IWalkways helps the designer minimize the local cost for a node in the walkway network. This is done by highlighting a node a shade of light blue as its cost becomes unacceptable (see figure 1).
The system colors a node a shade of light blue according to the following formula:
C*[(cost of current segment)-(optimal cost of current segment)]/[(optimal cost of current segment)]*(cost sensitivity slider value)
where C is a constant.
figure 1: the interior node is expensive to build since it is an unecessary detour between the two end nodes.
One way of correcting the problem while editing is to drag the node to the center, which is shown in figures 2 and 3.
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figures 2 and 3: improving the interior point to a good value |
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Another way that the interior node can achieve an acceptable value is to attatch a new node so that the interior node becomes more central (figure 4).
figure 4: similar to figure 1, but the interior node is not costly since a third node is attatched to it and makes it a central node.
Although it might seem that the optimal cost of a segment is a trivial funcion, its non-trivial derivation is as follows (which will be implemented when time permits- for now a simpler approximation is being used):
optimize the distance equation (when n>2*):
f(x0,y0)=C1*[(x1-x0)2+(y1-y0)2]1/2+C2*[(x2-x0)2+(y2-y0)2]1/2+ ... + Cn*[(xn-x0)2+(yn-y0)2]1/2
where CN is the edge width between nodes N and 0, and xN and yN are points connected to x0, y0
with partial derivatives:
df/dx0=
CN*2*(x0-x1)/2*[(x1-x0)2+(y1-y0)2]2 + ... = 0
and
df/dy0 yields a similar result.
which we then solve for numerically.
* When n=2, we have an infinite number of soultions- namely any point on the line connecting the two adjacent nodes. When n is 1 we have a degenerate case where the most cost efficient place for a new node attatched to only one other vertex is on that vertex.
