Graphics Reference
In-Depth Information
X
y = x 2
y = mx + b
axis
midpoints
Q
parallel
chords
(a)
(b)
Figure 2.21.
Midpoints of parallel chords for parabola are parallel to axis.
Solution. Since all parabolas are affinely equivalent we may restrict ourselves to the
special case of the parabola defined by the equation y = x 2 and the family of chords
determined by the lines y = mx + b, where m is fixed and b ≥ 0. See Figure 2.21(b).
To find the intersection of the lines with the parabola, we must solve the equation
mx + b = x 2 . The two solutions are
2
2
mm b and
+
+
4
mm b
-
+
4
x
=
x
=
.
1
2
2
2
The midpoint Q = (u,v) of such a chord is defined by
xx m
+
1
2
u
=
=
2
2
and
2
mx
++
b
mx
+ =
b
m
+
2
b
1
2
v
=
,
2
2
which proves the result.
Finally, note that one could have developed affine geometry without first coordi-
natizing points. We could make points, lines, etc., undefined terms and use axioms to
define their properties. This is the synthetic geometry approach. Coordinates could be
introduced at a later stage. The point is that, in the context of affine geometry, the
exact lengths of geometric figures are not important. At most it is relative size that
counts, that is, the ratios of segments.
2.4.1
Parallel Projections
Definition. Let v be a nonzero vector in R n and let W be the family of parallel lines
with direction vector v . Let L p denote the line in W through the point p . If X is a
hyperplane in R n
not parallel to v , then define a map
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