 Welcome back everyone to our lecture series Math 3130, Modern Geometry for Students at Southern Utah University. As usual, I'm your professor, Dr. Andrew Misledine. It's good to have you here today. This lecture entitled the Hyperbolic Parallel Postulate is going to be loosely based upon section 6.3, the Parallel Postulate and some implications from the textbook Roads to Geometry by Wallace and West. If you remember from our previous lectures, we've been talking about recently the Euclidean Parallel Postulate, equivalence of Euclidean Parallel Postulate in neutral geometry and just Euclidean Geometry in general. So we've spent the last two lectures really focus on Euclidean Geometry, but it's not the intention of me or for this course to actually prove all of the well-known theorems of Euclidean geometry or even come close to a reasonable subset of these theorems. With a few exceptions, the reader has certainly had some great exposure and familiarity with these theorems through probably past courses aka high school geometry, maybe trigonometry classes, calculus, even linear algebra usually has tons of Euclidean geometry in it even though it might not be obvious. Because of these reasons for the remainder of this lecture series here, I really don't want to focus on Euclidean geometry whatsoever, I actually want to focus on non-Euclidean geometries. Particularly, we're going to talk about the hyperbolic geometry, and later on we'll talk about elliptic geometry as well. Any future reference Euclidean geometry will really only serve as a comparison to these non-Euclidean geometries hyperbolic and elliptic. If students or viewers here are interested in learning more about the development of Euclidean geometry from its axioms, I highly encourage you to study the remaining sections of the book, Roads to Geometry, which does a really good job documenting and developing Euclidean geometry. Of course, there's many other geometric textbooks out there. Greenberg's geometry books pretty good as another notable reference, but there's no shortage on good textbooks on Euclidean geometry. Our rigorous treatment of axiomatic geometry should have well-prepared everyone's. I mean, if you participate in this lecture series up to this point, you're in a really good situation where you could study the topics, theorems, proofs that are conveyed in these Euclidean geometry textbooks, and you could probably do it individually. I actually encourage you to believe in yourself, and I bet you could do it. Give it a try. Lifelong learning is not just something that takes place in a classroom or through a YouTube video with an instructor. It rests in your hands and so I encourage you to try to do that. For our purposes, we are going to be focusing the next couple of lectures on hyperbolic geometry. The definition is in front of us, we say that a neutral geometry is hyperbolic, if it satisfies the hyperbolic parallel postulate, which remember the hyperbolic parallel postulate tells us that we have a multiplicity of parallel lines. If we have a line L and a point off of that, there could be multiple parallel lines going through that point. I mean, for hyperbolic, that's exactly the case. I do want to remind the viewer of probably the two most famous models of hyperbolic geometry. There is what we'll call the half plane model where if we take the x-axis and we look at all the points above the x-axis in the Euclidean plane, then there's two types of lines we define in this situation. There is one line, which would be a Euclidean half vertical ray that goes upward, and so we consider this a hyperbolic line, and then we also will take semicircles whose center lives on the x-axis there. Now, in terms of this half plane model, the half plane model, with the half plane model, the points on the x-axis do not actually belong to the model whatsoever and also anything below the x-axis doesn't belong. This model consists of only those Euclidean points who have a strictly positive y-coordinate. With these examples right here, they're kind of like there's two types of lines here, these semicircles and these rays, but really we don't really want to think of these as anything distinguishable whatsoever. These rays are always going to be perpendicular to the x-axis and also these semicircles are chosen in such a way that the semicircle is also perpendicular. And what we mean by a line and a circle being perpendicular to each other, of course talking about the tangent line associated to that semicircle at those end points is going to be perpendicular to the line as well. And so with this half plane model, we can actually talk about parallel lines. And so if there was a point, say right here, P, and this is our line L, we can actually find lots of lines parallel. One of them would be this ray here. Another one might just be a semicircle. We could get something like maybe the following. This would give us a second and we can actually have more steeper and steeper sipping circles, that is circles with much larger radii that would give us more and more parallel lines. And so in this model, there will be an infinitude of parallel lines through P that