# generalized circle

## English

### Noun

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 ${\displaystyle \zeta =1/z}$ the generalized circle (53.3) goes into
(53-4)    ${\displaystyle \beta \left|\zeta \right|^{2}-p\zeta -{\overline {p}}{\overline {\zeta }}+\alpha =0}$,
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).