Math 559 MODERN GEOMETRY SOLUTION FOR QUIZ – I (09/11

Math 559 MODERN GEOMETRY
SOLUTION FOR QUIZ – I (09/11)
September 18 (Thu), 2014
Instructor:
Yasuyuki Kachi
Line #: 13191.
[I] (4pts)
(1)
Homogenization of
◦ Alternatively,
(2)
◦ Alternatively,
(1)
with respect to Z is
(2)
y, z
X : Y : Z
=
X : Y
=
Homogenization of
[II] (4pts)
′
x, z
: Z
′
X : Y : Z
=
X : Y
=
: Z
De-homogenization of
−3 8
1 :
1
8
:
1 5
−3 .
8
is
5 : 10 : 2 .
1
2
: 1 :
1 .
5
X : Y : Z
=
−1 : 1 : 4
Homogenization of
=
with respect to
Y is not feasible.
[III] (2pts)
is
8 : 1 : −3 .
−1
1 .
,
4
4
X : Y : Z = 1 : 0 : 3
x′′ , y ′′
De-homogenization of
1
,
=
8
1
=
,
2
11 ′
x +
4
5 ′
z + 1 = 0
4
11 X + 4 Y + 5 Z = 0.
1
is
[IV] (6pts)
(1)
De-homogenization of
X − Y + Z = 0
with respect
to Y is
x′ + z ′ − 1 = 0.
(2)
De-homogenization of
X − Y
= 0
with respect to Z is
x′′ − y ′′ = 0.
[V] (3pts)
Suppose
y, z
=
√
2, 1
is the coordinate reading of a point of UX using the affine coordinate of UX . Then
the coordinate reading of the same point using the affine coordinate of UZ is
√ x′′ , y ′′
=
1,
2 .
[VI] (5pts)
UY
The conversion formula to express the affine coordinate
in terms of the affine coordinate y, z of UX :
x′ =
[VII] (6pts)
(1)
Suppose
1
,
y
z′ =
x′ , z ′
of
z
.
y
− 5 x′ + 7 z ′ + 12 = 0
is the equation of
a line in UY using the affine coordinate of UY . Then the equation of the same line
using the affine coordinate of UZ is
− 5 x′′ + 12 y ′′ + 7 = 0.
(2)
Suppose
y = 0
is the equation of a line in UX using the affine coordinate
of UX . Then there is no equation of the same line using the affine coordinate of UY .
The line is outside of UY .
2

Download Report