1.1 Introduction to Points, Lines, and Planes
Geometry is a theoretical, i.e., ideal, representation of space that is primarily concerned with establishing numerical properties of figures. Geometry represents an imaginative creation of a model of the physical world, a creation that is disciplined by logic. As such, it turns out to be remarkably and surprisingly useful in a vast array of activities and interests.
The building blocks of geometry are points, lines, and planes. These are analogous to the primary colors red, blue, and yellow in that just as all colors represent some combination of those three colors, so all the figures in geometry are made up of points, lines, and planes in one way or another. It is difficult to define these building blocks. For example, we may define a point as a location in space. But then when we define space, we are likely to say that it is made up of points. We have defined a point in terms of space and space in terms of points and such circular reasoning should make us feel uncomfortable. So we take these building blocks as undefined terms. A more sophisticated reason for taking points, lines, and planes as undefined is that rigorous thinking requires careful thinking. Every so often mathematicians discover they have used unstated, implicit assumptions in their reasoning that led them into a false conclusion or blinded them to the power and range of the mathematical ideas they were working with. In effect, they were working with a definition without realizing it. The story of the discovery of non-Euclidean geometry in the 1830's and 1840's is an instance of the degree to which previous mathematicians were blind to the kinds of models of space that geometry actually described.
Although we consider these building blocks to be undefined, we still understand certain things about them. For example, does a point take up room or not? If it takes up room, then a point behaves like a bead and a line segment would be a string of beads. That creates problems. Not only would each line segment have a finite number of points, but, for example, the line segment below consisting of 4 beads has no midpoint or midbead, while the line segment consisting of 5 beads has a midpoint.
Do we want a geometry in which some segments have midpoints and others don't? Since that would really complicate our work, it is better to insist that a point take up no room. So we'll think of a point as being a dimensionless location in space, i.e., a location with no length, width, or depth. That will mean that each line segment has a midpoint, but it also implies that each line segment, no matter how short or long, consists of an infinite number of points. Naturally, that result raises its own issues. For example, does a line segment of length 1 have the same number of points as a line segment of length 2, or does it have fewer? That is a thorny issue that wasn't resolved until the late 1800's. The issue deserves its own course; regretfully, we will pass over it. However, remember this: below the seemingly calm surface of geometry there is debate, there are arguments, sometimes bitter arguments that arise as we humans struggle to create a logically satisfying theory.
Representing Building Blocks:
We can represent our building blocks with both diagrams and letters as shown below:
We use $\overleftrightarrow{AB}$ for the line and $\overline{AB}$ to represent the line segment with endpoints A and B. Sometimes we'll use a lower case letter such as $\ell$ to name a line and will refer to planes with capital letters as in plane $M$. Notice that the diagrams have properties that the building blocks do not have. For example, a line has no thickness, but the diagram of a line does, and a plane does not have edges since it extends infinitely, but the diagram of a plane does.
Conventions Regarding Diagrams and Descriptions
To represent planes we use parallelograms. A horizontal plane is represented by $M$ and a vertical plane by $N$ in the diagram below. Note that with $M$ the observer is slightly above the plane; with $N$, the observer is slightly to the side.
We draw a line $m$ intersecting a plane in one point $P$ as shown below. We assume that if the line intersects the plane in two points, then it lies wholly in the plane as shown by the drawing of line $k$. The dotted line indicates that the line is passing under the plane. It is as if the plane were opaque and the line partially obscured. But when the line clears the edges of the parallelogram, we represent it as a solid line, knowing, of course, that for the observer, the plane would forever obscure the lower part of the line.
We draw two planes intersecting as shown below. Note that the line of intersection $k$ must be shown and although it extends infinitely, we don't draw it with arrows.
We draw two parallel planes in much the same way as we draw a picture of a box, the top and bottom representing two parallel planes.
Collinear points are points that lie on a line. Coplanar points are points that all lie on some plane. Shown below are collinear points $A$, $E$, and $B$, as well as coplanar points $A$, $B$, $C$, and $D$. Note that points $A$, $B$, $C$, $D$, and $F$ do not form a coplanar set of points.
To show that a plane and a line don't intersect, we draw the line $m$ above plane $M$ as shown. A line such as $k$ actually lies on the plane, since the plane extends infinitely.