CHAPTER II: AFFINE AND EUCLIDEAN GEOMETRY - OCW UPM

45 Pages·2011·0.18 MB·English

AFFINEANDPROJECTIVEGEOMETRY, E.Rosado&S.L.Rueda
CHAPTER II: AFFINE AND EUCLIDEAN GEOMETRY
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
2. AFFINE TRANSFORMATIONS
2.1 Definition and first properties
Definition
Let (A,V,φ) and (A(cid:48),V (cid:48),φ(cid:48)) be two real affine spaces. We will say that a map
f : A −→ A(cid:48)
is an affine transformation if there exists a linear transformation f¯: V −→ V (cid:48)
such that:
¯ A
f(PQ) = f (P)f (Q), ∀P,Q ∈ .
A
This is equivalent to say that for every P ∈ and every vector u¯ ∈ V we
have
¯
f(P + u) = f(P) + f(u).
¯
We call f the linear transformation associated to f.
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
Proposition
Let (A,V,φ) and (A(cid:48),V (cid:48),φ(cid:48)) be two affine subspaces and let f : A −→ A(cid:48) be an affine
transformation with associated linear transformation f¯: V −→ V (cid:48). The following statements
hold:
¯
1. f is injective if and only if f is injective.
¯
2. f is surjective if and only if f is surjective.
¯
3. f is bijective if and only if f is bijective.
Proposition
Let g: A −→ A(cid:48) and f : A(cid:48) −→ A(cid:48)(cid:48) be two affine transformations. The composition f ◦
g: A −→ A(cid:48)(cid:48) is also an affine transformation and its associated linear transformation is
¯
f ◦ g = f ◦ g¯.
Proposition
Let f,g: A −→ A(cid:48) be two affine transformations which coincide over a point P, f(P) =
¯
g(P), and which have the same associated linear transformation f = g¯. Then f = g.
Proof
Every X ∈ A verifies:
¯
f(X) = f(P + PX) = f(P) + f(PX) = g(P) + g¯(PX) = g(X).
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
2.2 Matrix associated to an affine transformation
Let (A,V,φ) and (A(cid:48),V (cid:48),φ(cid:48)) be two affine subspaces and let f : A −→ A(cid:48) be
an affine transformation with associated linear transformation f¯: V −→ V (cid:48).
We consider affine coordinate systems R = {O;B}, B = {e ,...,e } and
1 n
R(cid:48) = {O(cid:48);B(cid:48)}, B(cid:48) = {e(cid:48) ,...,e(cid:48) } of the spaces A, A(cid:48) respectively. Let us
1 m
assume that:
O(cid:48)f(O) = b e(cid:48) + ··· + b e(cid:48) ,
1 1 m m
 f¯(e ) = a e(cid:48) + ··· + a e(cid:48)
 1 11 1 m1 m
.
.
.
 f¯(e ) = a e(cid:48) + ··· + a e(cid:48)
n 1n 1 mn m
Let P be the point with coordinates (x ,...,x ) with respect to R and let
1 n
(y ,...,y ) be the coordinates of f(P) ∈ A(cid:48).
1 m
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
Then:
    
1 1 0 ··· 0 1
 y   b a ··· a  x 
 1  =  1 11 1n  1 
 ...   ... ... ... ...  ... 
    
