<s>
In	O
mathematics	O
,	O
an	O
implicit	B-Algorithm
curve	I-Algorithm
is	O
a	O
plane	O
curve	O
defined	O
by	O
an	O
implicit	O
equation	O
relating	O
two	O
coordinate	O
variables	O
,	O
commonly	O
x	O
and	O
y	O
.	O
</s>
<s>
Hence	O
an	O
implicit	B-Algorithm
curve	I-Algorithm
can	O
be	O
considered	O
as	O
the	O
set	O
of	O
zeros	O
of	O
a	O
function	O
of	O
two	O
variables	O
.	O
</s>
<s>
The	O
graph	B-Application
of	I-Application
a	I-Application
function	I-Application
is	O
usually	O
described	O
by	O
an	O
equation	O
in	O
which	O
the	O
functional	O
form	O
is	O
explicitly	O
stated	O
;	O
this	O
is	O
called	O
an	O
explicit	O
representation	O
.	O
</s>
<s>
Examples	O
of	O
implicit	B-Algorithm
curves	I-Algorithm
include	O
:	O
</s>
<s>
The	O
fifth	O
example	O
shows	O
the	O
possibly	O
complicated	O
geometric	O
structure	O
of	O
an	O
implicit	B-Algorithm
curve	I-Algorithm
.	O
</s>
<s>
In	O
practice	O
implicit	B-Algorithm
curves	I-Algorithm
have	O
an	O
essential	O
drawback	O
:	O
their	O
visualization	O
is	O
difficult	O
.	O
</s>
<s>
But	O
there	O
are	O
computer	O
programs	O
enabling	O
one	O
to	O
display	O
an	O
implicit	B-Algorithm
curve	I-Algorithm
.	O
</s>
<s>
Special	O
properties	O
of	O
implicit	B-Algorithm
curves	I-Algorithm
make	O
them	O
essential	O
tools	O
in	O
geometry	O
and	O
computer	O
graphics	O
.	O
</s>
<s>
An	O
implicit	B-Algorithm
curve	I-Algorithm
with	O
an	O
equation	O
can	O
be	O
considered	O
as	O
the	O
level	O
curve	O
of	O
level	O
0	O
of	O
the	O
surface	O
(	O
see	O
third	O
diagram	O
)	O
.	O
</s>
<s>
In	O
general	O
,	O
implicit	B-Algorithm
curves	I-Algorithm
fail	O
the	O
vertical	B-Algorithm
line	I-Algorithm
test	I-Algorithm
(	O
meaning	O
that	O
some	O
values	O
of	O
x	O
are	O
associated	O
with	O
more	O
than	O
one	O
value	O
of	O
y	O
)	O
and	O
so	O
are	O
not	O
necessarily	O
graphs	B-Application
of	I-Application
functions	I-Application
.	O
</s>
<s>
However	O
,	O
the	O
implicit	O
function	O
theorem	O
gives	O
conditions	O
under	O
which	O
an	O
implicit	B-Algorithm
curve	I-Algorithm
locally	O
is	O
given	O
by	O
the	O
graph	B-Application
of	I-Application
a	I-Application
function	I-Application
(	O
so	O
in	O
particular	O
it	O
has	O
no	O
self-intersections	O
)	O
.	O
</s>
<s>
If	O
the	O
defining	O
relations	O
are	O
sufficiently	O
smooth	O
then	O
,	O
in	O
such	O
regions	O
,	O
implicit	B-Algorithm
curves	I-Algorithm
have	O
well	O
defined	O
slopes	O
,	O
tangent	O
lines	O
,	O
normal	O
vectors	O
,	O
and	O
curvature	O
.	O
</s>
<s>
There	O
are	O
several	O
possible	O
ways	O
to	O
compute	O
these	O
quantities	O
for	O
a	O
given	O
implicit	B-Algorithm
curve	I-Algorithm
.	O
</s>
<s>
The	O
essential	O
disadvantage	O
of	O
an	O
implicit	B-Algorithm
curve	I-Algorithm
is	O
the	O
lack	O
of	O
an	O
easy	O
possibility	O
to	O
calculate	O
single	O
points	O
which	O
is	O
necessary	O
for	O
visualization	O
of	O
an	O
implicit	B-Algorithm
curve	I-Algorithm
(	O
see	O
next	O
section	O
)	O
.	O
</s>
<s>
Implicit	O
representations	O
facilitate	O
the	O
computation	O
of	O
intersection	B-Algorithm
points	O
:	O
If	O
one	O
curve	O
is	O
represented	O
