Maps Projections PDF

Shape of the Earth -- Geodesy Geodesy: the branch of mathematics dealing with the shape and area of the earth or large portions of it. A group of earths with different continents Description automatically generated Spheroid: The most accurate geometric shape of Earth, where it is slightly flatten...

Shape of the Earth -- Geodesy Geodesy: the branch of mathematics dealing with the shape and area of the earth or large portions of it. A group of earths with different continents Description automatically generated Spheroid: The most accurate geometric shape of Earth, where it is slightly flattened at the poles and bulging at the equator due to its rotation. Ellipsoid: A simplified mathematical model of the Earth that approximates its shape as an ellipsoid, useful for various calculations in geodesy. Earth's shape is usually considered on oblate (ellipsoid) when mapping Geoid: A more complex shape that represents the Earth\'s mean sea level and gravitational field. It accounts for variations in topography and density, making it irregular. Maps Projections Maps are used to represent the 3D earth - A cartographical map projection is a formal process which converts (mathematically and graphically) map features from a sphere or ellipsoid surface onto flat projection surface. - Some map projection designs are for special/specific purposes - Map projection will have distortion Equidistant Projection -- lengths of specified lines (often Meridians) are not distorted Refers to a property of certain types of projections where distances are preserved from the center of the projection to any other point on the map. This means that if you measure the distance between two points on the map, it will correspond accurately to the distance on the Earth\'s surface along a straight line from the center point. There are different types of equidistant projections, such as: 1. Equidistant Conic Projection: Preserves distances along certain lines, making it useful for mapping areas with a primarily east-west orientation. 2. Equidistant Cylindrical Projection: Maintains equal distances along the equator, but distortion increases as you move toward the poles. Equidistant projections are particularly useful for applications like air distance calculations or for specific geographic areas where accurate distance measurement is essential. However, they may distort shapes and areas, so they aren\'t ideal for all mapping purposes. Equal Area -- Equivalent/ Authalic -- areas of projections represent equal areas on the reference surface (sphere, globe) An equivalent projection (also known as an area-preserving projection) is a type of map projection that maintains the relative area of features on the Earth\'s surface. This means that the area of any given region on the map is proportional to its actual area on the Earth. The key characteristics of equivalent projections include: - Area Preservation: Shapes may be distorted, but the sizes of areas are accurately represented. For example, if two regions are shown to be the same size on the map, they are also the same size in reality. - Usefulness for Thematic Mapping: Equivalent projections are often used in thematic maps, such as population density maps or land use maps, where the accurate representation of area is critical for interpretation. While these projections preserve area, they may distort shape, angles, and distances, so the choice of projection depends on the specific needs of the map\'s purpose. Conformal or Orthomorphic projections -- angles on projection = angles on reference surface & orthomorphic = shape is maintained are types of map projections that preserve local angles and shapes, meaning that they maintain the geometry of small areas. This property allows for accurate representation of shapes, making them useful for navigation and for maps where local detail is important. Key Characteristics of Conformal Projections: - Angle Preservation: At any point on the map, angles are preserved, meaning that small shapes will appear as they do on the Earth\'s surface, although the overall size may be distorted. - Local Shape Accuracy: While larger areas may become distorted, small areas retain their shape. This makes conformal projections ideal for mapping regions where precise shape representation is necessary. - Use in Navigation: Because they preserve angles, conformal projections are useful for maritime and aerial navigation, where course angles are critical. Orthomorphic/ Conformal projections are useful for applications requiring accurate local angles and shapes they can distort areas, making them less suitable for thematic maps that require accurate area representation.  Types of Map Projection 1. Planer/Azimuthal projection - are a class of map projections that depict the Earth's surface from a specific point, usually projected onto a flat plane. This projection is useful for representing areas around a particular point, making it ideal for certain types of mapping. Key Characteristics of Azimuthal Projections: 1. Perspective: The projection is created by projecting points on the Earth's surface onto a flat plane from a specific point, typically from above (e.g., the North Pole) or from the center of the Earth. 2. Distance Preservation: Azimuthal projections can preserve distances from the center point to any other point on the map, making them useful for measuring straight-line distances. 