From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 98

C l G − 1

2 W l

E r G − 1

2 W r :=

:=

l r

versus

:= tr[(E l G − l ) T W l (E l G − l )] := tr [(E r G − r ) T W r (E r G − r )] ,

(1.309)

E l G − 1

2 W l

E r G − 1

2 W r

= =

l r

versus

=(vec[E l G − l ]) T W l (vec[E l G − l ]) = (vec[E r G − r ]) T W r (vec[E r G − r ]) .

Box 1.49 (Cauchy-Green distortion energy, Euler-Lagrange distortion energy).

(i) Cauchy-Green distortion energy:

2 d S l tr[C l G − l ]= 2 d S r tr[C r G − r ]=

1 1

(1st) versus

= 2 d S l ( Λ 1 + Λ 2 ) = 2 d S r ( λ 1 + λ 2 );

d S l det[C l G − l ]= d S r tr[C r G − r ]=

(1.310)

(2nd) versus

= d S l Λ 1 Λ 2 = d S 1 λ 1 λ 2 ;

d S l (ln Λ 1 +ln Λ 2 ) d S r (ln λ 1 +ln λ 2 );

(3rd) versus

(4th) d S l tr[(C l G − l ) T W l (C l G − l )] := versus d S r tr[(C r G − r ) T W r (C r G − r )] :=
:=

C l G − 1

2
W l

C r G − 1

2
W r

||| ||| := ||| ||| .

l r

(ii) Euler-Lagrange distortion energy:

d S l tr[E l G − l ]= d S r tr[E r G − r ]=

1
2

1
2

d S l ( K 1 + K 2 )= 1
2

d S r ( κ 1 + κ 2 );

= 1
2

(1st) versus

d S l det[E l G − l ]= d S r det[E r G − r ]=

= d S l K 1 K 2 = d S r √ κ 1 κ 2 ; (2nd) versus (1.311)

d S l (ln K 1 +ln K 2 ) sus 1
2

d S r (ln κ 1 +ln κ 2 );

1
2

(3rd)

d S l tr[(E l G − l ) T W l (E l G − l )] := d S r tr[(E r G − r ) T W r (E r G − r )] :=

1
2

1
2

:= ||| E l G l − 1

2
W l

:= ||| E 1 G − 1

2

||| |||

Wr .

(4th) versus

1