Simple conical projection with one standard parallel

Detail

1. The basic idea: a cone resting on the globe

• Conical projection: a map projection where the globe's net of latitudes and longitudes is moved onto a cone. The cone is then cut open and laid flat as a map.
• In the simple form, the cone touches the globe along only one parallel. This is a tangent cone (a cone that just touches the globe along one line and does not cut through it).
• Standard parallel: the parallel where the cone touches the globe.
• Along this line the map and the globe match exactly, so scale is true (the map distance equals the globe distance times the map scale).

• The farther you move from it, north or south, the more the map is distorted (shapes, areas and distances go wrong).

• The cone's tip (apex) lies above the pole, on the extended axis of the globe.

• Simple conical projection: another name for this one-standard-parallel projection. Spacing along the meridians is kept equal, so it is also described as an equidistant conic.

2. Construction: NCERT worked example (Fig. 4.3)

Given: an area from 10°N to 70°N and 10°E to 130°E, scale 1:250,000,000, graticule interval 10°. (A graticule is the network of parallels and meridians on a map.)

Step	What is done	Value
1. Reduced Earth radius (R)	Earth's radius taken as 640,000,000 cm, divided by the scale	R = 640,000,000 ÷ 250,000,000 = 2.56 cm
2. Choose the standard parallel	Middle latitude of 10°N–70°N	40°N
3. Choose the central meridian	Middle longitude of 10°E–130°E	70°E
4. Radius of the standard parallel	Draw a tangent to the small globe at 40°N until it meets the extended polar axis. This length is the slant length of the cone.	R × cot 40° ≈ 3.05 cm
5. Spacing between parallels	The arc distance for 10° of latitude on the small globe is marked along the central meridian, again and again	2πR × 10/360 ≈ 0.45 cm
6. Spacing between meridians	The arc distance for 10° of longitude on the standard parallel is marked along the standard parallel on both sides of 70°E	0.45 × cos 40° ≈ 0.34 cm
7. Draw meridians	Join each mark on the standard parallel to the cone's apex with straight lines	Meridians converge towards the pole

How the drawing builds up:

• Draw the standard parallel first.

• With the cone's apex as centre and a radius of 3.05 cm, draw an arc. This arc is 40°N.

• Add the other parallels.

• Mark points every 0.45 cm along the central meridian (70°E), above and below 40°N.

• With the same centre, draw arcs through these points for 10°N, 20°N … 70°N.

• Add the meridians.

• Mark meridians every 0.34 cm along 40°N, out to 10°E and 130°E.

• Join each mark to the centre with a straight line.

• Why the pole becomes an arc (worked from the same figures):

• The pole is 50° of latitude from 40°N, so along the meridian it sits about 50 × 0.045 ≈ 2.23 cm from the standard parallel.
• The standard parallel's radius is about 3.05 cm, so the pole lies about 0.8 cm short of the apex.
• The pole is therefore drawn as a small arc, not a point.

3. Properties

• Parallels: arcs that are equally spaced and share one common centre (they are concentric). The common centre is the cone's apex.
• Meridians: straight lines that meet at the pole (towards the apex). They cross every parallel at right angles.
• Scale is true:
• along all meridians, because the parallels are spaced at their true arc distance; and

• along the standard parallel only.

• Scale is wrong on every other parallel.

• Every parallel is drawn with the same meridian spacing as on the arc of the cone, so parallels away from the standard parallel come out too long.

• This stretching grows both towards the pole and towards the equator.

• The pole is shown as an arc, not a point (see the worked figures above).

• Neither equal area nor orthomorphic.
• Equal area (homolographic): the map keeps the true size of areas. This projection does not, because areas away from the standard parallel are enlarged.
• Orthomorphic (conformal): the map keeps the true shape of small areas, because scale is the same in every direction at a point. This projection does not: off the standard parallel, scale along the parallel is not equal to scale along the meridian.
• NCERT exercise 1(ii): this projection shows neither area, nor shape, nor direction correctly.

4. Limitations

• Useless for a world map.
• The cone touches only one hemisphere.

• In the opposite hemisphere, parallels lie far from the standard parallel, so distortion becomes extreme.

• Poor for large areas even within one hemisphere.

• A large north–south extent takes the map far from the standard parallel, and the parallels get badly stretched.
• A very wide east–west extent also makes the outer meridians bend away sharply.

