Graphics Reference
In-Depth Information
3.6.1.3. Theorem.
A conic can be found that passes through any given five points.
It is unique if no four of these points is collinear.
Proof.
We use homogeneous coordinates and need to show that we can always find
a nondegenerate equation of the form
2
2
2
ax
+++++=,
by
cz
dxy
exz
fyz
0
which is satisfied for the points. One such equation is
2
2
2
xyz y z z
xyz xy xz yz
xyzxyxzyz
xyzxyxzyz
xyzxyxzyz
xyzxyxzyz
2
2
2
1 1
11
11
1
1
1
2
2
2
2 2
22
22
2
2
2
=
0
.
2
2
2
3 3
33
33
3
3
3
2
2
2
4 4
44
44
4
4
4
2
2
2
5 5
55
55
5
5
5
For the rest, see [PenP86].
Next, we look at some problems dealing with fitting conics to given data. The
following fact is used in justifying the constructions.
3.6.1.4. Lemma.
If
C
1
and
C
2
are affine conics with equations C
1
(x,y) = 0 and
C
2
(x,y) = 0, then
(
)
=
(
)
+-
(
)
(
)
=
Cxy
,
l
Cxy
,
1
l
Cxy
,
0
l
1
2
or
•
(
)
=
(
)
-
(
)
=
Cxy Cxy Cxy
,
,
,
0
1
2
is the equation of a conic
C
l
that passes through the intersection points of the two
given conics. If
C
1
and
C
2
have exactly four points of intersection, then the family
C
l
,
lŒ
R
*, of conics consists of all the conics through these four points and each one is
completely determined by specifying a fifth point on it.
Proof.
See [PenP86].
Our design problems will also involve tangent lines and so we need to define those.
Tangent lines play an important role when studying the geometry of curves. There are
different ways to define them depending on whether one is looking at the curve from
a topological or algebraic point of view. The definition we give here is specialized to
conics. More general definitions will be encountered in Chapter 8 and 10. Our present
definition is based on the fact that, at a point of a nondegenerate conic, the line that
we would want to call the tangent line has the property that it is the only line through
that point that meets the conic in only that point.
