as i'm reading a paper "An Underdetermined Linear System for GPS" By Dan Kalman
i understand the paper but when i traced the equations there's something i don't understand ,may be my mathematics is not good,the thing is according to the Table 1. Satellite data. in the paper he begins to calculate equations which as is follow:
d = .047(t − 19.9). This same distance can be expressed in terms of (x, y, z) and the satellite’s position (1, 2, 0): d = sqrt((x − 1)2 + (y − 2)2 + (z − 0)2). Combining these results leads to the equation (x − 1)2 + (y − 2)2 + z2 = .0472(t − 19.9)2. (1)
Algebraic Solution. Let us focus again on the equation for the first satellite: (x − 1)2 + (y − 2)2 + z2 = .0472(t − 19.9)2. Expanding all the squares and rearranging leads to this version: 2x + 4y − 2(.0472)(19.9)t = 12 + 22 − .0472(19.9)2 + x2 + y2 + z2 − .0472t2. (here's my problem)
the equation which i denoted (here's my problem),i don't understand how it is simplified to this form and how the equation for t appears on both sides of the equation.
here's the link to the paper:http://mathdl.maa.org/images/upload_library/22/Polya/Kalman.pdf