There are three main systems of the coordinates used in geometry, theoretical mechanics, other sections of physics: Cartesian, polar and spherical. In these systems of coordinates each point has three coordinates. Knowing coordinates of two points, it is possible to define distance between these two points.
It is required to you
- Cartesian, polar and spherical coordinates of the ends of a piece
Instruction
1. Consider for a start the rectangular Cartesian system of coordinates. The position of a point in space in this system of coordinates is defined by coordinates x, y and z. From the beginning of coordinates to a point the radius vector is carried out. Projections of it radius vector on coordinate axes will also be coordinates of this point. Let you have two points with coordinates of x1, y1, z1 and x2, y2 and z2 respectively now. Designate for r1 and r2, respectively, the radius vectors of the first and second point. It is obvious that the distance between these two points will be equal to r vector module = to r1-r2 where (r1-r2) is a vector difference. R vector coordinates, obviously, will be the following: x1-x2, y1-y2, z1-z2. Then the module of a vector of r or distance between two points it will be equal: r = sqrt (((x1-x2) ^2)+ (y1-y2) ^2)+ (z1-z2) ^2)).
2. Consider now the polar system of coordinates in which the coordinate of a point will be set by the radial coordinate of r (radius vector in the XY plane), angular coordinate? (a corner between a vector of r and axis X) and the coordinate of z similar to z coordinate in the Cartesian system. Polar coordinates of a point can be transferred to Cartesian as follows: x = r*cos?, y = r*sin?, z = z. Then distance between two points with r1 coordinates? 1, z1 and r2? 2, z2 will be equal to R = to sqrt (((r1*cos? 1-r2*cos? 2) ^2)+ ((r1*sin? 1-r2*sin? 2) ^2)+ (z1-z2) ^2)) = sqrt ((r1^2)+ (r2^2)-2r1*r2 (cos? 1*cos? 2+sin? 1*sin? 2)((z1-z2) ^2))
3. Now consider the spherical system of coordinates. In it the provision of a point is set by three coordinates of r? and?. r - distance from the beginning of coordinates to a point? and? - azimuthal and antiaircraft corner respectively. Corner? it is similar to a corner with the same designation in the polar system of coordinates, and? - corner between radius vector r and axis Z, and 0 <=? <= pi. Let's transfer spherical coordinates to Cartesian: x = r*sin? *cos?, y = r*sin? *sin? *sin?, z = r*cos?. Distance between points with r1 coordinates? 1? 1 and r2? 2 and? 2 R = will be equal to sqrt (((r1*sin? 1*cos? 1-r2*sin? 2*cos? 2) ^2)+ ((r1*sin? 1*sin? 1-r2*sin? 2*sin? 2) ^2)+ ((r1*cos? 1-r2*cos? 2) ^2)) = (((r1*sin? 1) ^2)+ ((r2*sin? 2) ^2) - 2r1*r2*sin? 1*sin? 2 * (cos? 1*cos? 2+sin? 1*sin? 2) ((r1*cos? 1-r2*cos? 2) ^2))