are parallel to the original L. That gives you one model to think of hyperbolic geometry. The other commonly used model is the disk model, the disk model, in which case the definition is very similar. We'll take the unit disk that has all the points interior to the unit circle in the plane. And we take lines, hyperbolic lines to be the following. We either take diameters of the disk, that's one example, or we take semicircles, or we take circles which are perpendicular to this boundary disk. So we want that the angle between the arc that we drew with the boundary circle to be right angles right here. And so similar type thing here, if we take a line L, which I just drew right there, and we take points P, for example, we can find a multitude of parallel lines that go through this point P that are parallel to the original line L. I should mention the first diameter I drew there was not parallel to L, so we could kind of get rid of it. But the two lines you now see going through P are in fact parallel to L. And so these are both examples of hyperbolic geometry. They satisfy all the axioms of incidence, betweenness, congruence, and continuity. But we also have this multiplicity of parallel lines. Now, just like Euclidean geometry, there exist many equivalent forms to the hyperbolic parallel postulate. And several of these are listed in the book, Wallace and West Roads to Geometry. Other textbooks, again, like Greenberg, and other geometric textbooks can have those. I'm not going to write a comprehensive list or even come close to write a comprehensive list. We kind of did that for Euclidean equivalents a little bit. And essentially the negation of any of the equivalents that we saw of the Euclidean parallel postulate within the realm of neutral geometry are going to be equivalents to the hyperbolic parallel postulate. Like, for example, if you look inside the realm of the Sicarial Legendre theorem, which states that a neutral geometry, the angle sum of all triangles is less than or equal to 180 degrees, well, if you pick 180, that will give you Euclidean geometry. If you pick strictly less than 180, you get hyperbolic. And in the context of the Sicarial Legendre theorem, there's only the two options. You're either 180 degrees triangles or you're less than 180 degree triangles. The all or nothing theorem basically puts you into one of two categories. Inside gives you an example. And this occurs for basically all these other equivalent forms. We had talked about the right angle hypothesis before, that in a Sicary quadrilateral, the summit angles are right angles. Well, because of the Sicarial Legendre theorem, the summit angles of a Sicary quadrilateral have to either be right angles or acute angles. So if they're not right angles, they'll be acute angles. And thus the acute angle hypothesis is equivalent to the hyperbolic parallel postulate as it's the negation of the right angle, the right angle hypothesis in the context of the Sicarial Legendre theorem. And so for many, many, many of the equivalent forms we talked about, there is that dichotomy that comes from the Sicarial Legendre theorem. And the fourth angle of a Lambert quadrilateral is either always acute or it's always right. And that also is a distinction between hyperbolic and Euclidean geometry. But some equivalent forms aren't as obvious on what the negation would be. I mean, logically speaking, we know what the negation would be, but how do we interpret that in this geometric context? So the one I wanna talk about next is what about the converse of the alternate interior angle theorem? We know that the alternate interior angle theorem tells us that if alternate interior angles are congruent, then the two lines are parallel. The converse, and so the alternate interior angle theorem, this is true for neutral geometry, it's true for Euclidean and it's true for hyperbolic geometry. But the converse of the alternate interior angle theorem is equivalent to the Euclidean parallel postulate we talked about before. And remember the converse tells us that if two parallel lines are cut by a transversal, then alternate interior angles will be congruent. The negation of that says, I mean, because so if we go back to the converse of the alternate interior angle theorem says that for all parallel lines and all transversals, then all alternate interior angles will be congruent. There's a lot of universal quantifiers in that statement there, right? I mean, if we were to label it, we get something like for all, for all, for all, for all, for all, okay, maybe I exaggerate a little bit. But the point is there's a lot of universal quantifiers there and so when we negate it, so we take not the converse of the alternate interior angle theorem, then by just usual properties of inferential logic, each of those universal quantifiers when we negate them will become existential quantifiers that is there exists, there exists, there exists. So the negation of the converse of the alternate interior angle theorem would say that there exists a pair of parallel lines for which there exists a transversal such that alternate interior angles are not congruent. So the