<s>
A	O
biarc	B-Algorithm
is	O
a	O
smooth	O
curve	O
formed	O
from	O
two	O
circular	O
arcs	O
.	O
</s>
<s>
In	O
order	O
to	O
make	O
the	O
biarc	B-Algorithm
smooth	O
(	O
G1	O
continuous	O
)	O
,	O
the	O
two	O
arcs	O
should	O
have	O
the	O
same	O
tangent	O
at	O
the	O
connecting	O
point	O
where	O
they	O
meet	O
.	O
</s>
<s>
Biarcs	B-Algorithm
are	O
commonly	O
used	O
in	O
geometric	B-Application
modeling	I-Application
and	O
computer	O
graphics	O
.	O
</s>
<s>
They	O
can	O
be	O
used	O
to	O
approximate	O
splines	B-Algorithm
and	O
other	O
plane	O
curves	O
by	O
placing	O
the	O
two	O
outer	O
endpoints	O
of	O
the	O
biarc	B-Algorithm
along	O
the	O
curve	O
to	O
be	O
approximated	O
,	O
with	O
a	O
tangent	O
that	O
matches	O
the	O
curve	O
,	O
and	O
then	O
choosing	O
a	O
middle	O
point	O
that	O
best	O
fits	O
the	O
curve	O
.	O
</s>
<s>
This	O
choice	O
of	O
three	O
points	O
and	O
two	O
tangents	O
determines	O
a	O
unique	O
pair	O
of	O
circular	O
arcs	O
,	O
and	O
the	O
locus	O
of	O
middle	O
points	O
for	O
which	O
these	O
two	O
arcs	O
form	O
a	O
biarc	B-Algorithm
is	O
itself	O
a	O
circular	O
arc	O
.	O
</s>
<s>
In	O
particular	O
,	O
to	O
approximate	O
a	O
Bézier	O
curve	O
in	O
this	O
way	O
,	O
the	O
middle	O
point	O
of	O
the	O
biarc	B-Algorithm
should	O
be	O
chosen	O
as	O
the	O
incenter	O
of	O
the	O
triangle	O
formed	O
by	O
the	O
two	O
endpoints	O
of	O
the	O
Bézier	O
curve	O
and	O
the	O
point	O
where	O
their	O
two	O
tangents	O
meet	O
.	O
</s>
<s>
More	O
generally	O
,	O
one	O
can	O
approximate	O
a	O
curve	O
by	O
a	O
smooth	O
sequence	O
of	O
biarcs	B-Algorithm
;	O
using	O
more	O
biarcs	B-Algorithm
in	O
the	O
sequence	O
will	O
in	O
general	O
improve	O
the	O
approximation	O
's	O
closeness	O
to	O
the	O
original	O
curve	O
.	O
</s>
<s>
In	O
the	O
below	O
examples	O
biarcs	B-Algorithm
are	O
subtended	O
by	O
the	O
chord	O
and	O
is	O
the	O
join	O
point	O
.	O
</s>
<s>
Biarc	B-Algorithm
3	O
includes	O
the	O
straight	O
line	O
segment	O
.	O
</s>
<s>
Biarcs	B-Algorithm
1	O
–	O
4	O
are	O
short	O
in	O
the	O
sense	O
that	O
they	O
do	O
not	O
turn	O
near	O
endpoints	O
.	O
</s>
<s>
Alternatively	O
,	O
biarcs	B-Algorithm
5	O
,	O
6	O
are	O
long	O
:	O
turning	O
near	O
one	O
of	O
endpoints	O
means	O
that	O
they	O
intersect	O
the	O
left	O
or	O
the	O
right	O
complement	O
of	O
the	O
chord	O
to	O
the	O
infinite	O
straight	O
line	O
.	O
</s>
<s>
Biarcs	B-Algorithm
2	O
–	O
6	O
share	O
end	O
tangents	O
.	O
</s>
<s>
3	O
,	O
among	O
the	O
family	O
of	O
biarcs	B-Algorithm
with	O
common	O
tangents	O
.	O
</s>
<s>
3	O
shows	O
two	O
examples	O
of	O
biarc	B-Algorithm
families	O
,	O
sharing	O
end	O
points	O
and	O
end	O
tangents	O
.	O
</s>
<s>
4	O
shows	O
two	O
examples	O
of	O
biarc	B-Algorithm
families	O
,	O
sharing	O
end	O
points	O
and	O
end	O
tangents	O
,	O
end	O
tangents	O
being	O
parallel	O
:	O
</s>
<s>
In	O
particular	O
,	O
for	O
biarcs	B-Algorithm
,	O
shown	O
in	O
brown	O
on	O
shaded	O
background	O
(	O
lens-like	O
or	O
lune-like	O
)	O
,	O
the	O
following	O
holds	O
:	O
</s>
<s>
the	O
total	O
rotation	O
(	O
turning	O
angle	O
)	O
of	O
the	O
curve	O
is	O
exactly	O
(	O
not	O
,	O
which	O
is	O
the	O
rotation	O
for	O
other	O
biarcs	B-Algorithm
)	O
;	O
</s>
<s>
:	O
the	O
sum	O
is	O
the	O
angular	O
width	O
of	O
the	O
lens/lune	O
,	O
covering	O
the	O
biarc	B-Algorithm
,	O
whose	O
sign	O
corresponds	O
to	O
either	O
increasing	O
( +1	O
)	O
or	O
decreasing	O
curvature	O
( −1	O
)	O
of	O
the	O
biarc	B-Algorithm
,	O
according	O
to	O
generalized	O
Vogt	O
's	O
theorem	O
(	O
)	O
.	O
</s>
<s>
A	O
family	O
of	O
biarcs	B-Algorithm
with	O
common	O
end	O
points	O
,	O
,	O
and	O
common	O
end	O
tangents	O
(	O
1	O
)	O
is	O
denoted	O
as	O
or	O
,	O
briefly	O
,	O
as	O
being	O
the	O
family	O
parameter	O
.	O
</s>
<s>
Biarc	B-Algorithm
properties	O
are	O
described	O
below	O
in	O
terms	O
of	O
article	O
.	O
</s>
<s>
Degenerate	O
biarcs	B-Algorithm
are	O
:	O
</s>
<s>
Biarc	B-Algorithm
:	O
as	O
,	O
,	O
arc	O
vanishes	O
.	O
</s>
<s>
Biarc	B-Algorithm
:	O
as	O
,	O
,	O
arc	O
vanishes	O
.	O
</s>
<s>
Discontinuous	O
biarc	B-Algorithm
includes	O
straight	O
line	O
or	O
and	O
passes	O
through	O
the	O
infinite	O
point	O
:	O
</s>
<s>
Discontinuous	O
biarc	B-Algorithm
is	O
shown	O
by	O
red	O
dash-dotted	O
line	O
.	O
</s>
<s>
The	O
whole	O
family	O
can	O
be	O
subdivided	O
into	O
three	O
subfamilies	O
of	O
non-degenerate	O
biarcs	B-Algorithm
:	O
</s>
<s>
biarcs	B-Algorithm
are	O
shown	O
in	O
brown	O
,	O
</s>
<s>
biarcs	B-Algorithm
in	O
blue	O
,	O
</s>
<s>
and	O
biarcs	B-Algorithm
in	O
green	O
.	O
</s>
