Introduction to Plane Geometry
Solid geometry involves points, lines, and planes. The difference between plane geometry and solid geometry is the fact that there’s an extra dimension. This means greater freedom, such as we would enjoy if we had flying cars. It also means that things are more complicated, reflecting the expanded range of maneuvers we would have to master if we had flying cars.
Point
A point in space can be envisioned as an infinitely tiny sphere, having height, width, and depth all equal to zero, but nevertheless possessing a specific location. A point is zero-dimensional (0D). A point in space is just like a point in a plane or a point on a line.
Line
A line can be thought of as an infinitely thin, perfectly straight, infinitely long wire. It is one-dimensional (1D). A line in space is just like a line in a plane, but there are more possible directions in which lines can be oriented in space, as compared with lines confined to a plane.
Plane
A plane can be imagined as an infinitely thin, perfectly flat surface having an infinite expanse. A plane is two-dimensional (2D). A plane comprises a “flat 2D universe” in which all the rules of Euclidean plane geometry apply.
Space
Space is the set of points representing all possible physical locations in the universe. Space is three-dimensional (3D). The idiosyncrasies of time, often called a “fourth dimension,” are ignored in Euclidean space .
Time - 3D
An alternative form of 3D can be defined in which there are two spatial dimensions and one time dimension. This type of three-space can be thought of as an Euclidean plane that has always existed, exists now, and always will exist.
Hyperspace - 4D
If time is included in a concept of space, we get four-dimensional (4D) space, also known as hyperspace. We’ll look at hyperspace later in this book. It, as you can imagine, gives us “hyperfreedom.”
Naming Points, Lines, And Planes
Points, lines, and planes in solid geometry are usually named using uppercase, italicized letters of the alphabet, just as they are in plane geometry. A common name for a point is P (for “point”). A common name for a line is L (for “line”). When it comes to planes in 3D space, we must use our imaginations. The letters X, Y , and Z are good choices. Sometimes lowercase, nonitalic letters are used, such as m and n.
When we have two or more points, the letters immediately following P are used, for example Q, R, and S . If two or more lines exist, the letters immediately following L are used, for example M and N . Alternatively, numeric subscripts can be used. We might have points called P 1, P 2, P 3, and so forth, lines called L 1, L 2, L 3, and so forth, and planes called X 1, X 2, X 3, and so forth.
Point and Line Principles
Three Point Principle
Suppose that P, Q , and R are three different geometric points, no two of which lie on the same line. Then these points define one and only one (a unique or specific) plane X . The following two statements are always true, as shown in Fig. 7-1:
P, Q , and R lie in a common plane X- X is the only plane in which all three points lie
Fig. 7-1 . Three points P, Q , and R , not all on the same line, define a specific plane X . The plane extends infinitely in 2D.
In order to show that a surface extends infinitely in 2D, we have to be imaginative. It’s not as easy as showing that a line extends infinitely in 1D, because there aren’t any good ways to draw arrows on the edges of a plane region the way we can draw them on the ends of a line segment. It is customary to draw planes as rectangles in perspective; they appear as rectangles, parallelograms, or trapezoids when rendered on a flat page. This is all right, as long as it is understood that the surface extends infinitely in 2D.
Intersecting Line Principle
Suppose that lines L and M intersect in a point P . Then the two lines define a unique plane X . The following two statements are always true, as shown in Fig. 7-2:
- L and M lie in a common plane X
- X is the only plane in which both lines lie
Fig. 7-2 . Two lines L and M , intersecting at point P , define a specific plane X . The plane extends infinitely in 2D.
Line And Point Principle
Let L be a line and P be a point not on that line. Then line L and point P define a unique plane X . The following two statements are always true:
- L and P lie in a common plane X
- X is the only plane in which both L and P lie
Plane Regions, Half Planes, and Intersecting Planes
Plane Regions
The 2D counterpart of the 1D line segment is the simple plane region . A simple plane region consists of all the points inside a polygon or enclosed curve. The polygon or curve itself might be included in the set of points comprising the simple plane region, but this is not necessarily the case. If the polygon or curve is included, the simple plane region is said to be closed . Some examples are denoted in Fig. 7-3A; the boundaries are drawn so they look continuous. If the polygon or curve is not included, the simple plane region is said to be open . In Fig. 7-3B, several examples of open simple plane regions are denoted; the boundaries are dashed.