negation just says it allows for the possibility of having parallel lines within congruent alternate interior angles. It does not say that alternate interior angles can never be congruent. And so in hyperbolic geometry, it kind of begs the question under what circumstances might alternate interior angles still be congruent? Like under what circumstances will the converse of the alternate interior angle still be true? It won't always be true, but sometimes it will be, right? And so let's look at this right here. When can the converse of the alternate interior angle still be true? And the answer is actually simple. It turns out it's still true in the context of hyperparallel lines, which we'll define what that is a little bit later today, where the transversal intersects the midpoint of a common perpendicular. And so I wanna talk about this common perpendicular for a little bit. This is gonna be a big deal as we talk about parallel lines in hyperbolic geometry. So this theorem 632 that you've seen for our lecture here today, this is gonna be in the context of neutral geometry. Put a little comment right here. This is a neutral theorem. Honestly speaking, we don't need continuity here whatsoever. So really it's a congruence theorem. But this is gonna be important because this results quite trivial in the context of Euclidean geometry, but in hyperbolic geometry, it's gonna be a pretty big deal. So the theorem says that two parallel lines that are crossed by a transversal have congruent alternatory angles if and only if the transversal contains the midpoint of a segment that's perpendicular to both. This is an if and only if statement here. So there's really two proofs for the price of one. We're gonna first prove the first direction. That is, let's suppose we have two parallel lines. We're gonna call them L and M. So we have the line L and we have the line M and they're cut by a transversal, let's call the transversal T, and they're cut in such a way that alternatory angles are congruent to each other. So we're starting off with this assumption. If we have alternatory angles, well congruent alternatory angles, excuse me. Well clearly the lines would have to be parallel because the alternatory angle theorem holds in neutral geometry, but what more can we say about these parallel lines L and M? So we're gonna argue that these two lines L and M have a common perpendicular and this transversal T bisects that perpendicular line. All right, so for the sake of discussion, let's call the intersection of L and T the point C. Let's call the intersection of T and M the point D. And let's take M to be the midpoint, let M be the midpoint of the segment CD. So we've done this in such a way that C M is congruent to MD right there. All right, and let's also introduce some other points into the situation. We're gonna take the perpendicular dropped from M onto the line L. So take the perpendicular that drops from M down here, call the intersection of L and this perpendicular A, the so-called foot of the perpendicular. This gives us a line segment right here AC and by segment translation, let's take the segment, or let's take the point B, which is on the opposite side of the line T that A is, but we also want that the segment DB is congruent to AC. And we can connect the dots from M to B right here, but we can't quite say that this second segment we drew here is perpendicular to M, the line M. Then that's sort of biting off too much then we can get right now. We'll be able to prove it in just a moment, but we're not quite there yet. So what can we say? Well, recall by the constructions we have here, M was chosen to be the midpoint, so MC is congruent to MD. By construction, the segment AC is congruent to DB. And then also by assumption, the angle C is congruent to the angle D. These are where our congruent alternate here angles here. So then by our side angle side triangle condition we get that triangle ACM is congruent to the triangle B DM, like so. And so our corresponding parts of congruent triangles are in fact congruent. So we're gonna get that angle AMC is congruent to angle DMB. So if we write that down here, we get angle AMC is congruent to angle BMD. Now those sure look like vertical angles, which seems like duh, vertical angles are congruent, but we don't actually know that AM and B are collinear yet. This is somewhat of the problem we have here. Are they collinear or are they not? We don't know. And so what we're actually trying to do is make that argument that they are. And so for the sake of argument, let's suppose that AM and B are not collinear, then what we can do is we can find a line, right? Take the line determined by A and M, there's gonna be some point over here called B prime so that AM and B prime are collinear. In fact, M will be between A and B prime. And so in that situation, we get that angle DMB prime is in fact vertical to angle AMC. And so what did I say that angle was? We get angle B prime MD. That would be congruent to AMC and this happens from vertical angles, right? And so then we get that these angles right here, angle BMD and angle B prime MD. This BMD is congruent to B prime MD as