implicitly	O
and	O
the	O
other	O
parametrically	O
the	O
computation	O
of	O
intersection	B-Algorithm
points	O
needs	O
only	O
a	O
simple	O
(	O
1-dimensional	O
)	O
Newton	O
iteration	O
,	O
which	O
is	O
contrary	O
to	O
the	O
cases	O
implicit-implicit	O
and	O
parametric-parametric	O
(	O
see	O
Intersection	B-Algorithm
)	O
.	O
</s>
<s>
It	O
is	O
easy	O
to	O
generate	O
curves	O
which	O
are	O
almost	O
geometrically	O
similar	O
to	O
the	O
given	O
implicit	B-Algorithm
curve	I-Algorithm
by	O
just	O
adding	O
a	O
small	O
number	O
:	O
(	O
see	O
section	O
#Smooth	O
approximations	O
)	O
.	O
</s>
<s>
Within	O
mathematics	O
implicit	B-Algorithm
curves	I-Algorithm
play	O
a	O
prominent	O
role	O
as	O
algebraic	O
curves	O
.	O
</s>
<s>
In	O
addition	O
,	O
implicit	B-Algorithm
curves	I-Algorithm
are	O
used	O
for	O
designing	O
curves	O
of	O
desired	O
geometrical	O
shapes	O
.	O
</s>
<s>
If	O
one	O
starts	O
with	O
simple	O
implicit	B-Algorithm
curves	I-Algorithm
other	O
than	O
lines	O
(	O
circles	O
,	O
parabolas	O
,...	O
)	O
one	O
gets	O
a	O
wide	O
range	O
of	O
interesting	O
new	O
curves	O
.	O
</s>
<s>
(	O
product	O
of	O
two	O
circles	O
)	O
yields	O
smooth	O
approximations	O
of	O
the	O
intersection	B-Algorithm
of	O
two	O
circles	O
(	O
see	O
diagram	O
)	O
.	O
</s>
<s>
In	O
CAD	B-Application
one	O
uses	O
implicit	B-Algorithm
curves	I-Algorithm
for	O
the	O
generation	O
of	O
blending	O
curves	O
,	O
which	O
are	O
special	O
curves	O
establishing	O
a	O
smooth	O
transition	O
between	O
two	O
given	O
curves	O
.	O
</s>
<s>
To	O
visualize	O
an	O
implicit	B-Algorithm
curve	I-Algorithm
one	O
usually	O
determines	O
a	O
polygon	O
on	O
the	O
curve	O
and	O
displays	O
the	O
polygon	O
.	O
</s>
<s>
For	O
an	O
implicit	B-Algorithm
curve	I-Algorithm
one	O
has	O
to	O
solve	O
two	O
subproblems	O
:	O
</s>
<s>
For	O
the	O
solution	O
of	O
both	O
tasks	O
mentioned	O
above	O
it	O
is	O
essential	O
to	O
have	O
a	O
computer	O
program	O
(	O
which	O
we	O
will	O
call	O
)	O
,	O
which	O
,	O
when	O
given	O
a	O
point	O
near	O
an	O
implicit	B-Algorithm
curve	I-Algorithm
,	O
finds	O
a	O
point	O
that	O
is	O
exactly	O
on	O
the	O
curve	O
:	O
</s>
<s>
Because	O
the	O
algorithm	O
traces	O
the	O
implicit	B-Algorithm
curve	I-Algorithm
it	O
is	O
called	O
a	O
tracing	O
algorithm	O
.	O
</s>
<s>
If	O
the	O
implicit	B-Algorithm
curve	I-Algorithm
consists	O
of	O
several	O
parts	O
it	O
has	O
to	O
be	O
started	O
several	O
times	O
with	O
suitable	O
starting	O
points	O
.	O
</s>
<s>
If	O
the	O
implicit	B-Algorithm
curve	I-Algorithm
consists	O
of	O
several	O
or	O
even	O
unknown	O
parts	O
,	O
it	O
may	O
be	O
better	O
to	O
use	O
a	O
rasterisation	O
algorithm	O
.	O
</s>
<s>
If	O
the	O
net	O
is	O
dense	O
enough	O
,	O
the	O
result	O
approximates	O
the	O
connected	O
parts	O
of	O
the	O
implicit	B-Algorithm
curve	I-Algorithm
.	O
</s>
<s>
is	O
the	O
intersection	B-Algorithm
curve	O
between	O
a	O
sphere	O
and	O
a	O
cylinder	O
.	O
</s>
<s>
For	O
the	O
computation	O
of	O
curve	O
points	O
and	O
the	O
visualization	O
of	O
an	O
implicit	O
space	O
curve	O
see	O
Intersection	B-Algorithm
.	O
</s>
