1-2 Stretch or length distortion 11
Fig. 1.5. Special map projection of the sphere, called Hammer retroazimuthal projection, centered near
St.Louis (longitude 90
◦ W, latitude 40
◦ N), with shorelines, 15
◦ graticule, two hemispheres, one of which
appears backwards (they should be superimposed for the full map).
1-2 Stretch or length distortion
A second multiplicative measure of deformation: stretch or length distortion, Tissot portrait, simultaneous
diagonalization of two matrices.
The second multiplicative measure of deformation is based upon the scale ratio, which is also called
stretch, dilatation factor, or length distortion. One here distinguishes the left and right stretch:
left stretch:
right stretch:
Λ
2 dS
2 = ds
2 ,
ds
2
dS 2 = Λ
2 =: Λ
2
l ,
λ
2 ds
2 = dS
2 , Λ
2
r := λ
2 =
dS
2
ds 2 ,
subject to duality Λ
2 λ
2 = 1 .
(1.47)
Question.
Question: “What is the role of stretch {Λ
2 , λ
2
} in the context of the pair of (symmetric,
positive-definite) matrices {c MN , G MN }, {C l , G l }, and {C µν , g µν }, {C r , G r }, respectively?”
Answer: “Due to a standard lemma of matrix algebra, both matrices can be simultaneously
diagonalized, one matrix being the unit matrix.”
We briefly outline the simultaneous diagonalization of the positive-definite, symmetric matrices
{C l , G r } and {C r , G l }, respectively, which is based upon a transformation called “Kartenwechsel”:
left “Kartenwechsel”:
right “Kartenwechsel”:
versus
T : V l (U M 2
l
) → ˜
V l (U M 2
l
)
τ : V r (U M 2 r ) → ˜
V r (U M 2 r ) .
(1.48)
The commutative diagram shown in Fig. 1.6 illustrates this “Kartenwechsel”. Let us pay attention to
Theorem 1.1 and Corollary 1.3, and let us present the various transformations in the Boxes 1.2–1.8.
Fig. 1.5. Special map projection of the sphere, called Hammer retroazimuthal projection, centered near
St.Louis (longitude 90
◦ W, latitude 40
◦ N), with shorelines, 15
◦ graticule, two hemispheres, one of which
appears backwards (they should be superimposed for the full map).
1-2 Stretch or length distortion
A second multiplicative measure of deformation: stretch or length distortion, Tissot portrait, simultaneous
diagonalization of two matrices.
The second multiplicative measure of deformation is based upon the scale ratio, which is also called
stretch, dilatation factor, or length distortion. One here distinguishes the left and right stretch:
left stretch:
right stretch:
Λ
2 dS
2 = ds
2 ,
ds
2
dS 2 = Λ
2 =: Λ
2
l ,
λ
2 ds
2 = dS
2 , Λ
2
r := λ
2 =
dS
2
ds 2 ,
subject to duality Λ
2 λ
2 = 1 .
(1.47)
Question.
Question: “What is the role of stretch {Λ
2 , λ
2
} in the context of the pair of (symmetric,
positive-definite) matrices {c MN , G MN }, {C l , G l }, and {C µν , g µν }, {C r , G r }, respectively?”
Answer: “Due to a standard lemma of matrix algebra, both matrices can be simultaneously
diagonalized, one matrix being the unit matrix.”
We briefly outline the simultaneous diagonalization of the positive-definite, symmetric matrices
{C l , G r } and {C r , G l }, respectively, which is based upon a transformation called “Kartenwechsel”:
left “Kartenwechsel”:
right “Kartenwechsel”:
versus
T : V l (U M 2
l
) → ˜
V l (U M 2
l
)
τ : V r (U M 2 r ) → ˜
V r (U M 2 r ) .
(1.48)
The commutative diagram shown in Fig. 1.6 illustrates this “Kartenwechsel”. Let us pay attention to
Theorem 1.1 and Corollary 1.3, and let us present the various transformations in the Boxes 1.2–1.8.