<s>
In	O
geometry	O
,	O
an	O
intersection	B-Algorithm
is	O
a	O
point	O
,	O
line	O
,	O
or	O
curve	O
common	O
to	O
two	O
or	O
more	O
objects	O
(	O
such	O
as	O
lines	O
,	O
curves	O
,	O
planes	O
,	O
and	O
surfaces	O
)	O
.	O
</s>
<s>
The	O
simplest	O
case	O
in	O
Euclidean	O
geometry	O
is	O
the	O
line	B-Algorithm
–	I-Algorithm
line	I-Algorithm
intersection	I-Algorithm
between	O
two	O
distinct	O
lines	O
,	O
which	O
either	O
is	O
one	O
point	O
or	O
does	O
not	O
exist	O
(	O
if	O
the	O
lines	O
are	O
parallel	O
)	O
.	O
</s>
<s>
Other	O
types	O
of	O
geometric	O
intersection	B-Algorithm
include	O
:	O
</s>
<s>
Determination	O
of	O
the	O
intersection	B-Algorithm
of	O
flats	O
–	O
linear	O
geometric	O
objects	O
embedded	O
in	O
a	O
higher-dimensional	O
space	O
–	O
is	O
a	O
simple	O
task	O
of	O
linear	B-Language
algebra	I-Language
,	O
namely	O
the	O
solution	O
of	O
a	O
system	O
of	O
linear	O
equations	O
.	O
</s>
<s>
In	O
general	O
the	O
determination	O
of	O
an	O
intersection	B-Algorithm
leads	O
to	O
non-linear	O
equations	O
,	O
which	O
can	O
be	O
solved	B-General_Concept
numerically	I-General_Concept
,	O
for	O
example	O
using	O
Newton	O
iteration	O
.	O
</s>
<s>
Intersection	B-Algorithm
problems	O
between	O
a	O
line	O
and	O
a	O
conic	O
section	O
(	O
circle	O
,	O
ellipse	O
,	O
parabola	O
,	O
etc	O
.	O
)	O
</s>
<s>
Intersections	B-Algorithm
between	O
quadrics	O
lead	O
to	O
quartic	O
equations	O
that	O
can	O
be	O
solved	O
algebraically	O
.	O
</s>
<s>
one	O
gets	O
,	O
from	O
Cramer	O
's	O
rule	O
or	O
by	O
substituting	O
out	O
a	O
variable	O
,	O
the	O
coordinates	O
of	O
the	O
intersection	B-Algorithm
point	O
:	O
</s>
<s>
For	O
two	O
non-parallel	O
line	O
segments	O
and	O
there	O
is	O
not	O
necessarily	O
an	O
intersection	B-Algorithm
point	O
(	O
see	O
diagram	O
)	O
,	O
because	O
the	O
intersection	B-Algorithm
point	O
of	O
the	O
corresponding	O
lines	O
need	O
not	O
to	O
be	O
contained	O
in	O
the	O
line	O
segments	O
.	O
</s>
<s>
If	O
the	O
condition	O
is	O
fulfilled	O
one	O
inserts	O
or	O
into	O
the	O
corresponding	O
parametric	O
representation	O
and	O
gets	O
the	O
intersection	B-Algorithm
point	O
.	O
</s>
<s>
Remark	O
:	O
Considering	O
lines	O
,	O
instead	O
of	O
segments	O
,	O
determined	O
by	O
pairs	O
of	O
points	O
,	O
each	O
condition	O
can	O
be	O
dropped	O
and	O
the	O
method	O
yields	O
the	O
intersection	B-Algorithm
point	O
of	O
the	O
lines	O
(	O
see	O
above	O
)	O
.	O
</s>
<s>
if	O
If	O
this	O
condition	O
holds	O
with	O
strict	O
inequality	O
,	O
there	O
are	O
two	O
intersection	B-Algorithm
points	O
;	O
in	O
this	O
case	O
the	O
line	O
is	O
called	O
a	O
secant	O
line	O
of	O
the	O
circle	O
,	O
and	O
the	O
line	O
segment	O
connecting	O
the	O
intersection	B-Algorithm
points	O
is	O
called	O
a	O
chord	O
of	O
the	O
circle	O
.	O
</s>
<s>
If	O
holds	O
,	O
there	O
exists	O
only	O
one	O
intersection	B-Algorithm
point	O
and	O
the	O
line	O
is	O
tangent	O
to	O
the	O
circle	O
.	O
</s>
<s>
The	O
intersection	B-Algorithm
of	O
a	O
line	O
and	O
a	O
parabola	O
or	O
hyperbola	O
may	O
be	O
treated	O
analogously	O
.	O
</s>
<s>
The	O
intersection	B-Algorithm
of	O
two	O
disks	O
(	O
the	O
interiors	O
of	O
the	O
two	O
circles	O
)	O
forms	O
a	O
shape	O
called	O
a	O
lens	O
.	O
</s>
<s>
The	O
problem	O
of	O
intersection	B-Algorithm
of	O
an	O
ellipse/hyperbola/parabola	O
with	O
another	O
conic	O
section	O
leads	O
to	O
a	O
system	O
of	O
quadratic	O
equations	O
,	O
which	O
can	O
be	O
solved	O
in	O
special	O
cases	O
easily	O
by	O
elimination	O
of	O
one	O
coordinate	O
.	O
</s>
<s>
In	O
general	O
the	O
intersection	B-Algorithm
points	O
can	O
be	O
determined	O
by	O
solving	O