y b a ··· a x
m m m1 mn n
We will write
 
1 0 ··· 0
(cid:18) 1 0t (cid:19)  b a ··· a 
M (R,R(cid:48)) = =  1 11 1n 
f b M (B,B(cid:48))  ... ... ... ... 
f¯  
b a ··· a
m m1 mn
where b are the coordinates of f(O) in the coordinate system R(cid:48) and M (B,B(cid:48))
¯
f
¯
is the matrix associated to the linear transformation f taking in V the basis
B and in V (cid:48) the basis B(cid:48).
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
Example 1
A
Let ( ,V,φ) be an affine space with affine coordinate system R = {O;B =
{e ,e ,e }}, and let (A(cid:48),V (cid:48),φ(cid:48)) be an affine space with affine coordinate sys-
1 2 3
tem R(cid:48) = {O(cid:48);B(cid:48) = {e(cid:48) ,e(cid:48) }}. Is the transformation f : A −→ A(cid:48), f(x,y,z) =
1 2
(x − 2y + 5,x − z + 1) an affine transformation? Give its associated linear
transformation and obtain the matrix associated to f in the coordinate sys-
tems R,R(cid:48).
Solution.
To see if f is an affine transformation we have to check if there exists a
linear transformation f¯: V −→ V (cid:48) such that f(P)f(Q) = f¯(cid:0)PQ(cid:1) for every
A
pair of points P,Q ∈ . We take P(x ,y ,z ) and Q(x ,y ,z ) then
1 1 1 2 2 2
PQ = Q − P = (x − x ,y − y ,z − z )
2 1 2 1 2 1
and
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
f(P)f(Q) = f(Q) − f(P) = f(x ,y ,z ) − f(x ,y ,z )
2 2 2 1 1 1
= (x − 2y + 5,x − z + 1) − (x − 2y + 5,x − z + 1)
2 2 2 2 1 1 1 1
= ((x − x ) − 2(y − y ),(x − x ) − (z − z ))
2 1 2 1 2 1 2 1
¯
= f (x − x ,y − y ,z − z ).
2 1 2 1 2 1
Therefore, f is an affine transformation and its associated linear transfor-
¯
mation is f (x,y,z) = (x − 2y,x − z).
The coordinates of the origin O in the coordinate system R are the cooor-
dinates of the vector OO = (0,0,0) in the basis B where e = (1,0,0) ,
1 B
e = (0,1,0) and e = (0,0,1) .
2 B 3 B
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
Then:
f(O) = f(0,0,0) = (5,1),
¯ ¯
f(e ) = f(1,0,0) = (1,1),
1
¯ ¯
f(e ) = f(0,1,0) = (−2,0),
2
¯ ¯
f(e ) = f(0,0,1) = (0,−1).
3
So,
 
1 0 0 0
M (R,R(cid:48)) = 5 1 −2 0 .
 
f
1 1 0 −1
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
Example 2
A
Let ( ,V,φ) be an affine space with affine coordinate system R = {O;B =
{e ,e }}, and let (A(cid:48),V (cid:48),φ(cid:48)) be an affine space with affine coordinate system
1 2
R(cid:48) = {O(cid:48);B(cid:48) = {e(cid:48) ,e(cid:48) ,e(cid:48) }}. Determine the affine transformation f : A −→
1 2 3
A(cid:48), such that
f(1,2) = (1,2,3),
f¯(e ) = e(cid:48) + 4e(cid:48) ,
1 1 2
f¯(e ) = e(cid:48) − e(cid:48) + e(cid:48) .
2 1 2 3
Find the matrix associated to f in the coordinate systems R,R(cid:48).
Solution.
We know the value of f at the point P(1,2) and the linear transformation
associated to f, therefore we can determine f.
¯
f(1,0) = (1,4,0) ,
B(cid:48)
¯
f(0,1) = (1,−1,1) ,
B(cid:48)
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
To calculate the matrix associated to f we need to compute f(O). We have:
¯ ¯
f(P) = f(O) + f(OP) = f(O) + f(1e + 2e )
1 2
f is affine
¯ ¯
= f(O) + f(e ) + 2f(e )
1 2
¯
f is linear
= f(O) + (1,4,0) + 2(1,−1,1)
so
f(O) = f(1,2) − (1,4,0) − 2(1,−1,1)
= (1,2,3) − (1,4,0) − 2(1,−1,1)
= (−2,0,1).
therefore
 
1 0 0
 −2 1 1 
M (R,R(cid:48)) =  .
f  
0 4 −1
 
1 0 1