Parallel Lines Sometimes Meet REPACK

Download ->>->>->> https://tiurll.com/2t3npL

Perpendicular lines intersect at right angles (90 degree) while parallel lines are parallel to each other and never intersect. Graphically, parallel lines are two or more straight lines that do not touch each other even after extending them. However, perpendicular lines intersect each other at $90^{\\circ}$ .

Linear perspective is a rendering technique used by fine artists to create the illusion of depth on a flat surface. It is the most basic form of perspective in which parallel lines appear to converge in the distance at a vanishing point on the horizon line. (See illustration to the right.)

Also called orthogonal lines, convergence lines are when sets of parallel lines appear to get closer together as they recede into the distance and meet at a single vanishing point. All parallel lines will eventually converge at a vanishing point. Sometimes they can even represent the edges of objects, and some objects can have more than one set of parallels lines. An example of this would be a box or cube. Depending on where it is viewed from, we can see one, two, or three sets of orthogonal lines.

The point on the horizon line where all parallel lines appear to recede and converge is called the vanishing point. It is helpful to note more than one vanishing point can be present. This is called two-point and three-point perspective. There will be two vanishing points when there are two sets of parallel lines that appear to converge. If there are three sets of parallel lines, then there will be three vanishing points. See The Rules of Perspective for more information.

Of course we don't intend that "meet at infinity" talk to be taken literally. There is no place at infinity where the lines meet. All we really mean is that the lines go off indefinitely in the same northerly (or southerly) direction. The place at infinity where we imagine them meeting is just a reification of the idea that they persist indefinitely in going in the same direction.

It means that they don't meet, because as you correctly pointed out parallel lines never meet.Then what's the point in saying "they meet at infinity" if they never meet? Because you can obtain a parabola by an ellipse with focal distance $d$ in the limit where $d\rightarrow\infty$. In the ellipse rays from one focus get reflected to the other one, and the same happens for a parabola, but one focus is at infinity (therefore you'll never reach it, and therefore the rays won't meet).

As the object gets closer to the focus, the image (where the rays meet) gets farther and farther away, without any bound. You can make the image be as far away as you want, by bringing the object close enough. When the object is exactly at the focus, the rays are parallel, and thus never meet.

Why do we do this? The intuition here is this. If you look down a pair of extremely long parallel lines extending far from you, the two appear to your eyes like they would eventually converge at some point in the distance, even if they don't "in reality". The idea here is to take this apparent limiting point and make it real in the idealized mathematical world. Then the mathematical lines intersect there. Moreover, if you move laterally from side-to-side, so that you are looking down the parallels from a different "origin", the distant point never shifts.

A Hey, these two lines are parallel!B Indeed.A I wonder where they meet?B Follow the lines, and you will see them meet when you have travelled an infinite distance.A But I will never complete a journey of an infinite distance!B Exactly.

Another is to treat optics as taking place in a projective space. The claim that parallel lines never intersect is true in standard Euclidean space, but in projective space, parallel lines do intersect (in the projective plane, lines are dual to points; just as every two points define a line, every two lines define a point). The point at which they intersect is often refer as being "infinity", but there are constructions of projective space that don't make any reference to "infinity". Also, different pairs of parallel lines intersect at different points, so if the points are referred to "infinity", it has to be understood that there are different "infinities".

But that's what's happening here: you've got two rays that approach each other by an infinitely small amount, such that they'd never meet over a finite distance. But when we start talking about infinitely-large distances, then it's sorta like saying$\begin{align}\infty \times \frac{1}{\infty} &= \infty \times 0 \\& \Downarrow \\\frac{\infty}{\infty} &= 0 \,,\end{align}$where the silliness of the proposition becomes apparent.

In Euclidean geometry parallel lines always remain at a the same distance from each other no matter how far you extend those lines. But in non-Euclidean geometry, parallel lines can either curve away from each other (hyperbolic), or curve towards each other (elliptic). This looks like this:

We see that the lines form 90-degree angles, just like in the first black and white sketch. These lines are parallel at the zoomed in point. But if we zoom out again, we see that the black lines are curving away from each other, meaning that the black lines we are seeing look like they are hyperbolic.

