| | 53 | So, if θ,,test,, is less than 90, θ,,i,, = - abs(θ,,i,,). Otherwise, θ,,i,, = abs(θ,,i,,). |
| | 54 | |
| | 55 | == Basis vector separation angle == |
| | 56 | |
| | 57 | In this section, the method to calculate θ,,ij,, is presented. This is similar to the method for the calculation of θ,,i,,, but it is accomplished with respect to the rotated reference frame of '''i,,b,,''' and '''i,,bp,,''' instead of the x and y axes. The figure which represents this setup is as follows: |
| | 58 | |
| | 59 | [[Image(calc_theta_ij.png)]] |
| | 60 | |
| | 61 | The first step is to calculate the magnitude of θ,,ij,,, the angle between '''i,,b,,''' and '''j,,b,,'''. |
| | 62 | |
| | 63 | [[Image(thetamag_ij.png)]] |
| | 64 | |
| | 65 | Next, we need to determine the sign of θ,,ij,, in a manner similar to how the sign for θ,,i,, was determined. The angle θ,,ij,, always represents the angle ''from'' '''i,,b,,''' ''to'' '''j,,b,,''', and is positive counterclockwise for consistency with a right-handed coordinate system. To do this, we first need to calculate '''i,,bp,,''', which is perpendicular to '''i,,b,,''' and forms a [http://en.wikipedia.org/wiki/Cartesian_coordinate_system#Orientation_and_handedness right-hand coordinate system] with '''i,,b,,'''. Observe that '''i,,bp,,''' is '''i,,b,,''' after a 90 degree counterclockwise rotation. |
| | 66 | |
| | 67 | [[Image(basisvector_ip.png)]] |
| | 68 | |
| | 69 | Now we can determine the size of the angle between '''j,,b,,''' and '''i,,bp,,'''. In this situation, any angle less than 90 degrees means that '''j,,b,,''' is on the same side of '''i,,b,,''' as '''i,,bp,,'''. An angle more than 90 degrees means it lies on the opposite side. |
| | 70 | |
| | 71 | [[Image(thetatest_ip.png)]] |
| | 72 | |
| | 73 | So, if θ,,test,, is more than 90 degrees, θ,,ij,, = - abs(θ,,ij,,). Otherwise, θ,,ij,, = abs(θ,,ij,,). |
| | 74 | |
| | 75 | |
