Points
Points are the basis of all Geometry. There are so many things you can do with the little buggers that the possibilities are endless. Points are zero-dimensional. That basically means that they have no height, length, or width. They are just there.
There are four main definitions of a point. They are the dot, the exact location, the ordered pair, and the node. A point has four definitions because, over the years, many different mathematicians have come up with their own ideas as to what a point should be. Since their ideas were all equally true, the point was given four main definitions instead of a single definition. In fact, the point is considered undefined for that reason (among others). When being written out, points are always represented by a capitol letter. If a point is on a line, it is often represented by the same letter.
The first definition of a point is the dot. This was probably the first kind of point ever thought up. You see, a dot has size - it has a definite, measureable, length and width. Probably the best example of a dot today would be the pixel. Yup, that's right - a pixel. Those tiny spots of color that make up your computer screen. A matrix is a rectangular array made up of lots of pixels, so that's what your computer screen is. As you most likely know, the more pixel in a computer or TV screen, the better the resolution.
The second definition of a point is an exact location. The exact location is the perfect example of the normal, zero-dimensional point. No matter how much you zoom in, there will always be another point in between two others. This definition of a point was discovered sometime between 550 B.C. and 150 A.D. One example of where these are used in real life is in measuring distances, especially between two cities. Some cities are more than a mile across, so mapmakers have to pick one exact location in the city to measure from. One use of the exact location ties in with the the next definition, the ordered pair. The number line, or coordinatized line, is a line where every point is represented by a number and vice versa.
The third definition of a point is the ordered pair. The ordered pair was discoverd around 1630 A.D. by two mathematicians named Pierre de Fermat and René Descartes. You've probably heard of the latter before. Basically, an ordered pair is a pair of numbers in parentheses used to locate an exact location on a coordinate plane. The first number, represented by the variable x, tells you how far along the x-axis the point is(to the right or left, depending on the "polarity"/direction of the variable). The second number does the same, just along the y-axis (up and down). Ordered pairs that consist of whole numbers are called lattice points. Numbers in a number line, coordinate plane, or coordinate space, have both magnitude and direction. Magnitude is its distance from the origin, and direction is its positivity or negativity. The origin is the center of any graph. It marks the place where the measurements start. Everything one one side is positive, and everything along the other side is negative. The origin extends its "neutrality" along lines called the x, y, and z-axes. These lines need only one variable to represent them. The variable whose name they carry is the one that represents them, and its value is always zero.
The last definition of the point is the node. A node is a type of point thatis zero-dimensional, and two nodes can have more than one line between them. Nodes exist only in networks, which are a series of nodes and arcs. Arcs are lines that may curve and aren't dense. Arcs only contain their endpoints. There is a special kind of network in which all arcs can be crossed without going over one more than once. It is called a traversible network. Another kind of network is a graceful network. This kind of network is really hard to make is and is great for extra credit projects. To make a network graceful with x arcs, you must label each node with a number from 0 to x so that, if you find the positive difference of two connecting nodes and label their common arc with it, all arcs are labeled from 1 to x. If a node has an even number if arcs going through it, it is called an even node. An odd node is just the opposite. Nodes are sometimes called vertices.
Four important characteristics help to distinguish the different definitions of points:
- Unique line - Do the points determine a single line?
- Dimension - Are the points without size?
- Number line - Can the points of a line be put into a one-to-one correspondece with real numbers?
- Distance - Is there a unique distance between two points?
- Points as dots don't have #2 and #3 and may or may not have the others. Nodes don't have or #3, but have #2 and may or may not have the others. Points as locations and ordered pairs have all four.
A line is a one-dimensional figure. That is, a line has length, but no width or height. Basically, a line is made up of an infinite number of points. Points in the same line are called colinear. Between each point is another point. This continues on forever. You can never run out of points to discover in a line. However, when you are talking about points as dots, you can get something called a discrete line. A discrete line is a line made up of dots with space between the centers of the dots. A dense line is a line that is the shortest path between two points. The number line, or coordinatized line, is a line where every point is represented by a number and vice versa. The number line is a one-dimensional graph. See the paragraph about nodes to find out about networks and arcs.
If you have two points A and B, the line that contains them is the set of points consisting of the distinct points A and B, all of the points between them, all points for which A is between them and B, and all points for which B is between them and A. A line like that would be written . A line, if not made up by previously known points, can be represented by a single lowercase letter. This is as a contrast to the uppercase letters that represent points. A line segment is the set of points consisting of A, B, and all points between them. A line segment is written . If you have two points A and B, the ray that contains them is the set of points consisting of the distinct points A and B, all of the points between them, and all points for which B is between them and A. This is written.
Every line is either horizontal, vertical or oblique. Horizontal and vertical speak for themselves, and an oblique line is any line that isn't horizontal or vertical. Horizontal lines have a slope of zero. Vertical lines are said to have infinite slope, because they just go straight up and not over. People just can't stand that zero in the denominator. Here's something I bet you didn't know: In space, vertical lines never meet (they just go straight up/down), but it is possible for horizontal lines to meet (check out the corner of the ceiling - two horizontal lines meet there (the edge of the two walls)). Okay, so maybe you did know that.
There are four different relationships that two lines can have. Lines can be identical, intersecting, parallel, perpendicular, or skew. Identical lines are lines that coincide. Therefore, they are the same line. The second one is the most obvious. Intersecting lines are lines that share a point. Parallel lines are coplanar lines that never intersect. They always have a certain distance between them and always have the same direction. See the page on parallel lines for more information. Perpendicular lines are lines that intersect in one point and form a 90 degree angle while they're at it. They have a page of their own, too. Skew lines only happen in space. They are noncoplanar lines that never intersect. Unlike parallel lines, however, they don't always have a set distance between them, nor do they always have the same direction.
Planes are two-dimensional. A plane has length and width, but no height, and extends infinitely on all sides. Planes are thought of as flat surfaces, like a table top. A plane is made up of an infinite amount of lines. Two-dimensional figures are called plane figures. While this really should be in Algebra, coordinate planes are two-dimensional graphs that use the ordered pair to locate points. Another name for coordinate planes are Cartesian planes.
Space is the set of all points. It is made up of an infinite number of planes.Figures in space are called solids or surfaces. Coordinate space uses three coordinates. Instead of an ordered pair, an ordered triple is used. The new variable, z, measures the distance forwards or backwards that you move. The ordered triple looks like this: (x, y, z).