®
™
info@wearality.com
The Panorama Lens
™ in Wearality Sky represents five years of research optimizing wide field of view (FOV) optics for head-mounted displays (HMDs) in visual simulation and training. This patented and patent pending approach uses a coordinated pair of spherical-section Fresnel lenses to wrap input images around each eye, creating an immersive, wide field of view output image with broad focus, low chromatic aberration, and high brightness. Its telecentric arrangement allows use of prescription eyewear and supports personal variation in interpupillary distance (IPD) without adjustment.
The vision dome created by the Panorama Lens requires careful attention in measurement. Output field of view (light from the near lens that defines the eye’s field of view) must not be confused with the input extent, which is the greatest physical area that can be seen through the lens system from the eyepoint. The input extent defines the useful display screen size rather than the angular field of view over which that screen is seen by the human eye.
The 60mm lens in Wearality Sky outputs 130° horizontally for each eye using one half of a shared display, making it well-suited for mobile phone applications. The ultimate single-eye output FOV would be the full
180°–190° of the human visual system with an input extent closely matching a small display device screen.
The larger 90mm Wearality Panorama Lens outputs this uniquely wide 180° field using a tangential display device at each lens. Marginal input rays at the bottom of Figures
1a
and
1b
show how the Panorama Lens closely approaches this ideal.
This report details the output field of view of the Panorama Lens used in Wearality Sky by illustrating the physical geometry, reviewing key equations, systematically deriving view angles, and considering variations.
The values presented here are consistent with optical bench analysis of production units.
The design eyepoint of the Panorama Lens system in Wearality Sky is 18mm from the center of the near lens.
The outer 3mm, inner 2.8mm, upper 2.9mm and, lower 3.2mm are unused due to lens frame shadowing, with the remaining 55.6mm
×
50.1mm area forming the output image viewing geometry as shown in Figure
1 .
1

2
(a) Horizontal
(b) Vertical
Figure 1: Cross sections through the Wearality Sky Panorama Lenses
Single-eye partial view angles are readily determined from the measured dimensions using the right triangle identity,
θ = tan
−
1
( r / x )
, as shown in Figure
2
. Resulting primary view angles are shown in Figure
4 .
• Inward horizontal field of view with r
I
= 21 .
2 mm and x
I
= 14 .
8 mm:
θ
I
= arctan r
I x
I
= arctan
21 .
2
14 .
8
≈
55 .
0806
°
• Outward horizontal field of view with r
O
= 34 .
4 mm and x
O
= 9 .
0 mm: r
O
θ
O
= arctan x
O
= arctan
34 .
4
9 .
0
≈
75 .
3385
°
• Upward vertical field of view with r
U
= 26 .
5 mm and x
U
= 12 .
9 mm: r
U
θ
U
= arctan x
U
= arctan
26 .
5
12 .
9
≈
64 .
0435
°
• Downward vertical field of view with r
D
= 23 .
6 mm and x
D
= 14 .
0 mm: r
D
θ
D
= arctan x
D
= arctan
23 .
6
14 .
0
≈
59 .
3227
°
Single-eye total view angles combine the corresponding partial vew angles:
• Single-eye horizontal field of view spans inward and outward angles:
θ
H
= θ
I
+ θ
O
≈
130 .
4191
°
• Single-eye vertical field of view spans upward and downward angles:
θ
V
= θ
U
+ θ
D
≈
123 .
3663
°

3
Figure 2: Radial view angle Figure 3: Diagonal angle
θ
Figure 4: Single eye principal angles Figure 5: Single eye diagonal angle
Diagonal field of view is the great circle angle between points at the “corners” of the visible range. This angle is defined by the spherical law of cosines, which expresses the great circle distance
θ between Point 1
( ϕ
1
, λ
1
) and Point 2
( ϕ
2
, λ
2
) in terms of their latitude and longitude, as shown in Figure
3
.
θ = arccos
( sin ϕ
1 sin ϕ
2
+ cos ϕ
1 cos ϕ
2 cos
|
λ
1
−
λ
2
|
)
(
The single-eye diagonal view angle is the arc from
(
−
θ
D
θ
U
,
−
θ
I
) to
(
−
θ
D
, θ
O
) as highlighted in white in Figure
,
5 .
−
θ
I
) to
( θ
U
, θ
O
)
, or for the less fortunate, from
θ = arccos
( sin
−
= arccos
( sin
θ
U
θ
D
= arccos
(
− sin
θ
U
≈
156 .
6479
° sin
θ sin
−
θ sin
θ
U
D
D
+ cos
−
θ
+ cos
θ
U
D cos
θ
U cos
−
θ
D cos
|−
θ
I cos
|−
θ
I
+ cos
θ
U cos
θ
D cos
|−
θ
I
−
θ
−
θ
O
O
−
θ
O
|
)
|
)
|
)

4
With both eyes looking straight ahead to infinity, the total horizontal FOV is understood as the sum of the outer angles; left of the left eye and right of the right eye as detailed in Figures
6
and
7
, with the corresponding diagonal field of view shown as the white sector in Figure
8
.
• Total horizontal field of view spans left and right outward angles:
θ
H
= θ
O
+ θ
O
≈
150 .
6769
°
• Total vertical field of view spans upward and downward angles:
θ
V
= θ
U
+ θ
D
≈
123 .
3663
°
• Total diagonal field of view is the arc from
(
−
θ
D
,
−
θ
O
) to
( θ
U
, θ
O
)
:
θ = arccos
( sin
−
θ
D
≈
165 .
4652
° sin
θ
U
+ cos
−
θ
D cos
θ
U cos
|−
θ
O
−
θ
O
|
)
Figure 6: Dual eye total horizontal angle
Figure 7: Dual eye principal angles Figure 8: Dual eye diagonal angle

5
The output field of view of Wearality Sky is influenced by human factors including interpupillary distance, eye rotation, and distance from eye to lens. As shown in Figure
9
, nearer is wider and farther is narrower, though diagonal FOV remains greater than
150
° out to an eyepoint offset of 6.4891mm.
Perceived FOV may also be limited by input-side factors of image size and formation.
Figure 9: Total diagonal, horizontal, and Vertical FOV by offset (in mm) from design eyepoint
Wearality Sky single-eye partial output field of view exceeds:
• 75.3° outward and 55.0° inward horizontally
• 64.0° upward and 59.3° downward vertically
Wearality Sky single-eye total output field of view exceeds:
• 130.3° horizontally
• 123.3° vertically
• 156.6° diagonally
Wearality Sky dual-eye total output field of view exceeds:
• 150.6° horizontally
• 123.3° vertically
• 165.4° diagonally

6
Wearality’s unique Panorama Lens is a notable advance in the design and manufacture of optical systems for immersive applications. Figure
10
is a frame from time 4:44 in a customer video showing that the wide field of view optics of Wearality Sky significantly exceed the focus, chromatic aberration, and field of view qualities of well-known, expensive products of classical design.
Figure 10: Customer comparison of common HMD lens (bottom) and Wearality Sky lens (top)