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Under the many emails I have received from students and teachers not only in Mexico but in many other countries English-speaking countries (Argentina, Venezuela, Spain, Chile, etc.) than volume I can not answer individually (which is why I take here to ask for apologies for a lack of response from me), in which I requests to expand a little more this blog trying on the subject of light rays used as a tool for geometric paths that represent the shortest geometric distance (the geodesic) between two points and talk a little about other models of geometries hyperbolic addition to those already mentioned, and in a little deeper into the detail of minimum routes in a hyperbolic geometry, I have decided expand the blog with this additional input. Some of the material to be presented here was not included in the rest of the entries for the simple reason that it requires some basic knowledge of differential and integral calculus (calculus), which is not necessarily available to many of my potential readers have not completed their high school studies, why this material is being attached as a supplement. Anyway, if anyone is reading this left her studies because "it has no practical application" (analytic geometry, differential calculus, integral calculus), this supplement will give reasons why it is important to continue on with their studies, because in the development of this supplement will be seen that these materials do have a direct practical application to our understanding of the world around us


The
Light rays always follow the path that takes the least time possible


A well-known principle of physics, Fermat's principle (of the French mathematician and physicist whose last theorem without proof was left in check for centuries the best mathematicians in the world for several centuries until the British finally decided to Andrew Wiles 1993), set in 1679, tells us that a light beam to move always follows the route that will take the shortest time to reach your destination. If the media that moves the light beam is homogeneous, constant density, then to stay the same speed is equivalent to the beam of light takes the shortest route possible, which will take the shortest possible time. Like Euclid's fifth postulate, the principle of physics is not demonstrable, it is a fact of nature, confirmable mathematically and confirmed by our experience. This principle remains as valid today as since they are made. Even in the general theory of relativity in which Einstein raised to light as an absolute and unchanging reference (the speed of a light beam is exactly the same for all observers no matter how they are moving relative to each other) using it as a means to establish their equivalence principle (a observer enclosed in a sealed box will be unable to distinguish by any experiment if the box is moving with constant acceleration in one direction or if the box is static in the presence of a gravitational field that causes the same acceleration) and using light beams to trace the geodesic (minimum paths) in space, Fermat's principle is still valid. Using this principle and tools of calculus (maximum and minimum), we can derive the well known laws of reflection and refraction of light, which we now turn.

For law of reflection ( the angle of incidence equals the angle of reflection ), consider the following figure in which we start assuming that the angle of incidence of a ray from a point A and is reflected in point B to reach point A is not equal to the angle of reflection:

To simplify this demonstration, point A and C point will be made at the same height h reflectora.Llamaremos surface x to the horizontal projection of the light beam that travels from A B up, making it a variable. Applying the Pythagorean theorem for triangles, we need the lengths AB and BC be related to its horizontal projections on the reflecting surface as follows:

AB ² = x ² + h ²

BC ² = (lx) ² + h ²

total route length of the light beam to get from A point to point is given by:

L = AB + BC

Introducing the expressions for AB BC and function of the variable x , we have the value of total distance L depending on the variable x is:

then take the derivative of L about x :

and we set to zero:

L d / dx = 0

to obtain the condition gives us the minimum path after some algebra:

This tells us that the two triangles forming the light beam with the reflective surface to both get from A to B and B to go to C are equal, since the value of x for the minimum path is located right in the middle between triangles of the same height. Since the two triangles formed by the same extreme path having three equal sides, angles m and n must also be equal. We conclude that the angle of incidence equals the angle of reflection .

For the refraction of light, let us first the reasons why you should apply the hypothesis that it has to refract rather than follow a straight path from its point of origin to final destination. Consider the following figure in which a light beam at a point of origin A and reach the surface of a medium such as water at B is refracted to finally reach the point C :

