Difference between revisions of "Linkability Rules"

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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.
{{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:


==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
[[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.]]
  where D = diameter of the smallest bounding sphere of the collection of prim centers
* Whether two prims can be linked depends only on their scale and position.  Other properties (such as rotation, hollow, cut, etc.) are ignored.  This means that linking calculations are done as the prim were an undistorted rectangular box (with the same X, Y, and Z sizes, but everything else 'normal').  Editing any object and selecting the 'stretch' tool will display this box for that object.
  and N = number of prims in the collection
** Exception for megaprims: If any side is more than 10 meters, treat that side as if it were 10 meters.
* At the small extreme: two tiny prims can be linked if they are within about 2 meters from each other.
* At the large extreme: the largest linked object that can be created must fit within a sphere with a diameter of 54 meters.


A way to visualize the formula:
[[User:Andrew Linden|Andrew Linden]] 21:47, 02 December 2010 (UTC)
# Imagine transforming every prim in every object into the aforementioned boxes.
# For each existing (linked-together) object, imagine the smallest sphere that totally encloses the boxes that belong to it.  (For individual prims, just enclose that one box.)
# Triple the size of each sphere, then add another meter around it.  (Don't move the centers of the spheres.)
# You can link objects if their spheres intersect.  Link everything that can be linked at this point.
#* 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.
# Repeat this procedure with the newly linked objects until you can't link any more.


Kelly's Attempt at a Quick Overview:
If you are interested in how the old formula worked, see [[Linkability Rules/Havok 4]].
* Two pieces can link if their bounding boxes (seen by choosing the stretch tool) would completely fit within a "Linkability Sphere".
* The size of the Linkability Sphere is between 2m and 54m diameter, depending on the scale of the two pieces being linked.
** Specifically the size is 3 times the sum of both object's bounding box radius plus 2 meters, up to a maximum of 54m.
* 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. 
* Repeat until everything is linked.


Read the section below for details of the linkability formula, including examples and what actually gets linked if you can't link everything.
=== Notes ===
* [2011-08-27-16:52]  Andrew Linden: 54 was the minimum sphere that still contained all of the legacy linkable content.


==The details==
[[Category:Havok]]
Consider two primitives, A and B.  Whether they can be linked or not is determined
by measuring the ''span'' from the edge of one object's ''bounding sphere'' to
the far opposite edge of the other's bounding sphere and comparing that value to the
''maximum linkability span'' which is a function of the radii of the two bounding spheres:
 
    (1)    max_link_span = minimum( 3 * (radius_A + radius_B) + LINK_BONUS, LARGEST_MAX )
 
where:
 
    (3)    LINK_BONUS = 2.0 meters
    (4)    LARGEST_MAX = 54.0 meters
    (5)    radius_X = radius of the primitive X's ''bounding sphere'' (Figure A)
    (6)    minimum(C, D) = C if less than D, otherwise D
 
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
terms:
 
    (7)    A_can_link_to_B = ( length(center_A - center_B) + radius_A + radius_B <= max_link_span )
 
The '''bounding sphere''' is the smallest sphere that totally encloses the
primitive's ''local bounding box''.
 
The '''local bounding box''' is centered at the primitive's ''geometric center'' and
has sides that are equal to the primitive's ''scale''.  One exception to this
rule is that megaprim scale components greater than 10 meters are clamped to 10.
 
The '''geometric center''' of the primitive is its local symmetric center prior to
any ''cut'', ''shear'', ''twist'', ''taper'', or ''hollow'' operations.
 
'''Note''' that a primitive's bounding sphere is not necessarily the tightest sphere
possible for its shape, unless it is a simple box.  The bounding sphere
depends only on the primitive's position and scale, so any
severly cut and hollowed primitive will be significantly smaller than its bounding
sphere, and not necessarily near the center.  Also, a primitive with twist and/or
shear may have corners that extend outside of its bounding sphere.  Since the
linkability rules depend only on the bounding sphere, which is ultimately dependent
only on the primitive's position and scale, the linkability of two prims is independent
of changes to form and rotation.
 
TODO -- we need a new figure_A that shows ''radius'' instead of ''diameter''.
Also need new figures _B and _C for two linkable prims and two linkable multi-prim objects.
 
=== Linkability algorithm ===
 
The rules governing the '''linkability of multi-prim objects''' is very similar to
the two-primitive case.  The same formulae (1) and (7) apply, but the
bounding spheres of multi-prim objects are the smallest spheres that completely
contain all of the bounding spheres of the corresponding primitives.
 
When linking '''three or more objects''' the algorithm iterates over the candidate
objects until all linkable pieces have been found.  First the root object
is tested against each candidate object and the larger bounding sphere is
recomputed after a successful link.  Then any unlinked pieces are tested between
themselves and merged into larger collections according to the formulae.  The
root object is then re-tested against the modified candidates and the process
continues until all objects are linked, or no new links have been found.
 
=== Failure modes ===
 
If an '''unlinkable set''' is tested by the linkability algorithm then the final subset
of linkable parts is determined by the order in which the candidates were submitted.
The trivial proof for this is to consider a root primitive in the middle of an infinite
grid of other primitives.  It can't link to everything, but it were first tested against
all primitives west of it the final linkable subset of that first operation might not
link to any primitives to the east because of the LARGEST_MAX requirement (4).  If the
primitives to the east were tested first then the final result would be different.
 
If a '''linkable set''' is tested by the linkability algorithm then the final subset
of linkable parts is NOT affected by the order in which the candidates were submitted.
That is, if just the linkable subsets of the failure modes above are tested for
all permutations of sequence they will always link.  The proof of this is left
as an exercise for the reader.
 
=== Examples ===
 
==== 2 very small prims ====
radius_A = ~0.01
 
radius_B = ~0.01
 
LINK_BONUS = 2.0 meters
 
LARGEST_MAX = 54.0 meters
 
3 * (radius_A + radius_B) + LINK_BONUS = 2.06 meters
 
2.06 is less than 54
 
Thus max_link_span = 2.06m
 
==== one large prim and one small prim ====
radius_A = 5m
 
radius_B = ~0.01
 
LINK_BONUS = 2.0 meters
 
LARGEST_MAX = 54.0 meters
 
3 * (radius_A + radius_B) + LINK_BONUS = 16.03 meters
 
16.03 is smaller than 54.0
 
Thus max_link_span = 16.03m
 
==== 2 very large prims ====
The diameter of a bounding sphere is the square root of x^2 + y^2 + z^2
thus, the diameter of a 10m x 10m x 10m prim is the square root of (100+100+100) = ~17.3m
and the diameter of a 10m x 1m x 1m prim is square root (100+1+1) = ~10.1m
(The type of prim doesn't matter for this calculation. We only care about the dimensions.)
Let's take the case of two 10m x 10m x 10m prims.
 
radius_A = 8.66m
 
radius_B = 8.66m
 
LINK_BONUS = 2.0 meters
 
LARGEST_MAX = 54.0 meters
 
3 * (radius_A + radius_B) + LINK_BONUS = 53.96m
 
53.96 is smaller than 54
 
Thus max_link_span = 53.96m
 
[[Category:Havok4]]

Revision as of 16:57, 26 August 2011

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: 

 LINKABLE = D < 54  AND  N ≤ 256
 where D = diameter of the smallest bounding sphere of the collection of prim centers
 and N = number of prims in the collection

Andrew Linden 21:47, 02 December 2010 (UTC)

If you are interested in how the old formula worked, see Linkability Rules/Havok 4.

Notes

  • [2011-08-27-16:52] Andrew Linden: 54 was the minimum sphere that still contained all of the legacy linkable content.
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