John, Depending on how I understand the question, I think the answer is yes and/or no!
Suppose you chose to do your construction on a day for which the EoT is zero. Consider a flat dial plate with an isolated, small, slightly off-centre bump on the 12 o'clock line, created by drawing along the shadow at 12 o'clock. The line will have a bit of a "bulge" in it as it passes over the bump. However, at a different time of the the year, when the EoT does not equal zero, the shadow will be off to one side at 12 o'clock clock time. If the bump is small enough, the shadow will now miss it and the hour line will be entirely straight. Hence, depending on the time of year, the line you get will be different for a given clock time. On the other hand, if you are adjusting for EoT so that you are using true noon on whatever day, the sun will always be shining from the same direction and only its height will vary. The length of the shadow cast will vary, but its direction will be the same, hence the line on the ground will be the same. Different parts of the gnomon will account for different sections of the shadow throughout the year, but they will lie on the same "line". So, for a given sun time, the line you get will be the same. I hope that helps! David Higgon London England
