Points, Lines, Planes, and Space Help
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.
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.
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.
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 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
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.