Geometrically speaking, the shortest distance (Euclidean) between the two points A and C is the straight line and not the line "broken" ABC . If the light beam to follow the straight line from A to C ( through point B ') instead of the path ABC certainly travel a distance smaller. However, the distance B'C in water is greater than the distance BC. Since the speed of light is lower in water than in air, light can waste more time to travel the path B'C that would save time following the shortest distance AB 'AB that in the air. Since there is no reason to choose in advance, taking into account these considerations, which of the two routes will take you to a beam of light as quickly as possible, it is necessary to conduct the analysis complete from a mathematical perspective. Perched above

now proceed to obtain the law of refraction ( for all refracted ray passing from one medium to another, the sine of the angle of incidence between the sine of the angle of refraction is equal to a constant . To do this, first define the refractive index n , which depend dell value that has the speed of light in a medium where its velocity s was less than the speed of light in vacuum, settling here So the speed of light in vacuum is the maximum speed that can have a ray of light. The index of refraction of light in a medium is defined as the ratio of the speed of light c (about 300 thousand kilometers per second of elapsed time) in the gap between the speed v which takes a transparent medium (such as water or clear plastic):

n = c / v

By definition, the refractive index will always have a value greater than unity.

Consider the following figure, in which a beam of light traveling from point A in a medium with a refractive index n 1 transferred to another medium with a refractive index n 2 to the point B , refracted at point D :

Again, we turn to the Fermat principle of seeking the conditions for the path that takes the light beam is at go to A B is that it takes the shortest possible time t. In a homogeneous medium, the time required to travel a distance l is obtained by dividing this distance by the speed v :

t = l / v

But as in a medium with refractive index n speed is v given by the ratio mentioned above, the travel time will be:

t = l / (c / n) = nl / c

The total travel time of the light beam will be time later you go first from A to D and then from D to B :

_{t = t 1 + t 2}

n t = 1 ₁ l / c + n l _{2 2} / c

c n t = l _{1 1 2 + n 2 l}

As is a constant c universal, and t is the total travel time, the left side is what is commonly known as the optical path light beam. The ADB ray optical path is equal to:

Inspecting the figure, it is clear from the Pythagorean theorem that the lines AD DB and are given by:

Then the total path length of travel of A B to be:

The necessary condition for the optical path is external (minimum) is reduced again to l d / dx = 0. Then, as we did in deriving the geometrical law of reflection, we take the derivative of l about x and equate to zero:

But the figure we see that, by elementary trigonometry:

being i ₁ the incidence angle and i ₂ the angle of refraction, from which it follows that:

The sign of the second derivative can be used to confirm that this path is the minimum. With a slight rearrangement, and taking into account the ratio between the refractive indices of the media is equal to a constant:

2 n / n ₁ = constant = k this

can be expressed in the form that is usually taught in introductory physics courses in middle schools , the sine of the angle of incidence between the sine of the angle of refraction is equal to a constant :

sen i 1 / sin i ₂ = k

The Poincare Half-Plane

addition to the hard
investment, we can build other graphic representations to implement a hyperbolic geometry. We shall see one that is extremely useful to derive some important results and conclusions.

For convenience and clarity, we will begin setting up a system of rectangular Cartesian coordinates (x, y) as studied in analytic geometry. The horizontal axis x dimensional plane divided into two parts. For our purposes, we use only the upper level corresponding to positive values \u200b\u200bmeasured over of the coordinate and . Now imagine that in that half plane things behave in a curious way. Imagine that all objects are located in that half plane will be shrinking more and more as we approach the boundary line . Thus, an object (or person) to move up and down to half the distance that separates it from the boundary line will have a size half the original size. And if it continues moving downward again traveling half the distance that separates it from the boundary line, then shrinks back to half, finishing with a size that will be the fourth the size original, which had begun before going smaller. If it is a person, that person will not notice it, because all its rules, compasses, his clothes, his watch, his hands, everything will have been made in the same proportion. In the following picture:

can see that if a person walks horizontally from point to point E F covering a distance of one kilometer (its path is represented by a green arrow), then if you travel the same horizontal point no point starting E but C starting point to reach the point D As that person has contracted to half will have to go what she is a distance twice the distance between the point had E and point F . It will take twice as long to walk from C D to to walk from E F to . And take ten times longer to get from point A B point to get from point A to point E F .

