The textbook I use introduced the Taxicab "circle" using the set-notation, $$\{P\mid d_{T}(P,A)=1\}.$$ as the set of points, such that the distance between all the points and the center is equal to $1$ taxicab distance, effectively defining the unit-taxicab circle. A later exercise problem extrapolated this notation, and I am confused as to its meaning. I will show the problem, then give my own interpretation. I am looking for error correction and further explanations, either written or visual, if possible.
Graph $\{P\mid d_{T}(P,A)+d_{T}(P,B)=d_{T}(A,B)\}$.
This is the set of points such that the sum of the distances between $P$ and $A$, and $P$ and $B$ respectively, is equal to the distance between $A$ and $B$. Firstly, this is not a specific computation; rather, it is looking for something in the general case. Secondly, the set of points $P$ that is the answer is dependent on the distance between $A$ and $B$, thus I am to observe some property of the relationship between the sum of circles, which hardly makes sense to me — how does one "add" a geometrical figure? (a question perhaps for another time). The answer to this will be a taxicab circle in the end, arithmetically related to the two centers $A$ and $B$.
The furthest I've gotten is this. I suspect that the taxicab circles of centers $A$ and $B$ will overlap, and their overlapping will constitute some answer, but this is mere postulating, and I haven't any proof for my statement, and no bearing as to whether or not I am even in the right direction.
To reiterate, I am looking for
- error correction, and/or
- a written/visual explanation.
