Gnomonic projection tangent to equator
Gnomonic projection tangent to equator
1. Select the tangent point
flowchart LR
A["Globe"] --> B["Equator"]
B --> C["Select central meridian 40°E"]
C --> D["Tangent point T at Equator ∩ 40°E"]
D --> E["Projection plane tangent at T"]
2. Establish the tangent plane
flowchart TB
O((O<br/>Earth centre))
T((T<br/>tangent point))
P["Tangent plane"]
O ---|"Radius OT"| T
T ---|"OT perpendicular to plane"| P
Note["The projection plane touches the globe only at T"]
P -.-> Note
A more geometrical side view is:
flowchart LR
A["Globe surface"] --> B["T"]
B --> C["Tangent plane"]
O((O)) ---|"OT ⟂ tangent plane"| B
3. Mark the globe meridians
flowchart TB
E["Equator"]
M1["20°E"]
M2["30°E"]
M3["40°E"]
M4["50°E"]
M5["60°E"]
E --> M1
E --> M2
E --> M3
E --> M4
E --> M5
M3 --> C["Central meridian"]
C --> T["Tangent point at Equator"]
The meridian arrangement is:
flowchart LR
A["20°E"] --- B["30°E"] --- C["40°E"] --- D["50°E"] --- E["60°E"]
C --- F["Central meridian and tangent point"]
4. Project the equator
flowchart LR
G["Point on globe equator"] --> O((O<br/>centre))
O --> P["Intersection with tangent plane"]
P --> Q["Transfer to plan"]
Q --> R["Projected equator"]
For several equatorial points:
flowchart TB
E1["Equator ∩ 20°E"] --> O1((O))
E2["Equator ∩ 30°E"] --> O2((O))
E3["Equator ∩ 40°E"] --> O3((O))
E4["Equator ∩ 50°E"] --> O4((O))
E5["Equator ∩ 60°E"] --> O5((O))
O1 --> P1["Tangent plane"]
O2 --> P2["Tangent plane"]
O3 --> P3["Tangent plane"]
O4 --> P4["Tangent plane"]
O5 --> P5["Tangent plane"]
P1 --> L["Join transferred points"]
P2 --> L
P3 --> L
P4 --> L
P5 --> L
L --> R["Straight projected equator"]
5. Construct the projected meridians
flowchart LR
A["Choose globe meridian"] --> B["Mark its equatorial point"]
B --> C["Join point to globe centre O"]
C --> D["Extend line to tangent plane"]
D --> E["Transfer intersection to plan"]
E --> F["Draw perpendicular to projected equator"]
F --> G["Projected meridian"]
For the complete set:
flowchart TB
A["Projected equator"]
M20["20°E point"] --> L20["Perpendicular line"]
M30["30°E point"] --> L30["Perpendicular line"]
M40["40°E point"] --> L40["Perpendicular line"]
M50["50°E point"] --> L50["Perpendicular line"]
M60["60°E point"] --> L60["Perpendicular line"]
A --> M20
A --> M30
A --> M40
A --> M50
A --> M60
L20 --> R["Five projected meridians"]
L30 --> R
L40 --> R
L50 --> R
L60 --> R
The central meridian is identified separately:
flowchart TB
A["Projected equator"]
B["40°E equatorial point"]
C["Draw perpendicular"]
D["Central meridian"]
E["Tangent point T"]
A --> B --> C --> D
D --> E
6. Project the parallel
flowchart LR
A["Select intersection of 30°N and a meridian"] --> B["Join point to globe centre O"]
B --> C["Extend ray to tangent plane"]
C --> D["Transfer projected point to plan"]
D --> E["Repeat for every meridian"]
E --> F["Join points with smooth curve"]
F --> G["Projected 30°N parallel"]
For the five required meridians:
flowchart TB
N20["30°N ∩ 20°E"] --> O20((O))
N30["30°N ∩ 30°E"] --> O30((O))
N40["30°N ∩ 40°E"] --> O40((O))
N50["30°N ∩ 50°E"] --> O50((O))
N60["30°N ∩ 60°E"] --> O60((O))
O20 --> P20["Projected point"]
O30 --> P30["Projected point"]
O40 --> P40["Projected point"]
O50 --> P50["Projected point"]
O60 --> P60["Projected point"]
P20 --> C["Smooth connecting curve"]
P30 --> C
P40 --> C
P50 --> C
P60 --> C
C --> R["30°N projected parallel"]
7. Project the parallel
flowchart LR
A["Select intersection of 30°S and a meridian"] --> B["Join point to globe centre O"]
B --> C["Extend ray to tangent plane"]
C --> D["Transfer projected point to plan"]
D --> E["Repeat for every meridian"]
E --> F["Join points with smooth curve"]
F --> G["Projected 30°S parallel"]
flowchart TB
S20["30°S ∩ 20°E"] --> O20((O))
S30["30°S ∩ 30°E"] --> O30((O))
S40["30°S ∩ 40°E"] --> O40((O))
S50["30°S ∩ 50°E"] --> O50((O))
S60["30°S ∩ 60°E"] --> O60((O))
O20 --> P20["Projected point"]
O30 --> P30["Projected point"]
O40 --> P40["Projected point"]
O50 --> P50["Projected point"]
O60 --> P60["Projected point"]
P20 --> C["Smooth connecting curve"]
P30 --> C
P40 --> C
P50 --> C
P60 --> C
C --> R["30°S projected parallel"]
8. Descriptive-geometry transfer
flowchart LR
A["Globe plan"] --> B["Project point vertically to elevation"]
B --> C["Locate corresponding point in elevation"]
C --> D["Join point to O"]
D --> E["Intersect tangent plane"]
E --> F["Project intersection back to plan"]
F --> G["Final map point"]
For each graticule intersection:
flowchart TB
A["Intersection on globe"]
B["Plan view"]
C["Elevation view"]
D["Line from O"]
E["Tangent-plane intersection"]
F["Transferred point on map"]
A --> B
B --> C
C --> D
D --> E
E --> F
9. Final construction sequence
flowchart TD
A["Choose tangent point at 40°E on Equator"]
B["Draw tangent plane"]
C["Mark globe meridians 20°E–60°E"]
D["Project equatorial points"]
E["Draw five projected meridians"]
F["Mark 30°N intersections on globe"]
G["Project 30°N points"]
H["Join them smoothly"]
I["Mark 30°S intersections on globe"]
J["Project 30°S points"]
K["Join them smoothly"]
L["Complete equatorial gnomonic graticule"]
A --> B --> C --> D --> E
E --> F --> G --> H
H --> I --> J --> K
K --> L
10. Final schematic graticule
This is only a conceptual Mermaid representation; it is not geometrically to scale.
flowchart TB
N["30°N curved parallel"]
E["Equator"]
S["30°S curved parallel"]
M20["20°E"] --- M30["30°E"] --- M40["40°E"] --- M50["50°E"] --- M60["60°E"]
N --- E --- S
M20 --- M30 --- M40 --- M50 --- M60
M40 --> C["Central meridian"]
C --> T["Tangent point at 40°E"]
The operational rule represented by all the diagrams is:
Select a point on the globe, join it to the globe’s centre, extend that line to the tangent plane, and transfer the intersection to the map. Repeat for every required graticule intersection.