1-7 Angular shear 39
˙
X 1, ˙
X 2 ∈ T U 0 M
2
l
˙
x 1, ˙
x 2 ∈ T u 0 M
2
r
˙
X 1 =
∂X
∂U M
˙
U
M
1
˙
X 2 =
∂X
∂U N
˙
U
N
2
˙
x 1 =
∂x
∂u µ ˙
u
µ
1
˙
x 2 =
∂x
∂u ν ˙
u
ν
2
Ψ l
Ψ r
U 0
u0
Fig. 1.19. Angular measure of deformation, left and right shear.
Example 1.8 (Angular shear or angular distortion, f : E
2
A 1 ,A 1 ,A 2
→ S
2
r ).
Let us tak e reference to Example 1.3, where we analyze the isoparametric mapping f = id from an
ellipsoid-of-revolution M
2
l = E
2
A 1 ,A 1 ,A 2
to a sphere M
2
r = S
2
r . Here, we shall continue the analysis by
computing angular shear or angular distortion of two parameterized curves in M
2
l = E
2
A 1 ,A 1 ,A 2
as well
as their images in M
2
r = S
2
r .
Left surface, parameterized curves:
R ight surface, parameterized curves:
( i)parallel circles:
(
i)parallel circles:
U
1 = Λ = t l , U
2 = Φ = constant ;
u
1 = λ = t r , u
2 = φ = constant ;
( 1.135 )
( ii)meridians:
( ii)meridians:
U
1 = Λ = constant , U
2 = Φ = t l .
u
1 = λ = constant , u
2 = φ = t r .
( 1.136 )
U 1 (t l )=
Λ(t l )
Φ(t l )
=
t l
constant
,
t r
constant
=
λ(t r )
φ(t r )
= u 1 (t r ) ,
U 2 (t l )=
Λ(t l )
Φ(t l )
=
constant
t l
.
constant
t r
=
λ(t r )
φ(t r )
= u 2 (t r ) .
( 1.137 )
End of Example.
Φ
φ
Λ
λ
Fig. 1.20. Angular shear, isoparametric mapping E
2
A 1 ,A 1 ,A 2 → S
2
r , left and right parameterized curves of type
{ellipsoidal parallel circle, ellipsoidal meridian} and {spherical parallel circle, spherical meridian}.