Circle packing

Definition

Let $(C_i)_{i \in I}$ be a family of circles of positive radius in the plane $\R^2$ , and let $D_i$ be the closed disc bounded by $C_i$ . The family is a circle packing if the discs $D_i$ have pairwise disjoint interiors. Its tangency graph has vertex set $I$ , with an edge between distinct $i$ and $j$ when $C_i$ and $C_j$ are tangent. A packing of equal circles is one in which all radii are equal.

Remarks

Example: the circles of radius $1$ centred at $(0, 0)$ and $(2, 0)$ . The distance between centres is $2$ , the sum of the radii, so the discs meet in the single point $(1, 0)$ and their interiors are disjoint. The tangency graph is one edge. Non-example: circles of radius $1$ centred at $(0, 0)$ and $(1, 0)$ ; the distance $1$ is less than $2$ , so the open discs overlap. The same notion is used on the sphere and in the hyperbolic plane. A packing of equal circles of radius $r$ is the same as a sphere packing in $\R^2$ with radius $r$ .

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Circle concept
Closed ball concept
Circle packing definition

Tag	`JJ0PQZ7VZX`
URI	`m2://JJ0PQZ7VZX`
Kind	definition
Writing	W0 stub
MSC	52C15, 52C26, 05B40
Defines	Circle packing
Uses	Circle, Closed ball

Circle packing

Definition

Remarks

Reading path

Further reading

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