in this video we will see the position and the orientation of a rigid body we will let's start with the position how to represent the position of a rigid body here we have a rigid body and a reference frame so what we do is we assign a point an arbitrary point on the rigid body and we represent the position of this rigid body through the position of a point that is fixed to the body in this case this position vector will be ope how many coordinates are needed to represent the position if we are in two dimensions we need two coordinates as we saw before which are two degrees of freedom if we are in 3d we need three coordinates because we have three degrees of freedom for the position in 3d there are several forms of representation for example Cartesian cylindrical or spherical coordinates first the Cartesian coordinates which are the most obvious points we have here the reference frame we have the point P and we have the coordinates x y and z which is a vector then we have cylindrical coordinates for cylindrical coordinates we do the projection of this vector on the XY plane and we define this theta and over here and we defined the length of the projection vector which is Rho and we'll use this Rho this angle theta and this height Z to represent cylindrical coordinates we can make some relations we can establish relation between these two coordinates from a geometric point of view you can do it by yourself and we get this this is a relation from cylindrical coordinates to Cartesian coordinates and this is from Cartesian coordinates to cylindrical coordinates we also have a spherical coordinates where we use this length R we use this angle Phi and we use this angle theta and we represent it as R theta and Phi where R is going to be always positive this angle theta is going to be just from 0 to PI so here too roughly here and file is going to be from 0 to 2pi so it can go all the way around the relation with Cartesian coordinates can be again obtained in a geometric way and we have this relation this is from the spherical coordinates to Cartesian coordinates and this equations go from Cartesian to a spherical coordinates with cylindrical coordinates we have this this is from a spherical to cylindrical and this is from cylindrical to a spherical now here be careful because we call in the spherical coordinates theta this angle but in cylindrical coordinates the angle theta is this one so that's why here I puts Tita sub C then let's see the orientation of a rigid by how do we represent the orientation of a rigid body again we have this reference frame and we have the body and in this case it has some orientation what we do is we assign a reference frame to this body and then we represent the relation between this frame and this frame and the relation between frames is going to define the orientation of the rigid body so that's why we will be interested in defining the relations between frames how many coordinates we need for the orientation in two dimensions we just need one coordinate because we only have one degree of freedom in three dimensions we need three coordinates as we also saw before because we have three degrees of freedom in general in n dimensions we have n times n minus one divided by two coordinates you can check this formula by replacing two and three and seeing that we are getting these numbers in general orientation is not as straightforward as position the problem of orientation is a topology of the space that describes the orientation this space is not a gradient for example you can recall this a sphere which is a service of it to the sphere in the 3d space we could represent this for example with latitude and longitude but then we run into problems whatever when we get to the poles because there are similarities we have some similar problem in the orientation but in this case it's not a 2-d sphere that is rather a 3d sphere so the problems are even worse fear to represent the orientation we have the rotation matrix which is an implicit representation and gives constraints because we use more parameters and the needed ones and one related concept here or exponential coordinates which we'll cover also in a later video we have some parametrizations of orientation we have the axis have a representation which uses three parameters and is minimal and is somehow equivalent exponential coordinates we have rope with your angles which has three parameters and is also minimal because Euler angles which also has three parameters and is also minimum these are minimal because they used three parameters and orientation has three degrees of freedom in the space and we also have quaternions quaternions have four parameters so this is not minimal it is an implicit representation because it has four parameters but we have one extra constraint