5. Uses: narrow belts in the mid-latitudes

• Best fit: areas in the mid-latitudes (between the tropics and the polar circles) that are narrow north–south but long east–west. A strip like this stays close to one standard parallel along its whole length, so it has little distortion.
• Classic examples:

Feature	Country / region	Why it suits
Trans-Siberian Railway	Russia (Moscow to Vladivostok on the Pacific)	Long west–east line in the mid-to-high latitudes of Asia
Canadian Pacific Railway	Canada (Atlantic side to Vancouver on the Pacific)	West–east line across southern Canada
USA–Canada boundary	North America (its western part follows the 49°N parallel)	Boundary runs along roughly one parallel
Narmada valley	India (Madhya Pradesh, Maharashtra, Gujarat; river flows west into the Gulf of Khambhat)	Narrow rift valley that runs west to east, close to the Tropic of Cancer

• Rule for any such map: choose the middle latitude of the strip as the standard parallel and the middle longitude as the central meridian.

6. Beyond NCERT: conics with two standard parallels

• Secant cone: a cone that cuts through the globe along two parallels. This gives two lines of true scale. Distortion between and near them stays low, so it suits a whole mid-latitude country better.
• Survey practice: the US Geological Survey (USGS) once used only the Polyconic projection. It later moved to "several of the more common projections": conformal, equal-area and equidistant types [2].
• John P. Snyder's Map Projections: A Working Manual (USGS Professional Paper 1395, 1987) gives the forward and inverse equations for these projections. It treats the Earth both as a sphere and as an ellipsoid [2].

Projection	Standard parallels	What it keeps true	Typical use
Simple conic (NCERT)	One (tangent cone)	Scale along meridians and the one standard parallel; neither area nor shape	Narrow east–west strips (railways, boundaries, valleys)
Lambert conformal conic	Usually two (secant cone)	Shape (orthomorphic)	Mid-latitude countries, topographic and aeronautical maps
Albers equal-area conic	Usually two (secant cone)	Area (equal area)	Mid-latitude countries for maps of distribution, such as land use or population

Prelims Hooks

• The simple conical projection uses one standard parallel, and scale is true only along it and along all meridians.
• Parallels are equally spaced concentric arcs. Meridians are straight lines that converge and cross the parallels at right angles.
• The pole appears as an arc, not a point. This trap is often confused with the polar zenithal projection, where the pole is a point.
• The projection is neither equal area nor orthomorphic. It shows neither area, nor shape, nor direction correctly (NCERT exercise 1(ii)).
• NCERT example: R = 640,000,000 ÷ 250,000,000 = 2.56 cm. For 10°N–70°N and 10°E–130°E, the standard parallel is 40°N and the central meridian is 70°E.
• The radius of the standard parallel equals the cone's slant length. Meridian spacing is measured along the standard parallel. Parallel spacing is measured along the central meridian.
• Unsuitable for a world map because distortion is extreme in the opposite hemisphere.
• Suitable for: the Trans-Siberian Railway, the Canadian Pacific Railway, the USA–Canada boundary and the Narmada valley. All are narrow north–south and long east–west in the mid-latitudes.
• Tangent cone = one standard parallel. Secant cone = two standard parallels (Lambert conformal conic: shape; Albers: area).
• Map Projections: A Working Manual is USGS Professional Paper 1395 (1987) by John P. Snyder [2].

Mains Points

• Matching the projection to the shape of the area.
• A one-standard-parallel conic suits east–west corridors such as the Trans-Siberian and Canadian Pacific railways, or the Narmada valley.

• Planners of linear projects (railways, pipelines, border fencing, river-basin schemes) get near-true distances along the corridor.

• The trade-off behind distortion.

• Flattening a sphere always distorts something.
• The simple conic keeps distances along meridians and the standard parallel. To do this it gives up area and shape.

• This is why it cannot be used for thematic maps such as those of population density or forest cover.

• Moving from one to two standard parallels.

• For whole mid-latitude countries, a secant conic spreads accuracy over a wider band.
• The Lambert conformal conic keeps shape and suits navigation and topographic sheets. The Albers conic keeps area and suits maps of resource distribution.

• Mapping agencies have moved away from a single all-purpose projection towards projections chosen by purpose [2].

• Geospatial relevance (GS-I maps, GS-III infrastructure). Choosing the right projection is a basic step in GIS work for corridor projects, boundary demarcation and watershed mapping.