angles. And as they live on the same side of the line, right? Notice that M sits between A and B prime right here and M lives on T. So we see that A and B prime are on opposite sides of the line. Like lens, so why are A and B on opposite sides? Well, that's how we chose the point B, right? B and A are on opposite sides of T. So by plane separation, this gives that B and B prime are on the same side of the line T. But as they're the same angle measure resting on the same side of M, we actually get the uniqueness of angle translation here that these two lines, these two rays I should say, the MB and the MB prime are in fact one in the same thing. So because the rays are actually equal, we get that M, the ray MB is equal to the ray MB prime. And so that actually does tell us that A, M and B were co-linear with each other. So I'm gonna get rid of sort of this extra hypothetical situation. Back up, back up, back up, voila. We do get that A, M and B is all one line right here. So A dash M dash B, M is between A and B. And so what's the significance of this? Why did we need this, why do we need this line to be connected like so? Well, some things to notice here is that because corresponding parts of congruent triangles are congruent, this congruence we used before, if angle A was a right angle, that means angle B was a right angle as well. And since this is all the same line, this line AB forms a common perpendicular. So I'm gonna slide this down a little bit so we can see the rest of the statement of the proof here. I'm gonna redraw some of the picture without some of the extra fluff here. So we had our line L, we had our line little M, we had a transversal T where M moves in the middle and then we had this common perpendicular right here. So A, C, B, D like so. And so we found a common perpendicular to L and M and we started off with the assumption we had congruent alternate here angles like so, right? But M was the midpoint of this transversal T to start off with. Well, continuing on what since corresponding parts of congruent triangles are congruent, again, the triangle ACM and BDM are congruent to each other. The corresponding side AM and MB are gonna be congruent to each other as well. And so this tells us that M is the midpoint, not just at the transversal, but it's the midpoint of this common perpendicular line that sits between them. And this improves the first direction that if two lines are cut by a transversal with congruent alternate here angles, then that transversal will bisect a common perpendicular and that perpendicular will go through the midpoint of the transversal there as well. That guess is the first direction. The next direction, I'm actually leaving it as a homework exercise for my students. And so those viewers who are also just viewing this for their own educational purposes, I would encourage you to try to work through this on your own as well. So we've now talked about under what conditions in neutral geometry can we have alternate here angles, right? So it has to do with this bisecting a common perpendicular line. And this common perpendicular is gonna be a big deal as we continue on because these common perpendicular are actually quite rare in hyperbolic geometry. And to explain what I mean by that, let's talk, let's actually introduce the notion of equidistance or some people say equidistance. So what does it mean for two lines to be equidistant from each other? So let's say we have a line L and we have a line L prime. And for this discussion, it's not required that L and L prime be parallel. These lines might intersect at some point. And so let's take points along the line L. Let's take three points. But admittedly, I mean, we could have a lot of things right here. So we have some set of points. A, B, C, D, whatevs. This is a subset of our line L. So for each of these points, A, B, C, D, whatever, construct the perpendicular dropped from those points onto the other line L prime. So take the perpendicular drop from A, the perpendicular drop from B, the perpendicular drop from C, the perpendicular drop from D. And associated to each of these perpendicular drops, there should be a foot. And so from the perpendicular drop from A we call its foot A prime. The foot of the perpendicular drop from B we'll call B prime. From C, we get C prime. And from D prime, we get the foot. Oh, sorry, from the point D, we get the point D prime there. So we say that the set S, so this set of points S, we say it's equidistant. It's an equidistant set of points. If for each of these perpendicular lines, A, A prime, it's congruent to B, B prime, congruent to C, C prime, congruent to D, D prime, congruent to E, E prime, however many there are. This definition kind of sounds like it's a finite set, but admittedly, this could be an infinite set of points, it could be an uncountable set of points. As long as that if you take, there's a correspondence between the one set of L, we construct the second set of L prime and of all those corresponding segments are congruent, we call this an equidistant set of points. Now we say that two lines are equidistant if we take the entire set of points, because