3. Directional Accuracy: They maintain accurate direction from the center point, which is beneficial for navigation and planning. Types of Azimuthal Map Projections 1. Orthographic Projection - Here's what the Earth would look like if you were thousands of miles away in space. Orthographic maps are common for inset maps as it's the one azimuthal projection people can commonly relate to for perspective. - The Orthographic projection geometrically projects the globe onto a plane with the point of projection as infinity. All the projection lines are orthogonal to the projection plane. - The orthographic projection distorts shape and area near edges due to perspective. - Directions are true from the point of projection, with scale defeating away from its radiating lines. - The orthographic projection isn't conformal nor equal area. 2. Stereographic Projection  - The key to understanding the stereographic projection is understanding its source of light. At the opposite end where the tangent plane touches the reference globe is the light source for the stereographic projection. - This map projection is commonly used for polar aspects and navigation maps because of how it preserves shapes (conformal). Despite how the scale is greatly stretched by perspective, it's been used to map large continents or oceans including the Arctic and Antarctic. - In terms of map distortion, the stereographic projection is conformal but the distortion of area and distance increases away from the center point of projection. - Direction is true from the center point with each straight line representing a great circle. The stereographic projection isn't equal in area nor equidistant. 3. Gnomonic Projection - Unlike the stereographic projection, the Gnomonic projection light source is located at the sphere center. This means that it can only present less than one hemisphere at a time. - Each great circle (geodesic) including meridians is mapped to a straight line. This makes the gnomonic projection easiest to plot out the shortest route. This is why navigators have used the gnomonic projection along with Mercator maps (cylindrical projections) for finding the shortest route between two points. Furthermore, seismologists use this map projection because seismic waves often travel down great circles. - Thales first introduced the Gnomonic projection in the 6th century BC and is one of the oldest map projections today. - The Gnomonic distortion: the Gnomonic projection isn't equal area, equidistant, or conformal as the distortion of these two properties increases away from the center point. - You should avoid using the Gnomonic projection for measuring distances. However, it's particularly useful for navigation as a straight line drawn on the map is a great circle (geodesic). Conical projection - Earth is wrapped around by flat sheet made into cone. Rays are traced onto paper and paper is unfolded  Types of Conic Map Projections 1)Albers Equal Area Conic Projection - The Albers Equal Area Conic projection is commonly used for displaying large countries that require equal-area representation. For example, the USGS uses this conic projection for maps showing the conterminous United States (48 states). - H. C. Albers introduced this map projection in 1805 with two standard parallels (secant). As the name states, the purpose was to project all areas on the map proportionally to all areas on Earth. Like all projections, the Albers Equal Area Conic Projection has map distortion as shown in the table below. A screenshot of a white box Description automatically generated 2. Lambert Conformal Conic Projection - The Lambert Conformal Conic is one of the many creations by Lambert in 1772 still widely used in the United States today (such as in the State Plane Coordinate System (SPCS)). It looks similar to the Albers Equal Area Conic, but graticule spacings differ so that it's conformal rather than equal area. - It uses a conic developable surface secant at two standard parallels, usually at 33° and 45° to minimize distortion. However, standard parallels vary depending on location. For example, Canada's standard parallels are usually 49ºN. and 77ºN.  3. Polyconic Projection Polyconic Conic Projection - This now obsolete map projection uses an infinite number of cones tangent to an infinite number of parallels. This type of protection is generally used for countries that span along a longitudinal extent. - In a polyconic projection, all meridians except the central one have curved lines. Only along the central meridian, distances, directions, shape, and areas are true. However, distortion increases away from its central meridian. Overall, this map projection compromises many properties. It is neither conformal, perspective or equal area.  3)Cylindrical Map Projection -- Earth is wrapped around by flat sheets made into a cylinder. Rays are traced onto paper and the paper is unfolded. Types of Cylindrical Map Projection 1\) Mercator Projection - The legendary Flemish cartographer Gerardus Mercator created the Mercator projection by mathematically projecting a vertically oriented cylinder tangent to the Equator. - Navigators used this type of map because any straight line on a Mercator map is a rhumb line (line of constant direction). However, navigators often combined this type of map with the Gnomonic projection because of how straight lines are great circles showing the shortest path between points. - Mercator map projections show the true direction between places the best but are not equal-area or equidistant. This is the projection of choice from Google Maps for this reason, despite how the south and north poles distort land size. - Directions along a Rhumb line are true between any two points on a map. Distances are true only along the Equator. Although it has a conformal property, areas are greatly distorted increasing size at poles. 