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To preface, this question might come off as a bit silly, but it would serve me well to understand the concepts, and to actually get a good answer for this.

How can I design an ideal metric for walking places. I know about taxi cab geometry, so I know how to do things for city streets, but what I am more concerned about is sunlight. I don't like the sun very much and burn pretty easily. Therefore, when I use heuristics when I walk place,s by default, I take routes that involve more shade. I was talking with a friend about how to make this into a metric. She has taken classes involving metrics and I have not, but I have a basic understanding of them. Basically I proposed a metric that said that distances in the shade are shortened by some constant (real) factor. The eventual conclusion was that if you define the distance to be the minimum distance over all possible paths, where distance in the sun is measured in the Euclidean metric and distance in the shade is measured by k*Euclidean metric where $0<k<1$. Why can't I make k infintesimal? I know this breaks the triangle inequality, but why? My intuituion says that making k infintesimal is equivalent to scaling the distance in the sunlight by a factor of infinity. Is there some weird non-standard analysis trick I can use to make this work? Additionally, I was also wondering if I could make something that makes going up stairs worse than going down. I was told this violates reflexivity, but again is there a trick around this? . Thanks so much.

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    $\begingroup$ Why do you specifically want a metric? For route planning, a weighted directed graph seems like the sort of thing you're looking for $\endgroup$ – Dan Rust Jun 18 '14 at 20:21
  • $\begingroup$ I haven't done Graph Theory formally, but from what I remember, I would need an infinite weight directed graph. Since I want to be able to calculate distance between any two arbitrary points on the plane. Though if one were to make it a multigraph it would solve the stairs issue, I'm still not sure this would help for arbitrary points in the plane. Would you mind elaborating? $\endgroup$ – Thoth19 Jun 18 '14 at 20:25
  • $\begingroup$ So a friend of mine was telling me perhaps a Pseudometric space is what I'm looking for en.wikipedia.org/wiki/Metric_(mathematics)#Generalized_metrics Would this work to make k = 0 (or infintesimal) a reasonable thing? Sine I haven't studied this stuff in detail, I'd love a brief description of what types of properties and guarantees we lose when we use this rather than a regular metric. $\endgroup$ – Thoth19 Jun 19 '14 at 3:25
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You can make this work with infinite scaling between shade and sun but this would not be a metric on the whole plane anymore.

Either you make the distance in the sun infinite. Than you get standard euclidean distance in the shade but there are no path that go into the sun. Different connected components of the shade have infinite distance to each other and each point in the sun is inifitely far away from everywhere else.

Alternatively you can make the distance in the shade zero. Then you get a proper metric on the sunny parts (which will be locally euclidean but not globally) but each connected component of the shady parts collapses to a single point. This might be most useful for your scenario because the shortest path between two points will be the one with the smallest distance in the sun, regardless of distance in the shade.

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