<s>
In	O
geometry	O
,	O
an	O
intersection	B-Algorithm
curve	I-Algorithm
is	O
a	O
curve	O
that	O
is	O
common	O
to	O
two	O
geometric	O
objects	O
.	O
</s>
<s>
In	O
the	O
simplest	O
case	O
,	O
the	O
intersection	B-Algorithm
of	O
two	O
non-parallel	O
planes	O
in	O
Euclidean	O
3-space	O
is	O
a	O
line	O
.	O
</s>
<s>
In	O
general	O
,	O
an	O
intersection	B-Algorithm
curve	I-Algorithm
consists	O
of	O
the	O
common	O
points	O
of	O
two	O
transversally	O
intersecting	O
surfaces	O
,	O
meaning	O
that	O
at	O
any	O
common	O
point	O
the	O
surface	O
normals	O
are	O
not	O
parallel	O
.	O
</s>
<s>
The	O
analytic	O
determination	O
of	O
the	O
intersection	B-Algorithm
curve	I-Algorithm
of	O
two	O
surfaces	O
is	O
easy	O
only	O
in	O
simple	O
cases	O
;	O
for	O
example	O
:	O
a	O
)	O
the	O
intersection	B-Algorithm
of	O
two	O
planes	O
,	O
b	O
)	O
plane	O
section	O
of	O
a	O
quadric	O
(	O
sphere	O
,	O
cylinder	O
,	O
cone	O
,	O
etc	O
.	O
</s>
<s>
)	O
,	O
c	O
)	O
intersection	B-Algorithm
of	O
two	O
quadrics	O
in	O
special	O
cases	O
.	O
</s>
<s>
For	O
the	O
general	O
case	O
,	O
literature	O
provides	O
algorithms	O
,	O
in	O
order	O
to	O
calculate	O
points	O
of	O
the	O
intersection	B-Algorithm
curve	I-Algorithm
of	O
two	O
surfaces	O
.	O
</s>
<s>
Wanted	O
:	O
A	O
parametric	O
representation	O
of	O
the	O
intersection	B-Algorithm
line	O
.	O
</s>
<s>
A	O
point	O
of	O
the	O
intersection	B-Algorithm
line	O
can	O
be	O
determined	O
by	O
intersecting	O
the	O
given	O
planes	O
with	O
the	O
plane	O
,	O
which	O
is	O
perpendicular	O
to	O
and	O
.	O
</s>
<s>
The	O
normal	O
vectors	O
are	O
and	O
the	O
direction	O
of	O
the	O
intersection	B-Algorithm
line	O
is	O
.	O
</s>
<s>
is	O
a	O
parametric	O
representation	O
of	O
the	O
line	O
of	O
intersection	B-Algorithm
.	O
</s>
<s>
In	O
special	O
cases	O
,	O
the	O
determination	O
of	O
the	O
intersection	B-Algorithm
line	O
by	O
the	O
Gaussian	B-Algorithm
elimination	I-Algorithm
may	O
be	O
faster	O
.	O
</s>
<s>
In	O
any	O
case	O
,	O
the	O
intersection	B-Algorithm
curve	I-Algorithm
of	O
a	O
plane	O
and	O
a	O
quadric	O
(	O
sphere	O
,	O
cylinder	O
,	O
cone	O
,...	O
)	O
is	O
a	O
conic	O
section	O
.	O
</s>
<s>
It	O
is	O
an	O
easy	O
task	O
to	O
determine	O
the	O
intersection	B-Algorithm
points	O
of	O
a	O
line	O
with	O
a	O
quadric	O
(	O
i.e.	O
</s>
<s>
line-sphere	B-Algorithm
)	O
;	O
one	O
only	O
has	O
to	O
solve	O
a	O
quadratic	O
equation	O
.	O
</s>
<s>
So	O
,	O
any	O
intersection	B-Algorithm
curve	I-Algorithm
of	O
a	O
cone	O
or	O
a	O
cylinder	O
(	O
they	O
are	O
generated	O
by	O
lines	O
)	O
with	O
a	O
quadric	O
consists	O
of	O
intersection	B-Algorithm
points	O
of	O
lines	O
and	O
the	O
quadric	O
(	O
see	O
