STApi is a requirement when calculating stationing for STApc and STApt simply because in most situations the point of intersection between two straight alignments is the only known position which can be referenced (i.e. assigned a stationing) in relation to either approaching line segments.
Before GPS, the method to establish a curve required calculating offset values starting from the STApc. This is done by establishing the stationing at the STApc (calculated from knowing the STApi and subtracting the tangent length) and determining the location of the next point along the curve alignment based a bearing and distance from the previous point. After the curve is staked out (from STApc to STApt). the final stationing is established along the curve and the STApi will no longer be an issue.
Be aware that the tangent length is not directly related to the tangent point of the curve (i.e. STApt), but simply serves as a means to determine the curvature point (STApc) based on the curve's radius.
most horizontal curves were laid out by occupying the PC & sighting the PI, then turning deflection angles accordingly to even stationing on the actual curve. Since the PC to the PI is a straight line (being the tangent (T) also called the 'back' tangent), the stationing is treated accordingly. However, the stationing at the PT can only be arrived at by knowing the length of curve (L). Even though the forward tangent (PI to PT) is the exact same length as the back tangent (PC to PI), you cannot simply add T to PI to get PT. You've changed direction with the curve (the distance being shorter than along the tangents) and you will always wind up with a bigger stationing than actually is present at PT if you were to add this way.
so the second part of your quest 'PT = PI + LC' makes absolutely no sense. PT = PC + L (L being length of curve). LC usually stands for long chord, which will definitely not give you PT when added to PI, since its length is longer than tangent (T).
third part...you simply add length of curve (L) to the PC (not staPI) to get the PT. So no, you can't add T to PI to get PT because as I stated above, the stationing # would exceed actual stationing derived along length of curve, PC to PT (curve is shorter distance than tangents added). The PI is not the midpoint station.
