In geometry, parallel lines are lines in a plane which do not meet; that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel.
By extension, a line and a plane, or two planes, in three-dimensional Euclidean space that do not share a point are said to be parallel. The expected answer would be that Because it is my name and that's it. Actually parallel lines cannot meet at a point or intersect because they are defined that way, if two lines will intersect then they will not remain parallel lines. When I was 16 or so I was bored trying to solve my math homework, so I played with a magnifying glass, and I noticed something interesting:.
If you look at the grid paper with a magnifying glass the lines remain parallel this is evident when they are more or less on the "top of the glass" , but all the lines meet at the edge of the glass. I went to my older brother, who was an engineering freshman at the time, and I told him that parallel lines can meet; but he replied that they cannot because it's an axiom. Some years later I learned that there is a thing called non-Euclidean geometry.
In his comment JackyChong has identified the preliminary problem. The definition of "parallel" is clear: lines that don't meet, so there are no parallel lines that meet. The real question is the definition of "line". Geodesic is the most natural for geometry on the sphere. Then in this geometry there are no parallel lines.
For projective geometry one definition is to add a "point at infinity" on each line, and then make the added points into a "line at infinity". With these extra points and lines there are no parallel lines. Two that are parallel in the Euclidean plane share their point at infinity.
The railroad track analogy helps here. In hyperbolic geometry there are multiple lines parallel to a given line through a point not on that line. If you get that far with the "someone who knows no mathematics" you can show him or her the Poincare model. You cannot convince any mathematician let alone your friend as it is false. They can however grapple with the idea that for purely theoretical calculations we assume that 2 parallel lines can meet at a point in infinity.
Should you wish to travel to infinity to prove this point then please send a postcard when you get there. Someone who knows basic mathematics can understand it easily since according to the definition of parallel lines Parallel lines: The lines those distance is constant are called parallel lines [see the topmost definition]. Hence, as coinciding lines touch each other implies parallel lines also touch. See the video here also.
I would draw two parallel lines on a piece of paper. Then I would bring all the edges together into one point and I would show that the parallel lines would touch into that point.
A sentence said to me by my mother when I was a kid and that I didn't understand at that time was:. This statement and its explanation though art as suggested by Pedro Tamaroff in the comments and by showing how many statements and proofs are simplified if the person knows more math is a good introduction to the idea that thinking on parallel lines as lines meeting at infinity makes sense and can be useful.
This, as you can also see in Yves Daoust's answer, can be seen whenever you look a straight road going to the horizon. For example, see Wikimedia Commons for the source, The above image shows the reason why this conception was in the beginning a concept originated in art as it was really useful for representations of how we see things.
An striking application of this can be seen in the Santa Maria presso San Satiro , where projective geometry was used emulating an absent space in a church.
However, there are many other examples. In usual geometry i. This can be the definition in the plane, but it is generally in higher dimensions a result derived from the definition of parallel lines as lines with the same direction.
However, the reason why we can still make sense of the above statement about intersection at infinity is because affine geometry can be put inside projective geometry. When doing this, the points outside the affine space are called "points at infinity", parallel lines intersect at them and become the same as intersecting lines simplifying enormously many statements and proofs by permitting one to not distinguish cases.
An example can be Pappus's hexagon theorem. In conclusion, don't try to convince or show that parallel lines touch. Just try to explain the usefulness of thinking of parallel lines as lines that intersect at infinity.
In the Renaissance, you have many examples of why this a useful statement from the point of view of representing reality and perspectives well; in mathematics, there are many examples of how this is really useful for simplifying statements and proofs in geometry.
In the flat Poincare disk model circle segment geodesic parallels meet tangentially only at infinitely distant points on the boundary of the "horizon" or boundary circle. The lines can be seen to touch and move on the boundary seen in the Wiki link.
An animation can be also found elsewhere semi-circles touch on x-axis while changing semi-circle size during movement. One way to explain this is to think of the Euclidean plane as a picture of part of an ambient 3-dimensional world. This corresponds to how our vision actually works: light from the three-dimensional world is projected through the lens of an eyes onto the surface at the back of our eye. In more detail: Place a plane anywhere in three-dimensional space as long as it doesn't pass through the origin.
Call this the "picture plane". Any plane through the origin pierces the picture plane in exactly one line again, as long as the plane is not parallel to the picture plane. See below in which the picture plane is shown in blue.
These are points and this line can be considered "at infinity", but really all this means is that they do not have images in the picture plane.
You can experience this visually by holding your finger at arm's length in front of and a few inches above. As you draw your arm closer while maintaining its height the location of your fingertip seems to get "higher" or "farther away" in your field of view, as your eyes have to strain more and more to see it. Eventually when the fingertip is directly above the eye, the finger "disappears". It's not really gone, of course, but the "line of sight" from the fingertip to the lens of your eye is parallel to the retina, so that the fingertip cannot be seen.
Perceptually, the fingertip has receded "infinitely far away" in the picture plane -- but it is actually just a couple of inches away in 3-space. However , we all know what to do in that circumstance: turn your head! This corresponds to choosing a new picture plane with a different orientation. Now the point that was "at infinity" snaps into view and is revealed as an ordinary point in the new picture plane and at the same time points that were previously in view have no disappeared.
Now that we have this dictionary set up, let's think about what parallel lines in the picture plane are. Those planes are not parallel; they intersect, but the intersection is a line that does not pierce the picture plane.
Of course if you choose a different picture plane, then the lines will no longer appear parallel, and their intersection will be plainly visible as a point in the new picture plane. The final image below shows this. In that image, the orange and yellow planes are the same ones as before, but the blue picture plane has been moved and reoriented; the images of the two planes are now non-parallel lines.
Visually or conceptually parallel lines converge over an extremely large distance, they NEVER intersect. The same thing can be said about ARCs if two arcs belong to two different circles with different radius lengths where both radii have the same starting position; another words concentric circles as in a bulls eye in a dart board or an archery board or if they are separated at a distance greater than the sum of their radii else if either of their radius are different lengths and the do not have the same center point then they can have an intersection point.
There may be cases where they don't have the same center point and still don't intersect. Of the next few images only the concentric circles and ellipses would be considered parallel, the rest either intersect or don't. The only one of these that can be considered parallel are the Concentric Circles since each arc has the same distances of separation from another circle.
The Same can be applied for ellipses - Except they have two radii of different measure, one for the x-axis and one for the y-axis. I've noticed that many have used the terminology or the statement saying: Place a Point At Infinity either in their comments or in their answers. Figure 1 Intersecting lines. Two lines that intersect and form right angles are called perpendicular lines. Figure 2 Perpendicular lines. Two lines, both in the same plane, that never intersect are called parallel lines.
Parallel lines remain the same distance apart at all times. Figure 3 Parallel lines. The boards sleepers between the rails on a track are also parallel because they do not intersect and are the same distance apart. Notice that the sleepers and the rails touch. This is called an intersection. The place that two lines intersect at is called an intersection point. Above is an illustration of a single point that acts as an intersection point for several lines.
An intersection point can have infinitely many lines going through it, however two lines can only intersect at one point. These two properties really states the same thing--two lines always having the same distance apart implies they will never meet, and vice versa. Contrary to parallel lines, the distance between intersecting lines varies at different points on the line and the lines intersect at one point. We call this point an intersection. Identifying Intersections and Parallel Lines - Expii When two lines or rays, or segments run into each other, it's called an intersection.
Parallel lines don't have any intersections with each other, since they'll never run into each other. Geometry Lines, Shape, and Measure in Geometry.
0コメント