(C, D) = C if less than D, otherwise D
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| If the measured span of the two bounding spheres is less than or equal to max_link_span then the corresponding primitives can be linked. Put in mathematical
| | [[Category:Havok]] |
| terms:
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| (7) A_can_link_to_B = ( length(center_A - center_B) + radius_A + radius_B <= max_link_distance )
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| {|align="right"
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| |[[Image:Primitive_diameter.png]]
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| |}
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| The '''bounding sphere''' is the smallest sphere that totally encloses the
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| primitive's ''local bounding box''.
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| The '''local bounding box''' is centered at the primitive's ''geometric center'' and
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| whose local-frame sides are equal to the primitive's ''scale''.
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| The '''geometric center''' of the primitive is its local symmetric center prior to
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| any ''cut'', ''shear'', ''twist'', ''taper'', or ''hollow'' operations.
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| '''Note''' that a primitive's bounding sphere is not necessarily the tightest sphere
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| possible for its shape, unless it is a simple box. The bounding sphere
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| depends only on the primitive's position and scale, so any
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| severly cut and hollowed primitive will be significantly smaller than its bounding
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| sphere, and not necessarily near the center. Also, a primitive with twist and/or
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| shear may have corners that extend outside of its bounding sphere. Since the
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| linkability rules depend only on the bounding sphere, which is ultimately dependent
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| only on the primitive's position and scale, the linkability of two prims is independent
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| of changes to form and rotation.
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| TODO -- we need a new figure_A that shows ''radius'' instead of ''diameter''.
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| Also need new figures _B and _C for two linkable prims and two linkable multi-prim objects.
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| === Linkability algorithm ===
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| The rules governing the '''linkability of multi-prim objects''' is very similar to
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| the two-primitive case. The same formulae (1) and (7) apply, but the
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| bounding spheres of multi-prim objects are the smallest spheres that completely
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| contain all of the bounding spheres of the corresponding primitives.
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| When linking '''three or more objects''' the algorithm iterates over the candidate
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| objects until all linkable pieces have been found. First the root object
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| is tested against each candidate object and the larger bounding sphere is
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| recomputed after a successful link. Then any unlinked pieces are tested between
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| themselves and merged into larger collections according to the formulae. The
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| root object is then re-tested against the modified candidates and the process
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| continues until all objects are linked, or no new links have been found.
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| === Failure modes ===
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| If an '''unlinkable set''' is tested by the linkability algorithm then the final subset
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| of linkable parts is determined by the order in which the candidates were submitted.
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| The trivial proof for this is to consider a root primitive in the middle of an infinite
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| grid of other primitives. It can't link to everything, but it were first tested against
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| all primitives west of it the final linkable subset of that first operation might not
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| link to any primitives to the east because of the LARGEST_MAX requirement (4). If the
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| primitives to the east were tested first then the final result would be different.
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| If a '''linkable set''' is tested by the linkability algorithm then the final subset
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| of linkable parts is NOT affected by the order in which the candidates were submitted.
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| That is, if just the linkable subsets of the failure modes above are tested for
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| all permutations of sequence they will always link. The proof of this is left
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| as an exercise for the reader.
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| === Examples ===
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| ==== 2 very small prims ====
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| radius_A = ~0.01
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| radius_B = ~0.01
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| SMALLEST_MAX = 1.0 meters
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| LARGEST_MAX = 54.0 meters
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| max_link_distance = 1.06m
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| ==== one large prim and one small prim ====
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| radius_A = 5m
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| radius_B = ~0.01
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| SMALLEST_MAX = 1.0 meters
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| LARGEST_MAX = 54.0 meters
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| 3 * (radius_A + radius_B) + SMALLEST_MAX = 15.03 meters
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| 15.03m is smaller than 54.0m
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| Thus the max_link_distance = 15.03m
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| ==== 2 very large prims ====
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| The diameter of a bounding sphere is the square root of x^2 + y^2 + z^2
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| thus, the diameter of a 10m x 10m x 10m prim is the square root of (100+100+100) = ~17.3m
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| and the diameter of a 10m x 1m x 1m prim is square root (100+1+1) = ~10.1m
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| (The type of prim doesn't matter for this calculation. We only care about the dimentions.)
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| Let's take the case of two 10m x 10m x 10m prims.
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| radius_A = 8.66m
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| radius_B = 8.66m
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| SMALLEST_MAX = 1.0 meters
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| LARGEST_MAX = 54.0 meters
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| 3 * (radius_A + radius_B) + SMALLEST_MAX = 52.96m
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| 52.96m is smaller than 54m
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| Thus max_link_distance = 52.96m
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