GRB 603 MAP PROJECTIONS EXERCISE 5 Question given

GRB 603: MAP PROJECTIONS EXERCISE 5

Question given: Draw a graticule of the area extending from 30◦W to 60◦E and 60◦N to 60◦S on Gnomonic Equatorial Zenithal Projection with R. F. 1: 200, 000 and interval 15◦

First, let us understand what is Gnomonic Equatorial Zenithal Projection. Lets go backwards: Zenithal: developable surface is tangent to the globe Equatorial: the developable surface is tangent at the equator Other two types are: Cyllindrical: developable surface is in the form of a cyllinder Conical: developable surface is in the form of a cone Other two types are: Polar: developable surface is tangent at the pole Oblique: developable surface is tangent anywhere between the equator and the pole.

Gnomonic: the source of light is at the centre of the earth The other two types are: Stereographic: the source of light is at the pole Orthographic: the source of light is at infinity Gnomonic Equatorial Zenithal Projection Developable surface Equator Source of light Meridians (all great circles) are represented as straight lines on the projection, parallels are curved lines convex to the equator, which is the only parallel that is represented as a straight line, it is also a great circle.

Let us now start construction: Step I: Radius of the reduced earth (r) = Actual radius of the Earth Denominator of the given R. F. 640, 000 ₌ 3. 2 cms 200, 000 Step II: Distance of meridians from the Central Meridian on the equator (d 1) = r. tan θ , where, θ = angular distance of the concerned meridian from the central meridian Meridians to be drawn are: 30◦ W, 15◦ W, 0◦, 15◦ E, 30◦ E, 45◦ E and 60◦ E Central Meridian 30◦ W 15◦ W Angular distance is =15◦ Angular distance is =30◦ Angular distance is =45◦ 0◦ 15◦ E 30◦ E 45◦ E 60◦ E Angular distance is =15◦ Angular distance is =30◦ Angular distance is =45◦

meridian θ tan θ d 1 = r. tan θ (cms) 30◦ W 45◦ 1 3. 20 15◦ W 30◦ 0. 58 1. 86 0◦ 15◦ 0. 27 0. 86 CENTRAL MERIDIAN 15◦ E 30◦ E 15◦ 0. 27 0. 86 45◦ E 30◦ 0. 58 1. 86 60◦ E 45◦ 1 3. 20 Step III: Distance of each parallel from the equator along each meridian (d 2) = r. sec θ. tan ϕ where, θ = angular distance of the concerned meridian from the central meridian ϕ = angular distance of the concerned parallel from the equator, - which is basically equals to its own value, since equator = 0◦ -this will always be true since in this projection the developable surface is tangent at the equator. - Each parallel that we draw will cut each meridian at a different distance from the equator

Parallels to be drawn are: 15◦ N and S, 30◦ N and S, 45◦N and S, 60◦ N and S and Equator Drawing 15◦N and S: Angular distance of 15◦N and S from the equator = 15◦ , therefore, ϕ = 15◦ , therefore, tan ϕ = tan 15◦ = 0. 27 15◦N and S parallels will cut through all the meridians, therefore, we need to find the point where it will cut each meridians θ sec θ tan ϕ = tan 15◦ r d 2 = r. sec θ. tan ϕ (cms) 30◦ W 45◦ 1. 41 0. 27 3. 2 1. 22 15◦ W 30◦ 1. 15 0. 27 3. 2 0. 99 0◦ 15◦ 1. 04 0. 27 3. 2 0. 90 15◦ E 0◦ (distance 1 0. 27 3. 2 0. 86 with itself = 0◦) 30◦ E 15◦ 1. 04 0. 27 3. 2 0. 90 45◦ E 30◦ 1. 15 0. 27 3. 2 0. 99 60◦ E 45◦ 1. 41 0. 27 3. 2 1. 22 In this way, we have to calculate for each set of parallels in both the hemispheres.