pictures	O
)	O
.	O
</s>
<s>
In	O
the	O
first	O
case	O
,	O
there	O
exists	O
just	O
one	O
intersection	B-Algorithm
curve	I-Algorithm
.	O
</s>
<s>
The	O
second	O
case	O
shows	O
an	O
example	O
where	O
the	O
intersection	B-Algorithm
curve	I-Algorithm
consists	O
of	O
two	O
parts	O
.	O
</s>
<s>
The	O
intersection	B-Algorithm
curve	I-Algorithm
is	O
self-intersecting	O
.	O
</s>
<s>
If	O
the	O
cylinder	O
and	O
sphere	O
have	O
the	O
same	O
radius	O
and	O
the	O
midpoint	O
of	O
the	O
sphere	O
is	O
located	O
on	O
the	O
axis	O
of	O
the	O
cylinder	O
,	O
then	O
the	O
intersection	B-Algorithm
curve	I-Algorithm
consists	O
of	O
singular	O
points	O
(	O
a	O
circle	O
)	O
only	O
.	O
</s>
<s>
One	O
possibility	O
to	O
determine	O
a	O
polygon	O
of	O
points	O
of	O
the	O
intersection	B-Algorithm
curve	I-Algorithm
of	O
two	O
surfaces	O
is	O
the	O
marching	O
method	O
(	O
see	O
section	O
References	O
)	O
.	O
</s>
<s>
The	O
first	O
part	O
is	O
the	O
curve	O
point	O
algorithm	O
,	O
which	O
determines	O
to	O
a	O
starting	O
point	O
in	O
the	O
vicinity	O
of	O
the	O
two	O
surfaces	O
a	O
point	O
on	O
the	O
intersection	B-Algorithm
curve	I-Algorithm
.	O
</s>
<s>
The	O
second	O
part	O
of	O
the	O
marching	O
method	O
starts	O
with	O
a	O
first	O
point	O
on	O
the	O
intersection	B-Algorithm
curve	I-Algorithm
,	O
determines	O
the	O
direction	O
of	O
the	O
intersection	B-Algorithm
curve	I-Algorithm
using	O
the	O
surface	O
normals	O
,	O
then	O
makes	O
a	O
step	O
with	O
a	O
given	O
step	O
length	O
into	O
the	O
direction	O
of	O
the	O
tangent	O
line	O
,	O
in	O
order	O
to	O
get	O
a	O
starting	O
point	O
for	O
a	O
second	O
curve	O
point	O
,	O
...	O
(	O
see	O
picture	O
)	O
.	O
</s>
<s>
The	O
marching	O
method	O
produces	O
for	O
any	O
starting	O
point	O
a	O
polygon	O
on	O
the	O
intersection	B-Algorithm
curve	I-Algorithm
.	O
</s>
<s>
If	O
the	O
intersection	B-Algorithm
curve	I-Algorithm
consists	O
of	O
two	O
parts	O
,	O
the	O
algorithm	O
has	O
to	O
be	O
performed	O
using	O
a	O
second	O
convenient	O
starting	O
point	O
.	O
</s>
<s>
Usually	O
,	O
singular	O
points	O
are	O
no	O
problem	O
,	O
because	O
the	O
chance	O
to	O
meet	O
exactly	O
a	O
singular	O
point	O
is	O
very	O
small	O
(	O
see	O
picture	O
:	O
intersection	B-Algorithm
of	O
a	O
cylinder	O
and	O
the	O
surface	O
)	O
.	O
</s>
<s>
The	O
intersection	B-Algorithm
curve	I-Algorithm
of	O
two	O
polyhedrons	O
is	O
a	O
polygon	O
(	O
see	O
intersection	B-Algorithm
of	O
three	O
houses	O
)	O
.	O
</s>
<s>
So	O
,	O
for	O
the	O
intersection	B-Algorithm
of	O
two	O
parametrically	O
defined	O
surfaces	O
,	O
the	O
algorithm	O
for	O
the	O
intersection	B-Algorithm
of	O
two	O
polyhedrons	O
can	O
be	O
used	O
.	O
</s>