the	O
equation	O
by	O
a	O
Newton	O
iteration	O
.	O
</s>
<s>
b	O
:	O
the	O
tangent	O
line	O
in	O
common	O
and	O
they	O
are	O
crossing	O
each	O
other	O
(	O
touching	O
intersection	B-Algorithm
,	O
see	O
diagram	O
)	O
.	O
</s>
<s>
Because	O
touching	O
intersections	B-Algorithm
appear	O
rarely	O
and	O
are	O
difficult	O
to	O
deal	O
with	O
,	O
the	O
following	O
considerations	O
omit	O
this	O
case	O
.	O
</s>
<s>
The	O
determination	O
of	O
intersection	B-Algorithm
points	O
always	O
leads	O
to	O
one	O
or	O
two	O
non-linear	O
equations	O
which	O
can	O
be	O
solved	O
by	O
Newton	O
iteration	O
.	O
</s>
<s>
The	O
intersection	B-Algorithm
points	O
are	O
( −	O
0.3686	O
,	O
0.9953	O
)	O
and	O
(	O
0.9953	O
,	O
−	O
0.3686	O
)	O
.	O
</s>
<s>
If	O
one	O
wants	O
to	O
determine	O
the	O
intersection	B-Algorithm
points	O
of	O
two	O
polygons	B-General_Concept
,	O
one	O
can	O
check	O
the	O
intersection	B-Algorithm
of	O
any	O
pair	O
of	O
line	O
segments	O
of	O
the	O
polygons	B-General_Concept
(	O
see	O
above	O
)	O
.	O
</s>
<s>
For	O
polygons	B-General_Concept
with	O
many	O
segments	O
this	O
method	O
is	O
rather	O
time-consuming	O
.	O
</s>
<s>
In	O
practice	O
one	O
accelerates	O
the	O
intersection	B-Algorithm
algorithm	O
by	O
using	O
window	O
tests	O
.	O
</s>
<s>
In	O
this	O
case	O
one	O
divides	O
the	O
polygons	B-General_Concept
into	O
small	O
sub-polygons	O
and	O
determines	O
the	O
smallest	O
window	O
(	O
rectangle	O
with	O
sides	O
parallel	O
to	O
the	O
coordinate	O
axes	O
)	O
for	O
any	O
sub-polygon	O
.	O
</s>
<s>
Before	O
starting	O
the	O
time-consuming	O
determination	O
of	O
the	O
intersection	B-Algorithm
point	O
of	O
two	O
line	O
segments	O
any	O
pair	O
of	O
windows	O
is	O
tested	O
for	O
common	O
points	O
.	O
</s>
<s>
In	O
3-dimensional	O
space	O
there	O
are	O
intersection	B-Algorithm
points	O
(	O
common	O
points	O
)	O
between	O
curves	O
and	O
surfaces	O
.	O
</s>
<s>
In	O
the	O
following	O
sections	O
we	O
consider	O
transversal	O
intersection	B-Algorithm
only	O
.	O
</s>
<s>
The	O
intersection	B-Algorithm
of	I-Algorithm
a	I-Algorithm
line	I-Algorithm
and	I-Algorithm
a	I-Algorithm
plane	I-Algorithm
in	O
general	O
position	O
in	O
three	O
dimensions	O
is	O
a	O
point	O
.	O
</s>
<s>
for	O
parameter	O
of	O
the	O
intersection	B-Algorithm
point	O
.	O
</s>
<s>
If	O
a	O
line	O
is	O
defined	O
by	O
two	O
intersecting	O
planes	O
and	O
should	O
be	O
intersected	O
by	O
a	O
third	O
plane	O
,	O
the	O
common	O
intersection	B-Algorithm
point	O
of	O
the	O
three	O
planes	O
has	O
to	O
be	O
evaluated	O
.	O
</s>
<s>
If	O
the	O
scalar	O
triple	O
product	O
equals	O
to	O
0	O
,	O
then	O
planes	O
either	O
do	O
not	O
have	O
the	O
triple	O
intersection	B-Algorithm
or	O
it	O
is	O
a	O
line	O
(	O
or	O
a	O
plane	O
,	O
if	O
all	O
three	O
planes	O
are	O
the	O
same	O
)	O
.	O
</s>
<s>
The	O
intersection	B-Algorithm
points	O
are	O
:	O
( −	O
0.8587	O
,	O
0.7374	O
,	O
−	O
0.6332	O
)	O
,	O
(	O
0.8587	O
,	O
0.7374	O
,	O
0.6332	O
)	O
.	O
</s>
<s>
A	O
line	B-Algorithm
–	I-Algorithm
sphere	I-Algorithm
intersection	I-Algorithm
is	O
a	O
simple	O
special	O
case	O
.	O
</s>
<s>
Like	O
the	O
case	O
of	O
a	O
line	O
and	O
a	O
plane	O
,	O
the	O
intersection	B-Algorithm
of	O
a	O
curve	O
and	O
a	O
surface	O
in	O
general	O
position	O
consists	O
of	O
discrete	O
points	O
,	O
but	O
a	O
curve	O
may	O
be	O
partly	O
or	O
totally	O
contained	O
in	O
a	O
surface	O
.	O
</s>
<s>
Two	O
transversally	O
intersecting	O
surfaces	O
give	O
an	O
intersection	B-Algorithm
curve	I-Algorithm
.	O
</s>
<s>
The	O
most	O
simple	O
case	O
is	O
the	O
intersection	B-Algorithm
line	O
of	O
two	O
non-parallel	O
planes	O
.	O
</s>
