Simple Conical Projection with one standard parallel

Introduction:

A simple circular cone touches the generating globe along a particular parallel on which parallel any kind of distortions is null. This is known as the standard parallel. Simple Conical Projection with one standard parallel is a perspective projection in which the parallels and meridians are projected directly on the inner surface of the cone with respect to a light source at the center of the generating globe.

Theory:

The radius of the generating globe, R = Actual radius of the earth ÷ Denominator of the R.F.

The division of the central meridian for spacing the parallel at i⁰ interval,

                                                 d₁=(πR/180⁰) ×i⁰

The radius of the standard parallel (ø₀₎, r_{0 =}Rcotø₀

The division on the standard parallel for spacing the meridians at i⁰ interval, d₂= (2π R cosϴ÷360⁰) ×i⁰

Properties:

1. Genetically it is a perspective projection. The parallels are concentric arcs of circles truly spaced on the central meridian.
2. Poles are also represented by arcs in this projection.
3. Radial scale is true along all the meridians.
4. Meridians are straight lines truly spaced on the standard parallel and converging at the vertex of the cone.
5. Tangential scale is true along the standard parallel only.
6. Deformation is positive towards the equator and negative deformation towards the pole.
7. It is an aphylactic projection, i.e., one that maintains neither area nor shape
8. It is suitable for smaller countries of mid-latitude or temperate regions.

Derivations:

There are three main formulae are used in this projection. The derivation of the formulae is as below.

EXAMPLE:1

Draw graticules on simple conical projection with one standard parallel for an area between 140⁰to 150⁰W latitudes & 20⁰S to 70⁰S longitudes at an interval of 10⁰; R.F.-1:80,000,000.

STEP-1:  Radius of the reduced earth (R) :

                               Earth’s actual radius/Denominator of the R.F.

                           R= 640,000,000/80,000,000

                              =8cm.

STEP-2:  Determination of central meridian and standard parallel:

CENTRAL MERIDIAN

140⁰ E, 150⁰E, 160⁰E, 170⁰E,  I    180⁰E, 170⁰W, 160⁰W, 150⁰W

                   Central Meridian = 175⁰E

STANDARD PARALLEL

20⁰S, 30⁰S, 40⁰S,    I    50⁰S,  60⁰S, 70⁰S

       Standard Parallel= 45⁰S

STEP-3:  Radius of the standard parallel:

                             Rcotø

                          =8cot45⁰

STEP-4:  Distance along the central meridian for placing the parallel:

                              (πR× Interval)÷180⁰

                           =(π×8×10⁰)÷180⁰

                            =1.396cm

STEP -5:  Distance along the standard parallel for spacing the meridians:

                                                                  (2π R cosø× Interval) ÷360⁰

                                                                =(2π8cos45⁰×10⁰)÷360⁰

                                                                =0.987cm

Exercise:1. Draw graticules on simple conical projection with one standard parallel for an area between 15⁰N -75⁰N longitudes & 125⁰E-145⁰W latitudes at an interval of 15 degrees with the R.F.-1: 85,000,000.

Example:2.  Draw graticules on simple conical projection with one standard parallel for an area between 40⁰N-70⁰N longitudes & 10⁰W-20⁰E latitudes at an interval of 5 degrees with the R.F.-1: 56,000,000.

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