after all, we could think of the line as a set of points. If you take S to be the entire line L, if that set is equidistant, then we would say that the two lines are equidistant. That is, we don't use, we use all the points, all right? And in Euclidean geometry, we're quite used to the notion of equidistant points. Many people so often define parallel lines as those lines which are equidistant from each other, all right? Every set of one point is gonna be equidistant automatically, it's kind of silly. And an empty set is also gonna be vacuously equidistant. So what's gonna be interesting is what about sets of two equidistant points? Are three or four or more than that? In Euclidean geometry, it's quite easy to get large sets of equidistant points. In hyperbolic geometry, it's actually gonna be quite rare. And so we actually get to our first main result for hyperbolic geometry. This is gonna be a strictly hyperbolic result. So we call this the at most two equidistant points theorem. And this is gonna be a result for hyperbolic geometry, not neutral geometry. So given two distinct lines in hyperbolic geometry, no equidistant set contains more than two points. That is, the largest equidistant set you can get is exactly two. And so the proof is actually pretty simple. Give them what we've already talked about before. Take two sets, the two lines. And again, these lines do not have to be parallel lines. For any pair of lines, equidistance maximizes at two points, all right? So let's take our line L and take our line, what do I call the other one, L prime still, right? And so let's prove this by contradiction. Assume we have at least three equidistant points. So let's take the point A, I'll put a little bit more space there, B and C. And let's construct the perpendicular drop from A so you get A prime, drop from C, you get C prime and drop from B, B prime. So let's consider this situation. Now this drawing, drawings can be very, very misleading in geometry. We have to be cautious of those things. In this situation right here, the drawing seems to suppose that B prime is between A prime and C prime. And as we're trying to, we're just taking three arbitrary points that are equidistant, we can make the assumption that sure, A or B is between A and C, just how we're gonna label them. But how do we guarantee that their feet have that same between this relationship? Let's back up for a second. What if, for example, B prime actually lived outside like this? That is what if we had B prime, A prime, C prime, but we had A dash B dash C. Could that be a possibility? Well, we're gonna get a problem here, right? I mean, the picture seems weird, but after all, hyperbolic geometry is kind of weird if you haven't seen much about it. So weird is not a good reason to argue why something's not true, because weird just means we don't have enough exposure to the subject matter yet. Let's look at these lines, these perpendicular lines, for example. Now I should mention that the perpendicular drop from B and the perpendicular drop from A, these lines are gonna be parallel to each other. How do we know that? Well, by the alternate interior angle theorem, if there's a common perpendicular, the two lines have to be parallel. And so L prime acts as a common perpendicular to these lines, the first line right here and the second line right here. Well, the diagram seems to suggest that, hey, wait a second, if they're a common perpendicular, then they have to be parallel, but the diagram seems to be suggesting that they intersect each other. Well, again, we have to be very careful. We do have to be careful about these type of things. How do we know that they, in fact, intersect? Again, the diagram seems to suggest they're intersecting lines, not parallel lines. So let's try to supply an argument why we get a contradiction if B prime was over here. Well, if B prime was on the left right here and A prime was between the two, this would suggest that B prime and C prime are on opposite sides of the line A prime, right? That's what that between the statement means right there. On the other hand, because B is between A and C and C is over here, then the between the statement also gives us that the segment BC doesn't intersect, doesn't intersect the line AB at all. So by plane separation of B and C are on the same side of A A prime and B prime and C prime are on opposite sides, that would tell us that B and B prime are on opposite sides of the line A A prime. And therefore, by plane separation, we get that there should be an intersection between the segment BB prime with the line AA prime. But as BB prime is parallel to AA prime, that's a contradiction. So we don't get that. We don't get that B prime is where it is. It can't be to the left of A prime and similar reasoning shows that it can't be to the right of C prime as well. So if we go back in time, we can remove our contradictory statement here. We get that, we actually are safe to assume that B prime sits between like this. So we actually do get, if B is between A and C, then we get that B prime is gonna be between A and C prime there. All right, so what does that have to do with