Hello, this part of the Geodesy lesson is dedicated to projections. A projection consist of moving a three-dimensional surface to a flat surface: the map. The problem with projection, lies with the fact that there are deformations. I transfer here a three-dimensional surface to a flat surface in 2D. In fact, this will generate deformations. If I take in my coordinated lines, I will have on the 3D surface a small difference, here, the latitude. a small difference in the longitude, I have here a small surface element and when I pass to the space on the map, my small surface element, here, is deformed. The maps and the planes that we utilize are planar representations of a portion of the surface of the Earth. I draw here the terrestrial sphere with a small difference of latitude, an small difference of longitude, so I have here a Δψ, here a Δλ. And on my map, at this small surface, I can have the element correspondant on the plane. That is a ΔX here and a ΔY here. So I found my small surface of the sphere or the ellipsoid and its correspondence on the plane. The projection consist of establishing the correspondenceses, meaning the formulas, in function of the latitude and of the longitude for the two coordinate axes X and Y of the plane. Therefore, there are infinite possibilities of establishing We have seen that all projection induce deformations. Here we take our small example with a surface element on the sphere, with a difference here, of latitude, a difference of longitude, I can trace, here, between a point P and a point Q a small linear element and I have compared coordinate lines, an angle, here, θ. I can also calculate the surface of this element, That I will name here σ. So I am here on the 3D surface, the sphere or the ellipsoid. I then project this element on a plane. So I return to the coordinate lines, a point P', so the image of point P, a point Q', the image of point Q. I find my small linear element here: an angle between the coordinate lines that I call θ'. And the surface here of this small element is σ. I have evidently here a ΔX and a ΔY. We have many types of projections. The first type, is the projections called compliant. These are the types that will keep the angles. In this case, the projection here by my small angle θ on θ' in the conserved plane. The projections called equivalents will retain the area. In this case, the little surface, here σ, is projected on σ' and is conserved in this case. The distances, are always deformed. There is therefore one impossibility to have a projection that is linearly neutral. The principle is to project elements onto the reference surface on a cylinder or cone surface. We find our terrestrial sphere here with its equator, the prime meridian. We will define, for the purpose of the projection, an origin. Here, we have point M, which is the origin of the projection system. And passing through this origin, we have a parallel contact. Then, the projection consist of projecting a point, for example here, the point P, on the surface of the cylinder, on the point P'. There exists different types of projections depending on where we place the projection surface. I have here my terrestrial surface I sketch here an axis of rotation. I have a cylinder with a projection called normal. By placing the cylinder here, parallel to the axis of rotation. We also have a possibility by placing a cylinder transversly. Here, in green. I have in this case a projection called transverse. We can also place a oblique cylinder and we have in this case a projection called oblique. To illustrate the principle of the projection, I resume with my example here the cone applied on a sphere. We have defined the major geometric elements of this figure. Now, we will develop a cone and obtain here a plane figure where we find the predefined elements. Namely, the parallel of the contact, as well as the original point, here, M' of my coordinate system. I have therefore now placed on M' a cartesian coordinate system, I place here an horizontal axis, the axe, here, which I name X, which points towards geographic north, and my axe, here, Y. I will now turn to the projection, here, of the point P', this point P' is defined here in polar coordinates. Namely, the coordinate r, which is a function of the latitude, and ε that is a function of the longitude. The projection consist now to define functions That will associate here r and ε. Respectively, latitude and longitude. I have a function f for the component X and a function g for the component Y. Thus I have my function of the projection defined here conceptually. Here is an example of a well known projection: the projection called the Mercator. It is a projection mostly utilized in the field of navigation. It is a compliant projection. So, that preserves the angles. And that is very practical because if we draw here a navigation route, I have this that we call a rhumb line, namely that my route, here, will cut the meridiens, always at the same angle. We see in this image that this projection generates substantial deformations at high latitudes. Switzerland, like most countries, has defined its geodetic reference and a projection system. This work was made in the early twentieth century, with the definition of the projection system based on a reference surface the Bessel ellipsoid. The Swiss projectopn is a compliant projection, so, which preserves the angles, and it is a double projection that will first project the Bessel ellipsoid reference on a sphere and then, the sphere on a cylinder that is oblique. The origin of the system is placed in Bern, here. And then, the cylinder is unrolled to have a Cartesian grid coordinate system. In this figure, we see our Cartesian axis system. So we have here an axis X that points towards north. This was defined by convention. Then, we have the axis, here, Y, which points towards the east. The origin of the system is in Bern. It is therefore the zero coordinates in this Cartesian system. We see in this image that the parallels and the meridians is not evidently straight on this projection, and there is an effect of deformation. We translated the origin of the system a way, to always have positive coordinates and the Y coordinates larger than the X. If I take this region of Lausanne, I have the coordinates on the axis, here, Y about 550 km, and on the X axis, towards north, about 150 km. So I have always my Y that is larger than X, which is 150 and here, 550. What are the main deformations of the Swiss projection. First, we consider the deformation called linear. I have here on this abacus the values of these deformations, starting here by the representation of the neutral axis. That means, it is the the axis where the distances are projected in real size. The more I move away from this axis, the more significant is the deformation. It is express here in ppm, namely in mm per km. I have here the scaling factor, in ppm. And I give you, some values here, representatives. 25 km from the neutral axis, I have 8 ppm, namely 8 mm per km, at 50, I will have 31 ppm, and at 100 km, I have 123 ppm. Meaning that for a point situated here, at EPFL, so 50 km from Bern, if I measure a distance of 1 000 m, it will elongate 31 mm. So we have, finally, a distance of 1 000,031 m. This, due to the deformation of distances. At the origin of the Swiss projection, meaning at Bern, the north on the map, and the geographic north are mixed. On the other hand, once I move away from this axis, I have a difference between the north on the map and the geographic north. We call this difference the convergence of the meridian. And on this figure, we find the values for this covergenece. Namely, for the region, here, of Geneva, a difference of about -1,1 degrees, And for the east of Switzerland, the region of Grisons, we have here +2,3 degrees. We must consider this convergenece of the meridian when we do calculations between the relative bearing and azimuth. In summary, projections are a mathematical application of a three-dimensional surfaces, sphere or ellipsoid, on a plane. All projections generate deformations. We distinguish the projections names complaint, which will retain the angles, and the projections named equivalents, which will retain the area. Each country has defined, with their geodetic reference, a projection system, to meet its needs, of the national measuring, or for the production of maps.