Drawing 30◦N and S: Angular distance of 30◦N and S from the equator = 30◦ , therefore, ϕ = 30◦ , therefore, tan ϕ = tan 30◦ = 0. 58 30◦N and S parallels will cut through all the meridians, therefore, we need to find the point where it will cut each meridians θ sec θ tan ϕ = tan 30◦ r d 2 = r. sec θ. tan ϕ (cms) 30◦ W 45◦ 1. 41 0. 58 3. 2 2. 62 15◦ W 30◦ 1. 15 0. 58 3. 2 2. 13 0◦ 15◦ 1. 04 0. 58 3. 2 1. 93 15◦ E 0◦ 1 0. 58 3. 2 1. 86 30◦ E 15◦ 1. 04 0. 58 3. 2 1. 93 45◦ E 30◦ 1. 15 0. 58 3. 2 2. 13 60◦ E 45◦ 1. 41 0. 58 3. 2 2. 62

Drawing 45◦N and S: Angular distance of 45◦N and S from the equator = 45◦ , therefore, ϕ = 45◦ , therefore, tan ϕ = tan 45◦ = 1 45◦N and S parallels will cut through all the meridians, therefore, we need to find the point where it will cut each meridians θ sec θ tan ϕ = tan 45◦ r d 2 = r. sec θ. tan ϕ (cms) 30◦ W 45◦ 1. 41 1 3. 2 4. 51 15◦ W 30◦ 1. 15 1 3. 2 3. 68 0◦ 15◦ 1. 04 1 3. 2 3. 33 15◦ E 0◦ 1 1 3. 2 30◦ E 15◦ 1. 04 1 3. 2 3. 33 45◦ E 30◦ 1. 15 1 3. 2 3. 68 60◦ E 45◦ 1. 41 1 3. 2 4. 51

Drawing 60◦N and S: Angular distance of 60◦N and S from the equator = 60◦ , therefore, ϕ = 60◦ , therefore, tan ϕ = tan 60◦ = 1. 73 60◦N and S parallels will cut through all the meridians, therefore, we need to find the point where it will cut each meridians θ sec θ tan ϕ = tan 60◦ r d 2 = r. sec θ. tan ϕ (cms) 30◦ W 45◦ 1. 41 1. 73 3. 2 7. 80 15◦ W 30◦ 1. 15 1. 73 3. 2 6. 31 0◦ 15◦ 1. 04 1. 73 3. 2 5. 75 15◦ E 0◦ 1 1. 73 3. 2 5. 53 30◦ E 15◦ 1. 04 1. 73 3. 2 5. 75 45◦ E 30◦ 1. 15 1. 73 3. 2 6. 31 60◦ E 45◦ 1. 41 1. 73 3. 2 7. 80

Let us now try drawing the graticules: 30◦ W 15◦ W 0◦ 15◦ E 30◦ E 45◦ E 60◦ N Step 1. : Draw a vertical line of any length Step 2: Draw a perpendicular through its centre of any length 45◦ N Step 3: Measure and mark d 1 on each side of the central meridian Step 4: Draw the meridians (don’t bother about their length) 30◦ N Step 5: Label the meridians Step 5: Measure and mark d 2 for 15◦ N and S on 30◦ W and 60◦ E = 1. 22 Step 6: Measure and mark d 2 for 15◦ N and S on 15◦ W and 45◦ E = 0. 99 Step 7: Measure and mark d 2 for 15◦ N and S on 0◦ and 30◦ E = 0. 90 3. 2 0. 86 15◦ N 0◦ 15◦ S Step 8: Measure and mark d 2 for 15◦ N and S on the central meridian, 15◦ E = 0. 86 30◦ S Step 9: Join the points with curved lines. 45◦ S Step 10: Similarly, draw all the latitudes and label them. 60◦ S

GNOMONIC EQUATORIAL ZENITHAL PROJECTION • Erase extra lines • Draw a neat border 60◦ E 30◦ W EXTENT: 30◦W to 60◦E 60◦N to 60◦S INTERVAL: 15◦ 60◦ N 45◦ E 15◦ W 0◦ 15◦ E 30◦ E 45◦ N • Write down the R. F, title and other details • Please try this one and you can share pictures of your projections on the group. • It requires only pencil and scale. • Try using a log table • Try not to use a calculator • Watch your time. 30◦ N • Try this question as well: 15◦ S 15◦ N 0◦ Draw a graticule of the area extending from 40◦W to 60◦E and 50◦N to 50◦S on Gnomonic Equatorial Zenithal Projection with R. F. 1: 160, 000 and interval 10◦ 30◦ S 45◦ S Read from: • An Introduction to Map Projections, J. A. Steers • Map Projections, George P. Kellaway 60◦ S R. F. : 1: 200, 000