Unformatted text preview: ight
in with deﬁnition of straight
line in ininwith deﬁnition which uses
line Cartesian space of straight
Ties Cartesian space which
a lineusesCartesian space which uses
point and a direction
in a point and a direction !
! a point and a direction Graphics Lecture 2: Slide 26 !
25 / 45 Adding two position vectors Adding two position vectors !
• If add two position vectors we obtain their their mid-point !
If wewe add two position vectors we obtainmid-point
Xa + Xb
(Xa + Xb ) / 2
B Ya C B Yb C B Ya + Yb C B (Ya + Yb ) / 2 C
B C+B C=B
@ Za A @ Yb A @ Za + Zb A @ (Za + Zb ) / 2 A
• This is reasonable since adding two position vectors has
This is reasonable since adding two position vectors has no real
no real meaning in vector geometry !
meaning in vector geometry Graphics Lecture 2: Slide 27 ! 26 / 45 The structure of a transformation matrix The structure of a transformation matrix ! The bottom row is always 0 0 0 1 • The bottom row is always 0 0 0 1 !
• The columns of a transformation matrix three direction
The columns of a transformation matrix comprisecomprise three
direction vectors and one
vectors and one position vector position vector !
qx rx sx Cx
Bqy ry sy Cy C
@qz rz sz Cz A
B qy C
@ qz A
B ry C
@ rz A
1 Graphics Lecture 2: Slide 28 !
27 / 45 Characteristics of Transformation matrices
CharacteristicsTransformation matrices a↵ected !
of transformation not
Characteristics ofZero, in the last ordinate ) matrices by the
Characteristics of Transformation matrices
10 1 0 1 Direction vector: Zero,rx the last ordinate ) not a↵ected by the
qx in sx Cx
Direction vector: Zero, in the last ordinate ⇒ ) not a↵ected by the
Zero, in the last ordinate not affected by the
B y ry sy Cy C B⇤C B⇤C
10 1 0 1
CB C B C
translation. ! 0 qx rx sx Cx1 0 ⇤1 = 0 ⇤1
@q z r z s...
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