Nature x axis for a coordinate value of zero and ( and = 0) is such that it is impossible to communicate between the upper half plane and the midplane lower, which will restrict our attention to the upper half plane, which is known as the flat- Poincaré upper half . or just flat- through Poincaré.

assume that the distance that measures a person living in this flat-means will be Euclidean. The distance between E F and will be one unit, but the distance traveled between A and B will increased by a factor of ten. Taking this into account, a first attempt to define a hyperbolic distance, limited to horizontal paths through the Poincaré plane, leads us to use the following definition:

= Hyperbolic space (Euclidean space) / y

This relationship successfully explains what we see is happening in the middle ground. Thus, the space in which it is a rule that placed one meter along the line EF : Hyperbolic space

horizontal = (1 meter) / 1 = 1 meter

seem to have doubled for a resident flat-half, having narrowed the rule to half its original size:

Hyperbolic space horizontal = (1 meter) / (0.5) = 2 meters

And while walking to the rule measuring the space between the points A and B , The space will appear to have increased tenfold: Hyperbolic space

horizontal = (1 meter) / (0.1) = 10 meters

Thus, this preliminary definition seems satisfactory. However, it does not provide recourse for measuring the hyperbolic space between the vertical points A and E or points C and E . To do this, we must modify the definition of distance (Euclidean) between any two points in Cartesian plane (x, y) using infinitesimal increments of distance (dx, dy) instead of finite increments. In Euclidean geometry, the infinitesimal distance between two points, according to their projection on the axes infinitesimal x and and is given by:

(ds) ² = (dx) ² + (dy) ²

This allows us to define the length of an infinitesimal segment hyperbolic straight line extending the original definition as follows:

(ds) ² = [(dx) ² + (dy) ²] / and

When it is understood that the infinitesimal length elements are being raised squared, the previous expression is written simply as:

ds ² = (dx ² + dy ²) / y

The total length of a line segment hyperbolic from a point P to a point Q is obtained by integrating infinitesimal line segments using the previous relationship. This is represented symbolically as follows:

We will use this relationship to calculate the figure above hyperbolic lengths of the lines AE, EF and AF . We'll start first with the line EF :

then calculate the hyperbolic length of the line AE :

Finally, calculate the hyperbolic length of the straight line between points A and F, for which we first obtain by usual analytical geometry the equation of the straight line connecting these points:

and = m + x
b
and = 0.9 + .1 x

whose derivative with respect to x is:

dy / dx = 0.9

dy dx = 0.9

thereby calculate the hyperbolic length of the line AF turns out to be:

The journey from point A to point F Staying within first since A E up (vertically) and then from E F up (horizontally) requires 3.3 units, while the A journey from F to follow the diagonal line requires 0,959 units. Obviously, the shortest distance between two points is a straight line joining them. However, the diagonal line between points A and F is the shortest distance between these points . In the Poincare half-plane, the shortest distance between two points is the arc of a circle between the points that have a center axis x . To be convinced of this, consider the following picture:

in which we have a Cartesian coordinate and drawn perpendicular to the axis x , which can specify a circle with the following equation:

x ² + y ² = 2

This is a circle whose circumference passes through the points A (-1.1) and B (1.1). The distance, the Euclidean and hyperbolic point A to point B following the blue line, is 2. Hperbólica calculate the distance along the arc of green, which corresponds to the equation given above. Taking into account that for the circle:

then the calculations for the hyperbolic arc length AB are:

It can be seen along the length arc AB is 1.76 units hyperbolic, which is certainly less than the Euclidean straight matches to A B and that is 2 units. The longer route has the shortest length! This is a result that would surely have perplexed geometers of ancient Greece. And indeed, this is the shortest distance that can exist between A and B . Euclidean length of the arc AB remains, of course, greater than the length of the line AB Euclidean, which checked with the following calculation:

The calculation to determine the length a general hyperbolic line segment PQ where P point is located at a distance to x axis and the point Q is located at a distance axis b x be in the same manner, considering an infinitesimal segment of vertical line at a distance d and the x axis, and carrying out the integration using the general relationship that defines a hyperbolic line:

thereby obtain the distance (hyperbolic) between points and P Q is: PQ

h = ln (b) - ln (a)

_{PQ = h} ln (b / a)

Consider now two line segments, PQ and QR situated one after another in the same vertical axis as follows:

