13.3 Capsules 629
13.3.3 Merging Capsules
If one capsule contains the other, just use the containing capsule. To determine if
this is the case, it is simple to formulate a test to see if a sphere of radius r
s
and
center C is contained in a capsule with medial segment M and radius r
c
.Letd be
the distance from C to M (distance from point to segment). The sphere is inside the
capsule as long as d + r
s
≤ r
c
.IfC is a point on M, then d = 0 and r
s
≤ r
c
is an
obvious condition for the sphere being inside the capsule. If C is not on M,letK be
the closest point on the segment to C. The extreme point of the sphere relative to
the segment is C + r
s
(C − K)/|C − K| and is a distance d + r
s
from M. To test if
one capsule is contained by another, it is sufficient to test ...