generalized circle

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generalized circle (plural generalized circles)

  1. (geometry, inversive geometry) A circle or a line, the two being regarded as types of a single form.
    • 1965, H. F. Weinberger, A First Course in Partial Differential Equations with Complex Variables and Transform Methods[1], page 247:
      We shall speak of lines and circles as generalized circles. A line is a generalized circle that passes through the point at infinity. Under the inversion the generalized circle (53.3) goes into
      (53-4)    ,
      which is a generalized circle in the ζ-plane.
    • 1999, David A. Brannan, Matthew F. Esplen, Jeremy J. Gray, Geometry, page 252,
      For any Apollonian family of circles defined by the point circles A and B, the Coaxal Circles Theorem states that every generalized circle through A and B meets each of the Apollonian circles at right angles.
    • 2001, Elie Zahar, Poincaré's Philosophy: From Conventionalism to Phenomenology[2], page 84:
      This is why it makes sense to call every circle and every straight line a generalized circle. Thus the geodesics of ds consist of all arcs of generalized circles which lie wholly within Ω and are orthogonal to w (see Figure 2).


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