Applying the formula obtained above, we have that in the Poincare half-plane lengths of each segment is: PQ

h = ln (b / a)

QR h = ln (c / b) Adding the two expressions

term:

_h PQ + QR h = ln (b / a) + ln (c / b)

and using the properties of natural logarithms:

_h PQ + QR = h ln [(b / a) • (c / b)]

_h PQ + QR h = ln (c / a)

_h PQ + QR h PR _{= h}

This result, as simple as it looks, is a very important issue, because consistent with the same result related to the sum of segments in Euclidean geometry, which in this geometry is derived without resorting to the fifth postulate (the parallel postulate). Have obtained a different result here, we would run into the first contradiction serious in our non-Euclidean hyperbolic geometry. So far, we found no contradiction.

Before continuing, try to derive a relationship which enables the location of the midpoint of a vertical line segment hyperbolic. In the following figure:

if M is the midpoint of AB hyperbolic, then the hyperbolic length AM segment should be equal to the length of the segment hyperbolic MB . That is,

_AM h = h MB

ln ( m / a ) = ln ( b / m )

m / a = b / m

m² = ab

Thus, the mean in a straight vertical hyperbolic is given by the geometric mean of to values \u200b\u200band b . If to b = 4 and = 9, Euclidean plane midpoint would be located at:

(4 +9) / 2 = 6.5 units

line x limit, ie the arithmetic. However, according to the relationship that we have just obtained, the midpoint M be hyperbolic placed to these same values \u200b\u200bat a distance of 6 units, the geometric mean .

shown above procedure to calculate hyperbolic lengths is quite general, there is no reason to restrict ourselves to straight lines and arcs. We superimpose a parable ordinary y = x ² in the flat-top half, matching the horizontal axis of Cartesian coordinates with the boundary line x flat-half, after which we can carry out the determination of the length hyperbolic between points we want to choose from the parable. For example, the hyperbolic length of the parabola segment between A points (2.4) and B (3.9) is determined as follows:

can demonstrate, by calculus of variations, all geodesics in the Poincare half-plane will necessarily arcs of circles centered on the boundary line (x axis ), so this allows us to define the only two "straight" hyperbolic possible that there may be on this plane half :

(1) circular arcs centered on the axis x where and

= 0 (2) vertical lines, perpendicular to the axis x , although some authors consider vertical lines as a limiting case of circular arcs with an infinitely large radius, thus effectively all geodesics in the median plane are defined as arcs of circles centered on the axis x . Based on this result, in the following figure:

all lines k, l , m, n and p are geodesics (straight hyperbolic). lines k and l are not parallel because they cross. Nor are the lines l and m for the same reason. Instead, the lines l and n are parallel, as are the lines p and n , as are the lines and m n . Hyperbolic lines k, m n and are designated as divergently parallel, while lines and p n are called asymptotically parallel . Given two points and A B any in the Poincaré half-plane, this knowledge allows us to find the geodesic between points A and B , all we have to do is draw an arc of a circle passing through these points and having its center on the boundary line ( and = 0). Knowing this, it should be no problem and construct a hyperbolic triangle in the Poincaré half-plane whose vertices are located at points A , B and C , as shown below:

Since the three circles which are based on the construction of hyperbolic triangle shown can travel together horizontally along equal distances, both left and right, here is the equivalent of Euclidean translational motion of a figure that leaves unchanged the geometric shape, changing only place. However, if the hyperbolic triangle is displaced vertically, it is clear that the size of the triangle will necessarily change.

The Poincare half-plane is of particular importance to us because is what fits the geometry developed by Lobachevsky axiomatically, the mathematics is applied within this model and the proofs of the theorems that can lead to out here correspond to the hyperbolic relationship discovered by Nicolai Ivanovich Lobachevsky.