equidistance? Well, we're assuming that these are all equidistant. So the segment AA prime is congruent to BB prime, which is congruent to the CC prime like those. And so what we're actually gonna get is a bunch of security quadrilaterals. So there's the AA prime B prime B security quadrilateral. There's gonna be the BB prime C prime C security quadrilateral. And then there's also going to be the much larger AA prime C prime C security quadrilateral. So if we have three equidistant points, we get these three security quadrilaterals. Well, because we have security quadrilaterals, the sub and angles are congruent. So angle A is congruent to angle B. Angle B will be congruent to angle C. And then angle A is gonna be congruent to angle C. And so there's a lot of congruence going on right there. So the two angle A's are congruent. The two angle C's are congruent. I mean, when I say they're congruent, I mean they're equal as angles, right? We're talking about the angle A prime AC and the angle C prime CA, those are congruent to each other because of the large rectangle. But because of the smaller rectangles, we get that the two angles B are congruent to each other. So what I mean here is angle A, B, B prime is gonna be congruent to C, B, B prime by transitivity of congruence. But if those two angles are congruent, the significance there is that there are also supplements of each other and congruent supplements mean we have right angles, all right? And so once one of these four angles is right, all four of them are gonna be right by transitivity because after all, they're all congruent to each other, right? That force is congruent right here. And so as we have supplements, they're gonna be right angles. And as some of them are right, they're all gonna be right. And so these security quadrilaterals are in fact rectangles. We have three rectangles in front of us. Existence of rectangles implies the Euclidean parallel postulate, which is a contradiction to the hyperbolic parallel postulate. So in hyperbolic geometry, there's at most two equidistant points between any two lines. So what does that then tell us here? In hyperbolic geometry, there are now two cases actually for parallel lines. What type of parallel lines have equidistant pairs and which ones don't, right? We can have, I mean, because like I said, every set of one point is equidistant automatically. So some parallel lines might have two equidistant points and some might not have any. Well, in the latter case, when you have no equidistant points, by Dedekin's axiom, we actually get the usual intermediate value theorem from calculus. And this is gonna imply that the distance between, let me say that again. By Dedekin's axiom, the intermediate value theorem applies to the distance between the parallel lines. Thus, two parallel lines are gonna be nowhere equidistant if and only if the distance between the lines is monotonic. That is, the distance is monotonically increasing or decreasing depending on how you order the lines. So that is, if you have a line, the parallel line might look something like this. Let's do a better picture. That is, the two lines will be asymptotic to each other in the usual sense of calculus. That the distance between them is decreasing or in decreasing depending on the direction you go. That is, there's nowhere equidistant points between them. And this is what we mean by asymptotic parallels. That is, this is gonna be the parallel lines that obtain the angle of parallelism that we had done in a previous lecture. It takes a little bit of calculus to establish this theorem and I don't wanna spend too much time on the calculus of things, but these asymptotic lines are gonna be those lines which approach each other and there's never equidistant points. On the other hand, we will say that two lines in hyperbolic geometry are hyperparallel if there exists a pair of equidistant points. So we have this dichotomy on parallel lines in hyperbolic geometry. Those which are monotonically approaching each other, asymptotic ones, they're nowhere equidistant, are these other ones which do have a pair of equidistant points. We call them hyperparallel. Hyperparallel is just an amalgamation of the word hyperbolic and parallel. And this gives us a type of parallel line that doesn't exist, so to speak, in Euclidean geometry. Because hyperparallels are those which have a pair of equidistant points. In Euclidean geometry, technically, all parallels have a pair, but it's actually, there's more than that. And so we don't really like to think of those as hyperparallel lines. And so as a theorem here, hyperparallel, in hyperbolic geometry, parallel lines are hyperparallel if and only if they have a common perpendicular. I don't wanna say too much about this necessarily right now because I'm leaving this as a homework question for students here to do. But the idea is gonna come down to the type of security quadrilateral as we had drawn before. We know that security quadrilaterals have their altitude, which is a common perpendicular between the summit and base of the security quadrilateral. And by construction, the summit and base have two equidistant