Graphics Reference
In-Depth Information
Line
. What constructors should the
Line
class have? What about a
Segment
class?
What methods should
Segment
have that
Line
lacks? Can you develop a way
to make rays, lines, and segments cooperate with the
ProjectiveTransformation
class, or are there insurmountable problems? Think about what happens when a
ray crosses the line on which a projective transformation is undefined.
Exercise 12.2:
General position of the points
P
i
(
i
=
1,
,4
)
was needed to
invert the matrix
B
in the construction of the
PointsToPoints
method for pro-
jective maps. We also assumed that the points
Q
i
(
i
=
1,
...
,4
)
were in general
position, but that assumption was stronger than necessary. What is the weakest
geometric condition on the
Q
i
that allows the
PointsToPoints
transformation to
be built?
Exercise 12.3:
Explain why the two characterizations of general position for
four points in the plane—that (a) no point lies on a line passing through another
pair and (b) the first three form a nondegenerate triangle, while the fourth is not
on the extensions of any of the sides of this triangle—are equivalent. Pay partic-
ular attention to the failure cases, that is, show that if four points fail to satisfy
condition (a), they also fail to satisfy condition (b), and vice versa.
Exercise 12.4:
Enhance the library by defining one-dimensional trans-
formations as well (
LinearTransformation1
,
AffineTransformation1
,
ProjectiveTransformation1
). The first two classes will be almost trivial. The
third is more interesting; include a constructor
ProjectiveTransform1(double
p, double q, double r)
that builds a projective map sending 0 to
p
,1to
q
,
and
...
to
r
(i.e.,
lim
x
→∞
T
(
x
)=
r
). From such a constructor it's easy to build a
PointsToPoints
transformation.
Exercise 12.5:
Enhance the library we presented by adding a constructor
TransformXYZYPRDegrees(Point3 P, float yaw, float pitch, float roll)
to
create a transformation that translates the origin to the point
P
, and applies the
specified yaw, pitch, and roll to the standard basis for 3-space.
Exercise 12.6:
By hand, find a transformation sending the points
P
1
=
(
2
,1
)
,
P
2
=(
1, 1
)
,
P
3
=(
2
,
∞
1
)
to the points
Q
1
=(
2
,
2
)
,
−
1
)
, and
P
4
=(
1,
−
Q
2
=
P
2
,
Q
3
=(
2
,
1
−
2
)
, and
Q
4
=
P
4
.