You see that these three lines, which are all parallel to each other at the top (the part we zoomed in on), all meet at the common point at the bottom of the basketball when you extend those lines along the natural curvature of the ball.

Sometimes those lines are exactly on top of each other. Other times they are spiraling around like diving birds on the summer wind. But parallel or intersections, we continue to move on together.

As a European, I find the (apparent) typical American view to be insular, isolationist, "everyone speaks English" (except Americans!!!), "no-one else matters" etc etc.I'm a committed European, speak four languages (two of them not very well at all), live on the Western side of my continent but love the Eastern side, ... as everything tends to sameness WE ALL LOSE!In databases everyone speaks relational ... you know my feelings on that ... in that field EVERYONE HAS LOST because - imho - C&D's first rule (you know, "data comes in rows and columns") has *even* *less* truth in it than Euclid's "parallel lines never meet" - and it's busy wreaking the same amount of damage ...Cheers,Wol Paralysis? Posted Dec 18, 2013 9:06 UTC (Wed) by micka (subscriber, #38720) [Link]

I'm not sure that's what you meant it to be.Parallel lines are defined as lines that don't meet (at least in dimension 2). Saying that parallel lines never meet is (tautologically) true.If you're talking about projective geometry, then you use a different definition of parallelism, it doesn't contradict Euclid as it doesn't talk about the same thing.And if you talk about Euclid fifth postulate, it doesn't say that. Paralysis? Posted Dec 18, 2013 19:36 UTC (Wed) by Wol (subscriber, #4433) [Link]

If there are more dimensions, then depending on their shape, parallel lines may eventually meet, or they may eventually diverge. The Universe might be on the 'inside' of a vast 4D bowl or sphere, or could be riding on the back of a gigantic 4D saddle shape. (we can't visualize these things well: these are the approximations that I've seen for two of the possible shapes of a hypothetical 4D universe.) The whole axiom of parallel lines not meeting carries a whole worldview with it: that the universe is three-dimensional, and perfectly flat. It's an axiom because nobody can prove it: it's defined as being true, but that doesn't mean it's actually true. That definition of truth, by imposing an underlying mental framework on anyone who accepted it, may have slowed the evolution of geometry. Paralysis? Posted Dec 19, 2013 8:43 UTC (Thu) by dlang (guest, #313) [Link]

And as soon as we get into the world of Relativity then parallel lines meet all the time ... for example all lines perpendicular to the surface of a black hole are mutually parallel!Mind you, the definition here of a straight line has changed too :-) in 3d space a straight line is the shortest *distance* between two points. In 4d space it's the shortest *time*.But until that postulate (thanks for giving me the right word) was proven not be an axiom, that assumption sat there blocking any possible advance towards relativity and quantum mechanics. If it were still there, modern science would not exist ...Cheers,Wol Paralysis? Posted Dec 18, 2013 20:29 UTC (Wed) by mathstuf (subscriber, #69389) [Link]

The unveiling of this 1,700-pound bronze statue in honor of Barry Goldwater was a special symbolic moment, not because the onetime senator from Arizona is regarded as the founding father of modem conservatism, not because he was an early and sometimes lonely supporter of contemporary causes such as gay rights, and not because the hard edges of political personalities almost always get worn away by the passage of time. This was an important moment because the unveiling won praise from men and women who seldom agree on anything, and whose view of our national passage , from Goldwater and Lyndon Johnson to the tea party and President Barack Obama, run along parallel lines that do not meet.

The other two objects in the Still Life represent different two-dimensional geometries. For over 2000 years mathematicians had assumed that a surface whose geometry satisfies four special rules, set out by the Greek mathematician Euclid ofAlexandria, also satisfies a fifth rule called the parallel postulate. This postulate can be stated in various ways; one of them says that given a straight line l and a point p not on the line, there is exactly one line through p that never meets l, in other words that is parallel to l. 2b1af7f3a8