Those who already read the previous posts made in this blog are familiar with the concept of the common perpendicular two parallel lines that share both a spherical geometry as in geometry Saccheriana. Then, looking at the lines "parallel" in the figure above, you may be wondering now where is the common perpendicular to each pair of parallel lines in the Poincaré half-plane? The answer is: all the lines (hyperbolic) drawn in the Poincaré half-plane have a common perpendicular with any of the other lines, and this common perpendicular, and is also a hyperbolic line, is also an arc of a parallel track to another, so that touching the two angles formed parallel with each of them. Below is the common perpendicular to pc parallel lines k and l :

This is the only common perpendicular is shown, for in-plane through two lines Poincaré hyperbolic divergent parallel may have one and only one common perpendicular . Remember that even in the Saccheri quadrilateral was more of a common perpendicular. Contrast this with the geometry Euclidean, where two parallel lines have an infinitely large common perpendicular can be drawn parallel to one another. And neither in spherical geometry can be traced more than a common perpendicular between two parallel (circles), we conclude that Euclidean geometry is the only geometry which can be traced over a common perpendicular to two parallel lines .

then be carried out show that in the Poincare half-plane is only possible to draw a common perpendicular to two parallel divergent. To simplify the demonstration of this important fact, we consider Hyperbolic two lines, one of which is a vertical line perpendicular to the axis x (which, as noted, can be considered as a limiting case of a circle in the plane-average centered on the axis x with infinitely large radius). The following figure shows the two parallel and divergent k l , having also drawn tangent to the semicircle AB passing through the tangent point T and P point at which the line touches the axis PQ x :

Then, as shown in the figure, making center with a compass point and drawing P an arc running from the line to k l straight, we have in that arc a common perpendicular to two lines and hyperbolic k l as being a perpendicular to the tangent and also be a vertical line perpendicular to PQ , at right angles to both lines and AB PQ. And as shown tangent is the only that can be traced to the line AB passing through the point P (tangent will pass any other point than P ) concludes that this is the only common perpendicular can be drawn between the two parallels.

Regarding the spherical geometry (elliptical), we saw that raising perpendicular to a line drawn along the ground and marking Ecuador equal distances on them to unite, forming a line equidistant from Ecuador by land, such a line could not considered as a line for not being part of a circle. Equidistant parallel lines do not exist on the surface of a sphere. We will now see that the same applies in the Poincaré half-plane. We begin with the following figure, which has drawn a line equidistant OB line OA :

PQ and RS are hyperbolic lines of the same length (measured hyperbolically with the mathematical relationship given to define the hyperbolic length). Therefore, Q points and S are equidistant from the line OA hyperbolic. But the line OB is not an arc and therefore is not a hyperbolic line. It follows that a line drawn equidistant hyperbolic Poincaré plane through hyperbolically will not be straight. Thus, the same is true in spherical geometry. Therefore, the only possible geometry where a line equidistant from a given line will also be a line is Euclidean geometry.

We use the Poincaré plane-through to carry out the proof of the following theorem

: The sum of the angles of any hyperbolic triangle is less than 180 degrees (two right angles) .

To perform this demonstration, consider the following figure:

Let's start by considering the hyperbolic triangle ABC , which in the Poincare half-plane is a triangle (The right angle is at point C ). to sides, and b c are, respectively, a segment perpendicular to the boundary line Euclidean x ( BC), an arc of the Euclidean circle with center M (CA ), and an arc of the Euclidean circle with center N (BA ). The vertex angle A equals the angle between the tangents of the circles and b point c A , or the angle between the spokes and MA NA of these circles . Moreover, by construction, the vertex angle B equals the angle BNM.

has been built in the BN segment, using a so-diameter, circumference Euclidean q, which has only one common point B the Euclidean circle c under its diameter is the radius of the circle c . For this reason A point is outside the limited circle the circumference q, and therefore the angle at the vertex A that equals the angle MAN is less than the angle MBN. Then, Given that, in the figure:

angle (MBN ) + angle ( B ) = a right angle (90 degrees)

we have:

The sum of angle ( A ) + angle ( B ) is less than a right angle (90 degrees)

and therefore, with the vertex angle C a right angle:

angle ( A ) + angle ( B) + angle ( C ) is less than two right angles (180 degrees) Demonstrated

this fundamental theorem Hyperbolic geometry in flat-top half, it is easy to show the following theorem

: The sum of the angles of hyperbolic quadrilateral is less than 360 degrees (four right angles) .

