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I know that in standard two-dimensional Euclidean space three non-collinear points have a unique circle that touches all three points. I saw that @dan uznanski gave a determinant form for finding the equation of of the circle using minor determinants of a specific matrix. I also saw a development similar to this in Felix Klein's elementry mathematics from an advance viewpoint book.

I was curious about circles in the taxicab geometry. I'm trying to make an analogue of the prime gap bound for Gaussian integers. I'm defining the prime gap through the biggest possible circle between primes in a taxicab geometry.

This led me to question a few things about taxicab cirlces. Notice that the points (-1,1), (0,0), and (1,1) fit on the circles |x-0|+|y-b|=b for b>1. These points are not collinear in a standard sense, but the point (0,0) falling on a corner of this taxicab circle makes them collinear. Is there a simple generalization of what it means to be non-collinear in a taxicab geometry so that we can have a unique taxicab circle fit through three non-collinear points? What are the general formulas for the center and radius of a taxicab circle that falls through three points that are not collinear under this more general definition of "collinear"? Is there a generalized concept for the determinant description for standard Euclidean circle centers and radius in context of the taxicab geometry? I'd love to hear your comments.

Kyle Bradford
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    It is not clear what you mean by "the point (0,0) falling on a corner of this taxicab circle makes them collinear". In what sense are you considering three points collinear? – Crostul Apr 23 '17 at 10:35
  • There's an important qualitative difference here from the Euclidean case. In the Euclidean case, a triple of points $(P, Q, R) \in (\Bbb R^2)^3$ generically determines a unique circle in the sense that the set of triples that do is open and dense, but this is not true in taxicab geometry. For $P = (0, 1)$ and $Q = (0, -1)$, there is a unique taxicab circle through $P, Q, R$ iff $|x| > |y|$ and $|x| + |y| > 1$, which is an open but not dense set. – Travis Willse Apr 23 '17 at 10:44
  • If there is something that fits into the scene, it is what is called "linear programming" (sometimes wrongly reduced to simplex algorithm). I will maybe develop these ideas later on. – Jean Marie Apr 23 '17 at 11:23
  • Are you aware of this ? – Jean Marie Apr 23 '17 at 17:49
  • @JeanMarie Thank you for the link. That was the stack exchange page that I reference for dan uznanski's determinant formulas. – Kyle Bradford Apr 24 '17 at 11:19
  • @Crosful I apologize for the confusion. I should clarify that one of the questions that asked is for a generalization of what it means to be collinear. Taxicab circles look like diamonds with equal side lengths. If three points can fit on the corners of a diamond, then there are an infinite number of taxicab circles that can touch these three points (like the example that I listed). If three points are on three opposite faces of the taxicab circle, then this circle is unique. I'm asking for a generalization of "collinear" so that I am guaranteed a unique taxicab circle. – Kyle Bradford Apr 24 '17 at 11:27

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