Graphics Reference
In-Depth Information
Find the cross-ratio of the points [1,0,1], [0,1,1], [2,-1,3], and [3,1,2] in
P
2
.
3.4.1.3.
The points
I
= [0,1,-1],
O
= [1,0,-2], and
U
= [2,-4,0] belong to a line
L
in
P
2
.
3.4.1.4.
(a) Find the coordinates of the point [1,2,-4] on
L
with respect to
I
,
O
, and
U
.
(b) Find the coordinates of the point [1,2,-4] with respect to
I
¢
= [1,0,-2],
O
¢
=
[0,1,-1], and
U
¢ = [1,1,-3].
(c) Find the transformation j that maps the coordinates with respect to
I
,
O
, and
U
to the coordinates with respect to
I
¢,
O
¢, and
U
¢.
Consider the points
I
= [0,1,-1],
J
= [1,0,1],
O
= [1,0,-2], and
U
= [2,-4,0] in
P
2
.
3.4.1.5.
(a) Find the coordinates of the point [1,2,-4] with respect to
I
,
J
,
O
, and
U
.
(b) Find the coordinates of the point [1,2,-4] with respect to
I
¢
= [1,0,-2],
J
¢
= [3,1,1],
O
¢ = [0,1,-1], and
U
¢ = [1,1,-3].
(c) Find the transformation j that maps the coordinates with respect to
I
,
J
,
O
, and
U
to the coordinates with respect to
I
¢,
J
¢,
O
¢, and
U
¢.
Section 3.4.3
Let T be the central projection that projects
R
2
onto the line
L
defined by 2x - 3y + 6
= 0 from the point
p
= (5,1).
3.4.3.1.
(a) Find the equation for T in two ways:
(1) Using homogeneous coordinates and projective transformations
(2) Finding the intersection of lines from
p
with
L
(b) Find T(7,1) and T(3,4).
Section 3.5.1
Let T be the central projection that projects
R
3
onto the plane
X
defined by x + y + z
= 1 from the point
p
= (-1,0,0).
(a) Find the equations for T in three ways:
3.5.1.1.
(1) Using the usual composites of rigid motions and central projections and
homogeneous coordinates
(2) Via the method of frames
(3) Finding the intersections of lines through
p
with the plane
(b) Find T(9,0,0) and T(4,0,5).
Section 3.6
3.6.1.
Consider the conic defined by the equation
(
)
(
)
2
2
31
x
-
10 3
xy
+
21
y
+
10 3
-
124
x
+
20 3
-
42
y
-
20 3
+
1
=
0
.
(a) Is the conic an ellipse, hyperbola, or parabola?
(b) Find its natural coordinate system.
(c) Determine its focus and directrix.
