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The circles in the image don't have the same center, so they are not concentric. Is there a word to describe circles that overlap (completely or not) so that enclosed circles are smaller than the outer circles?

non-concentric circles

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As you yourself called them: nested circles? That’s what I would call them. Nested, non-concentric circles if I wanted to be absolutely specific. –  Janus Bahs Jacquet Jun 8 at 20:27
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@JanusBahsJacquet since the question says "overlap (completely or not)" I inferred that they were interested in their eccentricity alone, rather than whether they were nested or imbricated. –  Jon Hanna Jun 8 at 20:49
    
The illustration given is of a set of circles with one tangent point in common, such that all circles but the largest are contained in another member of the set. This is a very special case of "nested circles that are not tangent"; to fulfil this description, the circles need not touch at all, and if they have tangent intersections, they needn't be at the same point. –  John Lawler Jun 8 at 21:36

5 Answers 5

up vote 6 down vote accepted

In your particular picture, those circles are tangent circles (i.e., they are all tangent to each other). More specifically, they are internally tangent circles. According to Wikipedia, because they all intersect at a single point, these are also known as "kissing circles" (informally, I presume).

Of course, there is more than one way to draw circles that are "nested but not concentric":

enter image description here

I would call the circles in A nested circles; and the circles in B and C internally tangent circles. Because the circles in B happen to be tangent at a single point, they could also be called kissing circles. I would call the circles in D nested circles, but, because not all of them are tangent, I would not call them tangent circles.

If you want more formal terminology, perhaps math.se would be the place to ask.

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I'm going a bit more into maths and geometry but the circle family in the image is called a parabolic pencil. These kind of circle sets are called pencil of circles. It is also a part of Apollonian circles.

A parabolic pencil (as a limiting case) is defined where two generating circles are tangent to each other at a single point . It consists of a family of real circles, all tangent to each other at a single common point. The degenerate circle with radius zero at that point also belongs to the pencil.

Also, they might be coaxal (or coaxial) circles.

Except for the two special cases of a pencil of concentric circles and a pencil of coincident lines, any two circles within a pencil have the same radical axis, and all circles in the pencil have collinear centers. Any three or more circles from the same family are called coaxal circles or coaxial circles.


Further information:

A pencil of circles is another name for specific families of circles who all share certain characteristics within the family.

The third kind of pencil is the parabolic pencil. This is the family of circles which all have one common point, and thus are all tangent to each other, either internally or externally. Also, the orthogonal set of circles to a parabolic pencil is another parabolic pencil.

enter image description here

Source: http://www.math.washington.edu/~king/coursedir/m445w04/as/ga2/gp2/groupassign2.html


enter image description here

Coaxal circles are circles whose centers are collinear and that share a common radical line. The collection of all coaxal circles is called a pencil of coaxal circles (Coxeter and Greitzer 1967, p. 35). It is possible to combine the two types of coaxal systems illustrated above such that the sets are orthogonal.


A representation of Apollonian circles:

enter image description here

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I believe this is the right answer and the right terminology. Your picture is a specific set of tangent circles and I gave the name of it. Also each different circle set is called a pencil of circles. There are nested ones and unnested ones. –  ermanen Jun 9 at 2:15

Eccentric: Not placed centrally or not having its axis or other part placed centrally.

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Technically correct according to the first definition, but confusing due to the second definition –  200_success Jun 9 at 0:51

If you are mostly interested in a mathematical jargon term that generalizes to other instances, I think the best term would be descending. Whenever you see 'descending' in mathematics, it usually means something is getting smaller or is being nested in something else. Some common turns of phrase include descending sequence or descending sequence of sets. With your picture, I'd be able to say descending set of circles to some colleagues and be perfectly understood (this also applies to all of the pictures in J.R.'s answer).

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Without the requirement that the circles be of decreasing size as shown, circles which overlap are simply intersecting.

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"Intersecting circles" doesn't capture the nesting requirement. They could very well intersect at more than one point. –  200_success Jun 9 at 0:46

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