2)Transverse Mercator Projection Transverse Mercator Projection - Lambert introduced the Transverse Mercator in 1772. It uses a horizontally oriented cylinder tangent to a Meridian. This is particularly useful for mapping large areas that are mainly north-south in extent. - The whole UTM grid system uses 60 horizontally oriented cylinders secant to the globe. While horizontal and vertical cylinders make up a Mercator and Transverse Mercator, an oblique aspect projection uses neither. - Outside of a 15° band, distortion increases significantly for size, distance and direction. Distances are true only along the central meridian but all distances, directions, shapes, and areas are reasonably accurate within 15º of the central meridian. - The Transverse Mercator m projection is **conformal** with shapes being true in small areas. While the equator is a straight line, other parallels are complex curves concave toward the nearest pole. 3)Miller Projection - It's Miller time. The Miller Projection was developed by O. M. Miller in 1942 using a cylinder projection developable surface tangent at the Equator. - The Miller map projection is very similar to the Mercator Projection, but straight lines are not Rhumb Lines. This means that it's not particularly useful for navigation, but more so for wall maps. - Poles are distorted without the same degree of bulging in the polar regions as the Mercator projection. This is why cartographers often use azimuthal projections for the polar regions. However, the Miller Projection increases the distortion of distances, areas, and shapes that occur at high latitudes. - Area and shapes are still distorted, but not as extreme as the Mercator projection. - The Miller projection is a compromise projection, which means that it isn't equal in area, equidistant, or conformal, and doesn't sacrifice either one at extremes. Directions distortion increases further away from the Equator at higher latitudes. Compromise projection: A map projection that is neither equal area, conformal, nor equidistant, but rather a balance between these geometric properties; often used in thematic mapping. Summary +-----------------------+-----------------------+-----------------------+ | Types of Map | Advantage | Disadvantage | | Projections | | | +=======================+=======================+=======================+ | azimuthal projection | When polar (normal) | Whether the plane is | | | projections are the | tangent (just | | | center point of the | touching it) or is | | | planar projection | secant (intersecting | | | surface, it results | it), you can minimize | | | in meridians as | the level of choosing | | | radial straight lines | standard lines. The | | | converging at poles. | tangent plane has one | | | This means that | point of contact (a | | | azimuthal projections | point of tangency), | | | present true | whereas the secant | | | direction (azimuth) | plane has an entire | | | from the mapmaker's | line of intersection. | | | center point of | A secant plane for a | | | choice. Further to | polar projection | | | this, a property of | results in a latitude | | | azimuthal projections | line as a standard | | | is that they have | line without any | | | straight geodesics | distortion. As a | | | through the map | result, distortion | | | center. This makes | increases away from | | | azimuthal projections | the point of tangency | | | well-suited for maps | or secancy. | | | of the Arctic, | | | | Antarctic, and | | | | hemisphere types of | | | | maps. | | +-----------------------+-----------------------+-----------------------+ | Conic Projection | more suitable for | generally not | | | mapping continental | well-suited for | | | and regional areas. | mapping very large | | | | areas. | | | good for representing | | | | continents | | +-----------------------+-----------------------+-----------------------+ | Cylinder Projection | A cylindrical | | | | projection does a | | | | decent job of | | | | representing the | | | | entire globe | | | | | | | | The Mercator | | | | projection is a | | | | popular choice for | | | | navigation because of | | | | how straight lines | | | | are Rhumb lines. The | | | | State Plane | | | | Coordinate System and | | | | UTM grids use a | | | | Transverse Mercator | | | | because it's ideal | | | | for large-scale | | | | mapping when you use | | | | the correct zone. | | +-----------------------+-----------------------+-----------------------+  Reasons for Projections 1. You have to use one form of projection due the curvature of the earth 2. Earth is a big blue marble that's the shape of a sphere (or close to it). This is why a globe is the best way to represent the Earth. 3. But globes are hard to carry in your suitcase and you can only see one side of the globe. On top of that, it's hard to measure distances and they're just not as convenient as paper maps. 4. This is why we use map projections on globes and flatten them out in two dimensions. But as you're about to find out, you can't represent Earth's surface in two dimensions without distortion. 5. On top of that, all types of map projections have strengths and weaknesses preserving different attributes. 6. When surveying small areas (\