points because it's the legs of the security quadrilateral. And essentially hyperparallel lines are the type of parallel lines one can build security quadrilaterals from. You can also do it with Lambert quadrilaterals. We know that Lambert and security quadrilaterals are parallelograms, the opposite sides are parallel, and those are gonna be hyperparallel lines in hyperbolic geometry. So with a slight, with a few closing remarks, I wanna make a comment about the hyperbolic models we had talked about before, right? So if we take the half plane model and we think of two lines, let's take a vertical line for a moment. This is a line. Well, some hyperparallels are gonna be some parallel, other vertical lines will be hyperparallel, but also semicircles that come nowhere close to that original parallel line, original line. These are gonna be examples of hyperparallel lines. And there's gonna be tons of them. There's gonna be infinitely many hyperparallel lines. On the other hand, asymptotic parallel lines are actually gonna be much more rare. The asymptotic parallel lines is gonna be those semicircles which come and touch the point at infinity. Remember, these points on the x-axis do not belong to the line. They don't belong to the geometry. And so if we take the semicircle which shares as its endpoint, the endpoint of the ray, those are gonna be lines which are, they're gonna be asymptotic to each other in hyperbolic geometry. They get closer and closer and closer. And there's gonna be two asymptotic parallel lines. There's gonna be a semicircle on one side or the other. But I mean, this is true if your line is a ray, a Euclidean ray. For semicircles, semicircles that don't intersect each other whatsoever will be hyperparallel. Rays will be hyperparallel to semicircles. But for a semicircle, there do exist these asymptotic parallel lines which are gonna look like rays. These are gonna look like rays in the hyperbolic geometry. Like so. And yeah. And so we can construct these asymptotic lines. These are gonna be lines that touch, that intersect each other at infinity. This sort of idea of lines at infinity will become much more prominent when we talk about elliptic geometry in the future. But it does come into play with these hyperbolic models we're talking about here. Similarly, if we think of the disk model of hyperbolic geometry, so maybe we think of a disk like this, we can talk about hyperparallels. So if we take a line L, a hyperparallel might be some semicircle that doesn't interact with the line whatsoever. On the other hand, so this one right here, would be an example of like a hyperparallel. The semicircles that don't intersect, but they do touch at infinity. Apologize for the crudeness of my drawings. This is mostly to give you an intuition, not necessarily exact drawings here, but you could get semicircles that touch the line at infinity, but they don't intersect in the model. Cause again, the circle on the boundary is not part of the geometry. And so these asymptotic parallel lines will be ones that touch, that intersect each other at infinity. The hyper ones means there's actually a sort of, a specific distance between the lines. There's a solid distance between them. So like for example, if you take these two vertical rays right here, there does exist a common perpendicular to them. And then what it's gonna do is it's gonna look like a semicircle. And these are gonna be right angles. And so this would act like a common perpendicular between them. These hyperparallel lines. Similar things happening here if you take this line and this line right here. There would exist a common perpendicular line. It itself will be a semicircle. It will look something like the following. And so this would be the common perpendicular that exists between all hyperparallel lines. And this common perpendicular is unique. There's only one perpendicular that lives between the two. And if we take any line, if we take any line that passes through the midpoint of this common perpendicular, we'll get alternate-tier angles. This is an example where the alternate-tier angles are right angles. We could also do something like take the midpoint right here. And so then take a circle that goes through that. And this situation, oh, I drew the wrong pictures before, I'm sorry. The alternate-tier angles should have been this angle right here with this angle right here. Or with our picture, the alternate-tier angle. I didn't quite draw this picture well enough, but it would be something like this and something like this. Those should be touching right there, not parallel. All right. And so that kind of could, that'll include our lecture for today. Give me this exposure to how parallel lines and hyperbolic geometry look kind of different. Next time I wanna talk about hyperbolic triangles, how they behave a lot differently than what you would expect from Euclidean geometry. So stay tuned. If you liked our lecture today, feel free to subscribe, post a comment below. If you have any questions, please post a comment and I would be glad to answer each and every one of them. Have a great day, everyone. I will see you next time. Bye.