To demonstrate just divide the quadrilateral into two triangles.



hyperbolic triangle trigonometry


The Breast Law and Law of Cosines (of which there are two versions, one for the sides and one for the angles) for hyperbolic triangles were derived simultaneously and independently by the Hungarian Janos Bolyai (1802-1860) and the Russian Nicolai Lobachevsky (1793-1856). This development is what makes a breakthrough on the geometry developed by Saccheri based on his ring, which was rather a futile attempt to prove the parallel postulate from Euclid's other postulates. In fact both Bolyai and Lobachevsky were also looking to prove Euclid's fifth postulate reduced to the status of a theorem, but seeing such a thing was not possible had the intellectual courage to investigate the consequences that result from the denial of Euclid's fifth postulate, discovering new ways from which it was not possible to find no contradiction. And that, ultimately, is what tests any mathematical theory, if based on certain principles of contradictions, then something is ruling principles, but even though it will look deriving an increasing amount of theorems among which there is no contradiction, then you must accept the fact that the principles are part of a coherent system. Of course, if someone obtains a contradiction from the axioms of spherical geometry or hyperbolic geometry, this would be enough to send down the entire structure. But a century of discoveries made by Bolyai and Lobachevsky, no one can find contradiction, and there is almost total certainty among contemporary mathematicians that this contradiction will be found. In relation to the formulas found by Bolyai and Lobachevsky, was the great similarity of the formulas found its counterpart in Euclidean geometry this was what finally convinced both of the correctness of hyperbolic geometry as well as total independence the same of the fifth postulate of Euclid. Another thing that is finished to convince them is that the formulas of hyperbolic geometry are reduced to the same formulas of Euclidean geometry where the relative distances within the geometry hyperbolic become extremely large, which has the effect of "flattening" the hyperbolic sheet. Let's look at some of the similarities, taking into account the accepted definitions for the three basic functions of hyperbolic trigonometry are defined similarly to the Euclidean trigonometric functions:

First, we have known Breast Act of Euclidean geometry:

Then we Hyperbolic sine law :

In Euclidean geometry, an important trigonometric identity tells us that for any triangle, the square of the sine of an angle added to the square of the cosine of that angle is equal to the unit or is:

sin ² (x) + cos ² (x) = 1

compare this formula with its counterpart hyperbolic:

cosh ² (x) - sinh ² (x) = 1

As an example of verification of hyperbolic trigonometric formulas, then we have a procedure to check the validity of this last formula:

As another example of the verification of hyperbolic trigonometric formulas, then we have a procedure to verify that the hyperbolic cosine of the sum of two hyperbolic angles x and and is the product of the hyperbolic cosine of each of these angles plus the product of the hyperbolic sine of these angles :

hyperbolic compare this formula with its Euclidean counterpart:

cos (x + y) = cos (x) cos (y) - sin (x) sin (y)

Now for the formula for the hyperbolic tangent of the sum of two angles x and and :

compare this formula with its Euclidean counterpart:

There is indeed a rule, known as the Osborne rule that says one can convert any trigonometric identity into a hyperbolic identity by expanding it completely in terms of integral powers of sines and cosines, changing sine ( sen) a hyperbolic sine (sinh ) and cosine (cos ) to hyperbolic cosine (cosh ), changing the sign of each term containing the product of two hyperbolic sinus.

It is instructive to do some calculations within the environment of hyperbolic geometry, which is why we do that then.

The first problem to solve is:

If we have a hyperbolic triangle whose internal angles are A = 10 °, B = 20 ° C = 40 °, what are the lengths (hyperbolic ) of its three sides to , b and c ?

To resolve this problem use the Hyperbolic Law of Cosines for angles in the hyperbolic length determination to side, with similar expressions for the hyperbolic lengths of the sides remaining:

cos ( A ) =-cos (B ) • cos ( C) + sin (B ) • sin (C ) • cosh ( to )

The solution proceeds as follows:

The second problem solve it:

If we have a triangle whose sides are to = 3, b = 4 and c = 5, what are the internal angles at the vertices of this triangle? What is the area (hyperbolic) of the triangle?

