# Interior, Exterior and Closure

We need the concept of interior, exterior and closure to fully appreciate the
discussion of regularized Boolean operators.
Intuitively, the interior of a solid consists of all points lying inside
of the solid; the closure consists of all interior points and all
points on the solid's surface; and the exterior of a solid is the set of all
points that do not belong to the closure.

Consider a sphere, *x*^{2} + *y*^{2} +
*z*^{2} = 1. Its interior
is the set of all points that satisfy
*x*^{2} + *y*^{2} + *z*^{2} < 1,
while its closure is *x*^{2} + *y*^{2}
+ *z*^{2} <= 1. Therefore, the closure is the
union of the interior and the boundary (its surface
*x*^{2} + *y*^{2} +
*z*^{2} = 1). Obviously, its
exterior is *x*^{2} + *y*^{2} +
*z*^{2} > 1.

A solid is a three-dimensional object and so does its interior and exterior.
However, its boundary is a two-dimensional surface.

### Formal Definitions

Recall that the open ball with center (*a,b,c*) and radius *r*
consists of all points that satisfy the following relation:
(*x - a*)^{2} + (*y - b*)^{2}
+ (*z - c*)^{2} < *r*^{2}

A point *P* is an *interior point* of a solid *S*
if there exists a radius *r* such that the open ball with center
*P* and radius *r* is contained in the solid *S*.
The set of all interior points of solid *S* is the *interior*
of *S*, written as **int(***S*). Based on this definition,
the interior of an open ball is the open ball itself.
On the other hand, a point *Q* is an *exterior point* of
a solid *S* if there exists a radius *r* such that the open
ball with center *Q* and radius *r* does not intersect
*S*. The set of all exterior point of solid *S*
is the *exterior* of solid *S*, written as
**ext(***S*).

Those points that are not in the interior nor in the exterior of a solid
*S* constitutes the *boundary* of solid *S*,
written as **b(***S*). Therefore,
the union of interior, exterior and boundary of a solid is the whole space.

The *closure* of a solid *S* is defined to be the union of
*S*'s interior and boundary, written as **closure(***S*).
Or, equivalently, the closure of solid *S* contains all points
that are not in the exterior of *S*.

### Examples

Here is an example in the plane. You should change all open balls to
open disks. Point *A* is an interior point of the shaded area
since one can find an open disk that is contained in the shaded area.
Similarly, point *B* is an
exterior point. Point *C* is a boundary point because whatever
the radius the corresponding open ball will contain some interior points and
some exterior points.

Note that a surface (a two-dimensional object) is never a solid
(a three-dimensional object).
In fact, a surface does not have any interior
point. Take any point of the surface (see figure below), the open ball with
arbitrary radius and center at that point always intersects the sphere in an
open disk (in pale green in the lower right corner). Therefore, no open
ball can be contained in the sphere, and, as a result, that point is not an
interior point of the sphere. Thus, we conclude that a surface does not
have any interior point.