Hello. This part of the lesson of Geodesy is dedicated to coordinate systems. We use coordinate systems everyday. Here is one small example of an ad for a public enquiry where we find various information about the location. First, I find an address. Here, "chemin de Villard". Then, I find a parcel number. And finally, I find the coordinates... ... a series of numbers here, which gives me an indication of the object's position on a map. We see with these three concepts of location things that we know, more specific things, such as a parcel number, and finally the figures that permits us to find a location in space. To find a location in space, there are several scales. If I am interested in finding this service station, I have many scales here: nominal, ordinal or cardinal. So for this example, we can first use the average, I would say, the most natural, ie a name with the address here, that is "rue de l'industrie", number 12. One can also use order. In this case, building number 12, is situated between number 10 and number 14. And finally, one can utilize a scale called cardinal... ... that make reference to a coordinate system. In this example, I draw here a small system and I can measure an offset here towards east, and north, and I should have here the east and north coordinates for my service station. In surveying and the systems that interests us, we have for most of the systems a spatial reference of nominal and cardinal scales. What is a coordinate system? A coordinate system it is primarily two axes,... one origin,... an orientation of these axes and a measurement scale along these axes. Our first example here, which is called a Cartesian system.. and that defines a plane. One can also define a coordinate system along an axis. If I take here an example with an axis of a road I can also link to this axis, a scale, an origin... and thus I can find an object or an event along this axis. If I take a little of the distance and return to my earthly sphere, I will define a spherical coordinate system... with the angular values relative to the reference planes. So I summarize here: I have here my spherical coordinate system, here my linear coordinate system, and here my coordinate system on the plane. A combined use of coordinate systems is the road sector. The interest in the road sector is that we have on one side a representation on a plane and essentially, as the road is a linear object, it can also be represented in a one-dimensional space. We speak here of a tensile axis type of representation, ie the geometry here on my path on the plane I can draw it, represent it, along a rectilinear axis, namely that the measured length on the plane will be found here and this tensile axis, which allow me to locate an event or an object on the map and to represent it here on this axis. The advantage of this method is to combine different types of information and to analyze specific locations. It is the object here in this example with a layer of information regarding the rate of accidents and another layer about the geometry of the road. And we see with this tensile axis that the high rate of accidents here is situated where there is a change of slope on the road. The spherical coordinates. Before defining spherical coordinates, we must characterize some geometric quantities on Earth. The first thing that comes to mind is the axis of rotation... that cut the Earth at two characteristic points, the north pole and south pole. Then, the reference plane, which is common knowledge, is the equator, and in the other direction we have the meridians parallel to equator, we have planes called parallels. With these geometric elements we can now define a spherical coordinate system. The spherical coordinates. The first spherical coordinate, that interests us, is the latitude. Latitude is the angle that is normal to the surface with the plane of the equator. It is the angle between the normal of the surface and the equator. The second spherical coordinate is the longitude,... which is the angle between the meridian planes, the prime meridian, passing through Greenwich, and the meridian of a place considered. So we have this angle here between these two meridian planes. In the figure, we see that, associated with these spherical coordinates, we also have a frame here Cartesian, geocentric, with a z axis, which is the axis of rotation of Earth, a x axis passing through the plane of the prime meridian and the equator, and a y axis, which is perpendicular to the xy plane. We will now look at some questions about the conversions of coordinates to proceed to the geocentric Cartesian space the spherical coordinates, latitude, longitude.