Polar Zenithal Stereographic Projection – In Details

Introduction

In Polar Zenithal Stereographic Projection, a 2-dimensional plane of projection touches the generating globe at either of the poles. It is a perspective projection, with the source of light lying at the pole diametrically opposite to one at which the projection plane touches the generating globe. the parallels are projected as concentric circles of varying radius while the meridians are projected as straight lines radiating from the poles.

THEORY:

• Radius of the generating globe, R= Actual radius of the earth / Denominator of R.F.
• Radius of any parallel(Ø),

r_Ø = 2R.tan [(90-Ø)/2

PROPERTIES:

• Parallels are represented by concentric circles of varying radius.
• Inter-parallel distance gradually increases toward the equator.
• Meridians are straight lines radiating from the poles at true azimuth apart.
• The direction between any two points is preserved.
• At any point, the radial scale is equal to the tangential scale.
• It is an orthomorphic projection, i.e., the shape of a map is truly preserved.
• It is commonly used for the map of the world in hemispheres.

EXAMPLE:1

Draw graticules from the northern hemisphere on polar zenithal stereographic projection at an interval of 15⁰ on a scale of 1:150,000,000.

STEP-1:  Radius of the reduced earth:

R=640,000,000 cm/150,000,000

=4.26666cm

STEP-2:  Distance of the parallel from the pole: 2R tan[(90⁰-Ø)/2]

Ø	(90⁰-Ø)/2	tan(90⁰-Ø)/2	R (in cm)	2Rtan(90⁰-Ø)/2 (in cm)
0⁰	45	1	4.26666	8.53332
15⁰	37.5	0.76732	4.26666	6.54778
30⁰	30	0.57735	4.26666	4.92671
45⁰	22.5	0.41421	4.26666	3.53458
60⁰	15	0.26794	4.26666	2.28641
75⁰	7.5	0.13165	4.26666	1.12341
90⁰	0	0	4.26666	0

EXAMPLE:2

Draw graticules on polar zenithal stereographic projection at an interval of 20⁰ extended from 40⁰E-160⁰E & 20⁰S-80⁰S when the distance of 20⁰ S parallels from the pole is 8 cm, calculate the R.F.

STEP-1:

2R tan( 90 degree – 20 degree)/2=8cm

R= 8cm / [2 tan {( 90 degree – 20 degree)/2}]

R= 5.71cm

STEP-2: Determination of R.F.:     640,000,00 cm / 5.71cm

                                                         =112,084,063

STEP-3: Distance of the parallel from the pole : [2R tan (90 degree – theta)/2

Ø	( (90 degree – theta)/2 )	R(in cm)	2R tan ( (90 degree – theta)/2 ) (in cm)
20⁰	35	5.71	7.22
40⁰	25	5.71	5.32
60⁰	15	5.71	3.06
80⁰	05	5.71	0.999

EXERCISE:2

               Draw graticules on polar zenithal stereographic projection at an interval of 10⁰ extended from 140⁰E-130⁰W & 15⁰S-85⁰S when the distance of 25⁰ S parallels from the pole is 7.5 cm, calculate the R.F.

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