This preview shows page 1. Sign up to view the full content.
Unformatted text preview: 0 p z
dz
qx 1 qz = = p z i + (p z q x p z i + (p z q x i.e.
i.e. 0 1 pz
px
px qz )j + px k0 @pz qx 1 qz A
=
pz
px A
@ px qz )j + px k = dx = pz dxdy= = ppz qx
z pz q x px q z px px q z dy dz= = z qpx px qz
px
d z = px
!
So we can write vector p completely in terms of d
So we can write vector p completely in terms of d !
0
1
So we can write vector p completely in terms of d
dz
0
1
!
@ dz 0 A
p=
A
p = @ 0 dx
dx Graphics Lecture 2: Slide 44 ! 43 / 45
43 / 45 Using the dot product
Using the dot product Using the dot product !
Lastly we can use the factthat the vectors vectors p a orthogonal
• Lastly we can use that the vectors p and q are orthogonal
Lastly we can use the factthe fact that the p and q are nd q are orthogonal! pq
p · ·q
) x qx + z qz
) ppx qx+ ppz qz =
=
=
= 00
00 • from the the product (previous slide)
AndAnd the cross cross product (previous
And from fromcrossproduct (previous slide) slide) !
ddy= ppz qx ppx qz
y = z qx
x qz
Sowe have have two linear equations to solve for q and for q it
So we two simple linear linear equations to solve write
So • wehave two simple simpleequationsto solve for q and write it
in terms write it in terms ofthe
and of the components d
in terms of the componentsofofd components of d ! Graphics Lecture 2: Slide 45 !
44 / 45
44 / 45 The ﬁnal matrix
The ﬁnal matrix !
The ﬁnal matrix
• Once we have expressions for p and q in terms of the
Once we have expressions for p and q in terms of the given vector
Once we vector d, we have ! and q in terms of the given vector
given have expressions for p
d,, we have
d we have
p
q
d
p
q
u=
v=
w= d
u = p
v = q
w = d
p
q
d
!
WeWe already know C as is also given. given. can we can write
• already know C as that that is also So we can write down
We already know C as that is also given. So we So write down
thedmatrix matrix !
own the
the matrix
0
1
!
0
1
ux uyy uzz
C ··u
ux u
u
Cu
B vx vy vz
C ··v C
B
C
C vC
B vx vy vz
B
C
@wx wy wz
A
C ··w A
@wx wy wz
Cw
0
1
00
00
0
1
Graphics Lecture 2: Slide 46 !...
View
Full
Document
 Spring '14

Click to edit the document details