Is there a formal mathematical name for the geometry of a circle, sphere, etc, regardless of dimension?

Question

In geometry of 2D and 3D, it's not uncommon for people to call a square or rectangle a Box in the field I work in. This makes naming things easier since it's clear what's in a folder of 'boxes'.

Does the similar name exist for a circle and a sphere? Interestingly we have circles in 3D that could be defined as the intersection of a geometric primitive with a sphere, like a plane slicing through a sphere to form a circle. However searching for a unifying name (even if it's not that good of a name) has been unfruitful.

Does a name exist for this classification? As in, a name for circular or spherical objects defined by some distance from (for convenience here) the origin?

Some things seem easy, like a hyperplane is going down a dimension, so it ends up working for almost any dimension. I was hoping there was something like this that would either work for 2 and 3 dimensions, or better, work for N dimensions.

Answer 1

We often refer to an "$n$-sphere", usually to say $S^{1}$ is a circle and $S^2$ is the surface of an ordinary sphere. An $n$-sphere sits in $R^{n+1}$. I sometimes see hypersphere used, but that is less common.

If you want to include the interior, the term is "$n$-ball".

By the way, the generic word for "length, area, volume" etc. is content.

Answer 2

There is a general geometric framework called the "space of spheres" covering spheres in any dimension, in particular

• ordinary circles for dimension 2 or even

• line segments $[a,b] \subset \mathbb{R}$ for dimension 1.

The important fact is that these spaces are endowed with a quadratic form that I am going to explain in the case of circles, but that can be extended to any dimension.

As a circle can be defined by kinds of equations:

$$(x-a)^2+(y-b)^2=R^2 \ \ \iff \ \ x^2+y^2-2ax-2by+c=0\tag{1}$$

we will consider that a circle is defined by coordinates $(a,b,c)$.

Due to the equivalence in (1), we have

$$R^2=a^2+b^2-c>0\tag{2}$$

The "set of circles" can be considered as a kind of 3D space deprived of the interior of a certain paraboloid.

Moreover, the RHS of (2) will give a kind of "norm" (please note the double quotes) for a circle:

$$\|(a,b,c)\|^2=a^2+b^2-c\tag{3}$$

or, introducing a 4th (projective) dimension $d$ for the homogeneization of expression (3).

$$\|(a,b,c)\|^2=a^2+b^2-cd=a^2+b^2+\tfrac12(c-d)^2-\tfrac12(c+d)^2$$

which is a true quadratic form with signature $(+,+,+,-)$ (Minkovski space structure)

For more about this space, see first my answer here, then the very nice article here showing application to Voronoi tessellations and Voronoi triangulations.

For more, see the paragraph "Space of spheres" in the marvellous book "Geometry II" of Marcel BERGER (Springer, 1987).