Kinematics - Galilean Transform

Assuming the relative motion 'v' is along the x dimension then x -> x0 + vxt

or if we have components of velocity in all dimensions the transform will be:

 t' x' y' z'
=
 1 0 0 0 vx 1 0 0 vy 0 1 0 vz 0 0 1
 t x y z

where:

• vx,vy,vz= relative velocity of the two reference frames in x,y and z directions.
• x,y,z= position in original frame.
• x',y',z'= position in transformed frame.

in other words point:

 t x y z
is transformed to:
 t x + vxt y + vyt z + vzt

The nature of this transform is a shear (also known as skew) transform:

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When doing this we choose to make time 'absolute' in that the time lines are left horizontal wheras the position lines are skewed althogh I guess that this is just a covention and we could have skewed the time and made the distance absolute.

Shear (skew) Transform Matrix

The shear transform has the following characteristics:

• determinant = 1
• trace = dimension

for example the two dimensional matrix

 a b c d

we have:

a+d=2