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Geometric Surfaces

Three Dimensions

Terms

Problems

Surfaces

Just like a curve is the basic building block for figures in a plane, a surface is the basic building block for figures in space. A surface is essentially a curve with depth. Curves and surfaces are analogous in many ways. If you think of a curve as being the trace of the motion of a point in a plane, a surface is like the trace of the motion of a curve in space. Surfaces are continuous, meaning that given two points on a surface, you can start from one and reach the other without leaving that surface. Just like a curve is still one-dimensional, a surface, although it exists in three dimensions, is still two-dimensional. For example, when you build a curve by tracing the motion of a point, that curve, although it spans both length and width, has no width of its own. The curve doesn't have area, it only has length, one dimension. Similarly, a surface can span more than one plane, but it still does not have depth of its own. It only has two dimensions, length and width. We will work mostly with the simplest surface, a plane. Below various surfaces are pictured.

Figure %: Surfaces in space

Surfaces can be classified as being closed or simple closed surfaces. The surfaces that form the boundaries of geometric solids are simple closed surfaces, so we'll focus on them. A simple closed surface is one that divides space into three distinct regions:

  1. The set of all points inside the surface (the interior of the surface).
  2. The set of all points outside the surface (the exterior of the surface).
  3. The set of all points on the surface.
A point is only interior if it can be joined to any other interior point by a segment of finite length. This is not true for exterior points--a segment joining exterior points could have infinite length, since the endpoints could be anywhere in space, and space is infinite.

A simple closed surface can also be either convex or concave. The rules are very similar to those we saw in Polygons. A convex surface is one in which any two points on that surface can be joined by a segment that lies either on the surface or in the interior of the surface. A concave surface has a segment between points on the surface that lies in the exterior of the surface.

One more note on surfaces: a surface, even if it is a simple closed surface, does not include the space in its interior. When a simple closed surface is united with its interior points, it is no longer a surface, it is a geometric solid.

Lines and Planes

So far we've only discussed parallelism and perpendicularity with respect to lines, but planes can be parallel and perpendicular, too. To understand relationships between planes, one must understand relationships between lines and planes.

A line and a plane are parallel if and only if they do not intersect. A line l and a plane are perpendicular if and only if the line l is perpendicular to every line in the plane that contains the intersection point of line l and the plane. These cases are drawn below.

Figure %: A line can be parallel to or perpendicular to a plane

Planes are parallel to each other if they don't intersect. Planes are perpendicular to each other if one of the planes contains a line perpendicular to the other plane. Here are the drawings:

Figure %: Planes can be parallel or perpendicular to one another

Polyhedra

A polyhedron is a simple closed surface that is the union of polygons. (Sound familiar? Remember that a polygon is a simple closed curve that is the union of segments. Much of three-dimensional geometry is only an extension of two-dimensional, or planar, geometry.) A polyhedron encloses a region in space. It does so with parts of intersecting planes, each part being a polygon. Each polygon that composes a side of a polyhedron is called a face of the polyhedron. The intersection of two faces is called an edge. The intersection of three or more faces is called a vertex. Below are pictured examples of polyhedra.

Figure %: Polyhedra
The following lessons will discuss certain kinds of polyhedra and their general forms.

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