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10. Spirals and Coordinates
As with most surveying stake out operations, having point coordinates makes for more flexible field operation.
Starting with directions of the incoming and outgoing tangents along with the PI coordinates, the process to compute point coordinates are:
a.TS and ST coordinates, Figure E17
Figure E17 TS and ST Coordinates 
These are computed using Equations E27 through E31.
Equations E27 through E29  
Equations E30 and E31 
b. Spiral point coordinates, Figure E17
Figure E17
Spiral Coordinates

Using the deflection angle notes, spiral point coordinates can be computed using Equations E32 through E37. Right deflection angles are positive, left are negative.
Entrance spiral  
Equations E32 through E34  
Exit spiral  
Equations E35 through E37 
c. Arc point coordinates, Figure E18
Figure E18 Arc Coordinates 
Arc points are computed using equations E38 through E40. Right deflection angles are positive, left are negative.
Equation E38 through E40 
Math check? The coordinates of the CS computed from the SC should match those computed from the ST.