From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 15

Box 1.3 (Left Tissot circle versus left Tissot ellipse, left Cauchy-Green deformation tensor:
Ricci calculus).

1 : 2

λ 1 ,λ 2

:
d S 2 = G MN U A U B d V A d V B =d s 2 = g μν u ν M u ν N U A U B d V A d V B =

Left Tissot circle S Left Tissot ellipse E

(1.50)

= δ AB d V A d V B = = Λ 1 (d V 1 ) 2 + Λ 2 (d V 2 ) 2 =

=(d V 1 ) 2 +(d V 2 ) 2 = Ω 1 + Ω 2 . = Ω 1 /λ 1 + Ω 2 /λ 2 .

Box 1.4 (Left Tissot circle versus left Tissot ellipse, left Cauchy-Green deformation tensor:
Cayley calculus).

1 : 1

λ 1 ,λ 2

Left Tissot circle S Left Tissot ellipse S :

d S 2 =

T F l G l F l Ω

d s 2 =

T F l C l F l Ω

Ω = Ω = (1.51)

T T diag( Λ 1 ,Λ 2 )

= Ω Ω ⇐⇒ = Ω Ω ⇐⇒

⇐⇒ F l G l F l =I . ⇐⇒ F l C l F l =diag( Λ 1 ,Λ 2 )=diag(1 /λ 1 , 1 /λ 2 ) .

Box 1.5 (The right Tissot ellipse versus the right Tissot circle, right Cauchy-Green deforma-
tion tensor: Ricci calculus).

Λ 1 ,A 2 1 :

Right Tissot ellipse E : Right Tissot circle S

d S 2 = G MN U μ U ν u α u β d v α d v β = d s 2 = g μν u α u β d v α d v β = (1.52)

=(d v 1 ) 2 /Λ 1 +(d v 2 ) 2 /Λ 2 = = δ αβ d v α d v β =

= λ 1 ω 1 + λ 2 ω 2 . =(d v 1 ) 2 +(d v 2 ) 2 = ω 1 + ω 2 .

Box 1.6 (The right Tissot ellipse versus the right Tissot circle, right Cauchy-Green deforma-
tion tensor: Cayley calculus).

1
Λ 1 ,A 2

1 :

Right Tissot ellipse E : Right Tissot circle S

d S 2 =

T F T

d s 2 =

T F T

ω C r F r ω = ω G r F r ω =

r r

T diag( λ 1 ,λ 2 ) T

= ω ω = ω ω (1.53)

⇐⇒ ⇐⇒

F T C r F r =diag( λ 1 ,λ 2 )=diag(1 /Λ 1 , 1 /Λ 2 ) . F T

G r F r =I .

r r