I crunched some numbers, had some peers look over what I have, and with some input, I think I've devised the optimal solution to a four-herogate system.
My previous model for the system put the Herogates in the middle of four square quadrants. I fleshed out the diagram a little bit in Figure 3:
The coordinates for the Herogates are, in this model, X=+/-1250, Z=+/-1250. This model
minimizes the maximum possible walking distance (1.8km) An example of such a distance is given by the red line in the Figure, and in all further figures. The issue that I have with this model is that while the maximum walking distance is so small, the area easily accessible from Sanctum (shown by the green diamond) is also very small. Specifically, it takes up 1563 sq km, only 6.25% of the total area of the surface (25,000 sq km). This would make travel more expensive than it needs to be, because most of the world is more easily traveled only after having used a Herogate
"But wait, Doreagarde! Wouldn't the easily-accessible area around Sanctum be round, and not the diamond shape? An "easy" distance should be the same no matter what direction you go in!" That's what I thought at first too, clever reader, and I'll refer you to Figure 4 for discussion:
Observe how the points of the circle that intersect the axes are equidistant from both Sanctum and the Herogates (1.25km, shown by the blue lines). That much is in favour of a "round" easy-access area around sanctum. Now look at the red line. See how it's easier to get to the point where the red line and green circle intersect by using the Herogate, and not Sanctum? In fact, that point of intersection is 0.88 km from Sanctum, and only 0.37 km from the Herogate. Thus we see that not all points in a circle around Sanctum are as convenient to the spawn as they are from the Herogates.
Punisher79 said:
I'd stick the Herogates in the extreme corners of the map, that way people have multiple routes to choose from....
Ok, and obviously it benifits me, who is stuck out in the EXTREME corner of the map... :icon_mrgreen:
I gave consideration to your model as well, and I want to discuss it before moving on to my final calculations. See Figure 5:
With the Herogates at X=+/-2500 and Z=+/-2500, we've managed to
maximize the usefulness of walking from Sanctum, by maximizing the area that is (relatively) easily-accessible from it. Specifically, that area is 12,500 sq km, or 50% of the surface. However, look what's happened to our maximum walking distance. Now, to reach the center of each border, we must walk the full 2.5 km, twice as far as the model in Figure 3. While not useful as a working model, this thought process did get me thinking about how to find a "happy medium", and I think I've done just that, in Figure 6.
Here, I've made a circle with R=2500 and placed the Herogates at the four points on the circle where abs(X) = abs(Z), which is 2500sin(45) and 2500*(1-sin(45)), or about +/-1768. Here, we have a
balance between the maximum walking distance and the usefulness of walking from Sanctum. The Sanctum walking area is 1768x1768, or 3123 sq km, 12.5% of the total area of the map. I wanted to bring this number up to 20%, or a fifth of the map (each of the other four fifths would have a Herogate going to it), but that would mean placing each gate at +/-2236, which would mean the maximum walking distance would still be too great (2.25 km). However, Figure 6 shows a maximum walking distance of only 1.91 km.
The gates probably can't go at exactly these locations, since there are likely towns and stuff in the way, and there might be more scenic locations nearby, like dense forests or beaches or the like, where the gates might look better. And I don't care if these results get thrown out anyway, I just found the problem a pleasant diversion. The closer the gates come to
+/-1768, however, the better.
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