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| Determining whether or not a set of prims/objects is linkable is somewhat complex. Generally, larger prims and objects may have greater distances between them to become linked, with a minimum of 2 meters and a maximum of 54. The distance depends on the position and the X/Y/Z size of each prim, but not their shapes, rotations, size changes due to cuts, hollows, etc.
| | {{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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| ==Quick overview== | | 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 |
| Kelly's Quick Overview:
| | where D = diameter of the smallest bounding sphere of the collection of prim centers |
| * Two pieces can link if their bounding boxes would completely fit within a "Linkability Sphere".
| | and N = number of prims in the collection |
| * The size of the Linkability Sphere is between 2m and 54m diameter, depending on the scale of the two pieces being linked.
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| ** Specifically the size is 3 times the sum of both object's bounding box radius plus 2 meters, up to a maximum of 54m.
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| * If linking more than two pieces, treat any 2 pieces that can link by the above rule as 1 piece whose scale encompasses both pieces.
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| * Repeat until everything is linked.
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| At the small extreme: two tiny prims can be linked if they are within about 2 meters from each other.
| | [[User:Andrew Linden|Andrew Linden]] 21:47, 02 December 2010 (UTC) |
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| At the large extreme: the largest linked object that can be created must fit within a sphere with a diameter of 54 meters.
| | If you are interested in how the old formula worked, see [[Linkability Rules/Havok 4]]. |
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| [[Image:Primitive_diameter.png|thumb|right|Figure A: The bounding spheres of two example prims, one of which has been cut. The two spheres are identical in size.]] | | === Notes === |
| * Whether two prims can be linked depends only on each prim's X/Y/Z scale and position. Other properties (such as rotation, hollow, cut, sculpt, etc.) are ignored. This means that linking calculations are done as if every prim were a rectangular box that is stretched but otherwise undistorted (i.e. with the X, Y, and Z sizes the same as the original prim, but everything else 'normal').
| | * [2011-08-27-16:52] Andrew Linden: 54 was the minimum sphere that still contained all of the legacy linkable content. |
| ** Editing and selecting any object and selecting the 'stretch' tool will display this "''local bounding box''" for that object.
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| ** Exception for megaprims: If any side is more than 10 meters, treat that side as if it were 10 meters.
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| A way to visualize the procedure:
| | [[Category:Havok]] |
| # Imagine transforming every prim in every object into the aforementioned boxes.
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| # For each existing (linked-together) object, imagine the smallest sphere that totally encloses the bounding boxes of the prims which belong to it. (For individual prims, just enclose that one box.)
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| # Triple the size of each sphere, then add another meter around each sphere. (Don't move the centers of the spheres.)
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| # You can link objects if their spheres touch or intersect. Link everything that can be linked at this point.
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| #* Exception: If any new object would need a sphere of bigger than 54 meters to enclose all its boxes, you can't make that object.
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| # Repeat this procedure with the newly linked objects until you can't link any more.
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| Read the section below for details of the linkability formula, including examples and what actually gets linked if you can't link everything.
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| ==The details==
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| Consider two primitives, A and B. Whether they can be linked or not is determined
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| by measuring the ''span'' from the edge of one object's ''bounding sphere'' to
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| the far opposite edge of the other's bounding sphere and comparing that value to the
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| ''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) + LINK_BONUS, LARGEST_MAX )
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| where:
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| (3) LINK_BONUS = 2.0 meters
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| (4) LARGEST_MAX = 54.0 meters
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| (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
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| terms:
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| (7) A_can_link_to_B = ( length(center_A - center_B) + radius_A + radius_B <= max_link_span )
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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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| has sides that are equal to the primitive's ''scale''. One exception to this
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| rule is that megaprim scale components greater than 10 meters are clamped to 10.
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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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| LINK_BONUS = 2.0 meters
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| LARGEST_MAX = 54.0 meters
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| 3 * (radius_A + radius_B) + LINK_BONUS = 2.06 meters
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| 2.06 is less than 54
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| Thus max_link_span = 2.06m
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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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| LINK_BONUS = 2.0 meters
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| LARGEST_MAX = 54.0 meters
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| 3 * (radius_A + radius_B) + LINK_BONUS = 16.03 meters
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| 16.03 is smaller than 54.0
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| Thus max_link_span = 16.03m
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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 dimensions.)
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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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| LINK_BONUS = 2.0 meters
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| LARGEST_MAX = 54.0 meters
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| 3 * (radius_A + radius_B) + LINK_BONUS = 53.96m
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| 53.96 is smaller than 54
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| Thus max_link_span = 53.96m
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| [[Category:Havok4]] | |