Fig. 7-3 . Plane regions. At A, closed; at B, open.
The respective regions in Figs. 7-3A and 7-3B have identical shapes. They also have identical perimeters and identical interior areas. The outer boundaries do not add anything to the perimeter or the interior area.
These examples show specialized cases, in which the regions are contiguous, or “all of a piece,” and the boundaries are either closed all the way around or open all the way around. Some plane regions have boundaries that are closed part of the way around, or in segments; it is also possible to have plane regions composed of two or more non-contiguous sub-regions. Some such plane regions are so complex that they’re hard even to define. We won’t concern ourselves with such monstrosities, other than to acknowledge that they can exist.
Half Planes
Sometimes, mathematicians talk about the portion of a geometric plane that lies “on one side” of a certain line. In Fig. 7-4, imagine the union of all possible geometric rays that start at L , then pass through line M (which is parallel to L ), and extend onward past M forever in one direction. The region thus defined is known as a half plane .
Fig. 7-4 . A half plane X , defined by two parallel lines, L and M . The half plane extends infinitely in 2D on the “M” side of L .
The half plane defined by L and M might include the end line L , in which case it is closed-ended . In this case, line L is drawn as a solid line, as shown in Fig. 7-4. But the end line might not comprise part of the half plane, in which case the half plane is open-ended . Then line L is drawn as a dashed line.
Parts of the end line might be included in the half plane and other parts not included. There are infinitely many situations of this kind. Such scenarios are illustrated by having some parts of L appear solid, and other parts dashed.
Intersecting Planes
Suppose that two different planes X and Y intersect; that is, they have points in common. Then the two planes intersect in a unique line L . The following two statements are always true, as shown in Fig. 7-5:
- Planes X and Y share a common line L
- L is the only line that planes X and Y have in common
Fig. 7-5 . The intersection of two planes X and Y determines a unique line L . The planes extend infinitely in 2D.
Parallel and Skew Lines
Parallel Lines In 3d Space
By definition, two different lines L and M in three-space are parallel lines if and only if both of the following are true:
- Lines L and M do not intersect
- Lines L and M lie in the same plane X
If two lines are parallel and they lie in a given plane X , then X is the only plane in which the two lines lie. Thus, two parallel lines define a unique plane in Euclidean three-space.
Fig. 7-4 . A half plane X , defined by two parallel lines, L and M . The half plane extends infinitely in 2D on the “M” side of L .
Skew Lines
By definition, two lines L and M in three-space are skew lines if and only if both of the following are true:
- Lines L and M do not intersect
- Lines L and M do not lie in the same plane
Imagine an infinitely long, straight, two-lane highway and an infinitely long, straight power line propped up on utility poles. Further imagine that the power line and the highway center line are both infinitely thin, and that the power line doesn’t sag between the poles. Suppose the power line passes over the highway somewhere. Then the center line of the highway and the power line define skew lines.
Points, Lines, Planes, and Space Practice Problems
PROBLEM 1
Find an example of a theoretical plane region with a finite, nonzero area but an infinite perimeter.
SOLUTION 1
Examine Fig. 7-6. Suppose the three lines PQ, RS , and TU (none of which are part of the plane region X , but are shown only for reference) are mutually parallel, and that the distances d 1 , d 2 , d 3 , . . . are such that d 2 = d 1 /2, d 3 = d 2 /2, and in general, for any positive integer n , d ( n +1) = d n /2. Also suppose that the length of line segment PV is greater than the length of line segment PT . Then plane region X has an infinite number of sides, each of which has a length greater than the length of line segment PT , so its perimeter is infinite. But the interior area of X is finite and nonzero, because it is obviously less than the interior area of quadrilateral PQSR but greater than the area of quadrilateral TUSR .
Fig. 7-6 . Illustration for Problem 1.
PROBLEM 2
How many planes can mutually intersect in a given line L?
SOLUTION 2
In theory, an infinite number of planes can all intersect in a common line. Think of the line as an “Euclidean hinge,” and then imagine a plane that can swing around this hinge. Each position of the “swinging plane” represents a unique plane in space.
Practice problems for these concepts can be found at: Points, Lines, and Planes Practice Test.
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From Geometry Demystified: A Self-Teaching Guide. Copyright © 2003 by The McGraw-Hill Companies, Inc. All Rights Reserved.
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