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x<br />
1<br />
4<br />
A<br />
4<br />
x<br />
3<br />
A<br />
3<br />
A<br />
5<br />
B<br />
N<br />
5<br />
maximal inscribed ball<br />
in its leftmost<br />
and upward position<br />
x5<br />
x5<br />
is an extremum<br />
of d(x0 ,x) below<br />
the planeΔ<br />
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£ ¦££ <br />
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surface of CA1<br />
CA1lies<br />
on<br />
this side<br />
dx<br />
maximal<br />
inscribed<br />
ball<br />
d z<br />
q<br />
0<br />
x 1<br />
x 2<br />
x<br />
z<br />
(a)<br />
θz<br />
y<br />
x3<br />
x 4<br />
CA4<br />
lies on<br />
this side<br />
spherical patch<br />
of B at x1<br />
flat finger A1<br />
spherical patch<br />
of B at x2<br />
surface of CA 4<br />
CA2<br />
lies on<br />
surface of CA2<br />
this side<br />
dx<br />
CA<br />
(c) 3 lies on<br />
(d) surface of CA this side<br />
(e)<br />
d z<br />
q 0<br />
3<br />
θz<br />
z<br />
x<br />
y<br />
(b)<br />
spherical patch<br />
of B at x3<br />
flat finger A4<br />
spherical patch<br />
of B at x4<br />
£ £ £ <br />
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dx<br />
d z<br />
q<br />
0<br />
θ z
maximal<br />
inscribed<br />
ball<br />
surface<br />
of CB<br />
CB31<br />
lies on<br />
this side<br />
31<br />
x 1<br />
x2<br />
dx<br />
x<br />
z<br />
(a)<br />
d z<br />
(c)<br />
y<br />
q 0<br />
θz<br />
x<br />
31<br />
x3 32<br />
x<br />
x31<br />
and x32<br />
generated by<br />
splitting x<br />
3<br />
this curve lies<br />
along spherical<br />
patch of B at x<br />
surface<br />
of CB<br />
3<br />
32<br />
CB32 lies on<br />
this side<br />
spherical patch<br />
of B at x<br />
spherical patch<br />
of B at x<br />
2<br />
θz<br />
1<br />
x<br />
z<br />
flat finger A<br />
d<br />
z<br />
q<br />
0<br />
y<br />
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spherical patch<br />
of B at x<br />
x<br />
(d)<br />
32<br />
(b)<br />
3<br />
flat finger A<br />
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£ <br />
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¥§¥ ¤
spherical patch<br />
of B at x 2<br />
spherical patch<br />
of B at x<br />
CA 1 = CA 2 lies<br />
on this side<br />
of CA1 = freespace is the region<br />
between the surfaces<br />
CA and CA3<br />
2<br />
x<br />
1<br />
surface of<br />
CA = CA<br />
1<br />
1<br />
x2<br />
x<br />
2<br />
dx<br />
z<br />
x<br />
32<br />
y<br />
θz<br />
(a)<br />
(c)<br />
saddle patch<br />
of B at x<br />
x 3<br />
3<br />
flat finger A2<br />
on hidden side<br />
CA3<br />
lies outside<br />
tobular surface<br />
surface<br />
of CA 32<br />
surface<br />
of CA<br />
surface<br />
of CA<br />
cylindrical finger A<br />
d x<br />
CA31 and CA32 lie outside<br />
tobular surface<br />
d x<br />
(e) (f)<br />
z<br />
31<br />
(b)<br />
cylindrical finger A<br />
q<br />
0 d y<br />
dy<br />
q<br />
0<br />
d x<br />
θ<br />
z<br />
q 0<br />
x 31<br />
d<br />
y<br />
flat finger A 1<br />
on hidden side<br />
£ ¨<strong>¡</strong>£ ¥ <strong>¡</strong>££ <br />
x<br />
3<br />
y<br />
(d)<br />
θ<br />
z<br />
θ<br />
z<br />
q 0<br />
32<br />
31<br />
surface of<br />
CA 1 = CA2<br />
dy<br />
£<br />
£ ¨£ £ <br />
§<br />
£<strong>¢</strong>£ ¥¤ §¥ ¦ ¨£ <br />
§ <strong>¢</strong>£ ¤ ¥<br />
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¦<br />
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CA 1,2<br />
dy y dy<br />
q<br />
0<br />
CA3<br />
σ = 0.2 σ = 0.3<br />
CA<br />
1,2<br />
σ = 0.2<br />
CA32<br />
d<br />
no freespace<br />
s<br />
d<br />
q<br />
0<br />
q0<br />
s<br />
s<br />
3<br />
σ = 0.4<br />
CA<br />
1,2<br />
y dy<br />
dy<br />
CA31<br />
CA32<br />
CA31<br />
CA31<br />
CA 1,2<br />
q<br />
0<br />
s<br />
no freespace<br />
freespace<br />
CA 1,2<br />
CA<br />
1,2<br />
σ = 0.3<br />
freespace freespace freespace<br />
CA<br />
no freespace<br />
£ ¤ £<strong>¢</strong> £ ¥ <br />
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£ £<br />
<strong>¡</strong> ££<br />
£<strong>¢</strong><br />
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¦<br />
¤<br />
CA 1,2<br />
σ = 0.4<br />
q 0<br />
q 0<br />
CA<br />
s<br />
32<br />
CA<br />
s<br />
3<br />
CA<br />
1,2<br />
σ = 0.5<br />
CA<br />
1,2<br />
CA<br />
31<br />
d<br />
y<br />
q<br />
dy<br />
0<br />
q 0<br />
no freespace<br />
s<br />
CA3<br />
CA32<br />
no freespace σ = 0.5 no freespace<br />
¥ ¨ © £§<strong>¡</strong> ¤ ££<br />
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££ ¥ £ ¤ ¤£ ¥¤ ¥¥<br />
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s<br />
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ottom half<br />
of ellipsoid B<br />
CA12<br />
concave<br />
at q<br />
0<br />
x<br />
3<br />
x<br />
(d)<br />
z<br />
(a)<br />
θ y<br />
contact points<br />
with inscribed ball<br />
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