To resolve this problem use the Hyperbolic Law of Cosines for Sides in determining the angle at the vertex A , with similar expressions for the angles at the remaining vertices:

cosh ( to ) = cosh (b ) • cosh (c ) - sinh (b ) • sinh ( c) • cos ( A )

The solution proceeds as follows:

Knowing the values \u200b\u200bof the three internal angles of a triangle hyperbolic expressed in radians, it is easy through the defect of the triangle δ get the same area when their internal angles are A , B and C :

δ = π - ( + A B C +)

δ Π = - (0.09178919299 + 0.2523642536 + 0.7463119988)

δ = 2.051127211

The proof of identity hyperbolic trigonometric be so much as is done in Euclidean trigonometry. For example, to demonstrate the following relationship, valid for all triangle (hyperbolic) with internal angles A , B and C with a right angle at the vertex opposite the side c :

cos ( A ) = cosh ( to ) • sin ( B )

demonstration procedure is as follows (in the second step uses the equivalent hyperbolic Pythagorean theorem ):

And for the following relationship for the same hyperbolic trigonometric triangle:

cot ( A ) • cot ( B ) = cosh ( c)

demonstration procedure is as follows:

Thus, we can develop a full course hyperbolic trigonometry, which can be taken even further by extending it to a hyperbolic geometry . For which, unfortunately, there is no space available even in many university undergraduate programs in mathematics. Observant readers will have noticed that the way in which the hyperbolic functions are defined, we can immediately obtain the derivatives thereof. And even in place of the real variable x we use as an argument in the hyperbolic trigonometric functions, we can use a complex variable z defined as :

z = x + i and

where i is the symbolic representation of the square root of -1 , which can develop many new relationships. As shown, the field to be filled is large, and here we have barely scratched the surface.



The signs of the curvatures of surfaces


Speaking surfaces in non-Euclidean geometries, it is common to assign a positive or negative sign to the areas represented in a sheet denoted here as letter ( manifold), using here the concept of assuming a sheet letter (non-Euclidean or Euclidean) infinitely large which is mounted in the region under study, whether a triangle, a quadrilateral, a polygon or other geometric figure. By convention, a surface has a positive curvature if it is "concave-concave" or "convex-convex, while a surface has a negative curvature if it is" concave-convex "or" convex-concave. " As this mathematical jargon itself is not very illuminating although very precise definitions of the signs become clearer as follows: Take the region under study and draw two straight on it so that they cross at right angles . If the two lines are twisted in the same direction , we say that an area bounded in the region has a positive curvature . Clearly, this is what happens on the surface of a sphere, which is why a region located within the spherical geometry is assigned a positive curvature. However, if the two lines are twisted in opposite directions (as would occur on the surface of a "saddle" as the letter-sheet-of hyperbolic paraboloid is among the closest thing we have to represent a hyperbolic surface in three dimensions) say that a region located within the hyperbolic geometry has negative curvature . And a zero curvature means that at least one of the straight lines on such a surface is actually a straight line in the Euclidean sense, no curvature of any kind, as in the surface of a cylinder in which, by way, allows the delineation of Euclidean parallel lines never cross. Euclidean geometry, the plane on which Euclid made all its shows, obviously has zero curvature. Following is a Comparing the three types of curvature (positive, negative, zero) in three different figures:

positive curvature :





negative curvature :




Curvature zero :

The last surface, with zero curvature, is known as a parabolic . However, there is actually a cylinder, a cylinder because the lines on its surface that are perpendicular to its axis of symmetry is found again to flip the enclosing circle, while the "cylinder" parabolic these lines are separated indefinitely without going to find. This curve is obtained by moving a type parabolic curve y = ax ² along an axis perpendicular to the plane in which the parable is.

The surface exhibits a negative curvature deserves special attention. This surface is generated by a flat curve known as the tractrix :

In the Cartesian plane, the curve is drawn by the following parametric formula:

x = 1/cosh (t )

and = t - tanh (t )

The presence of hyperbolic trigonometric functions in the generation of tractrix curve suggests that this may be important in the study of hyperbolic geometry. And indeed, it is. If it does turn around the horizontal axis tractrix x will generate an area known as the seudoesfera , which shows another different view:

The seudoesfera is also known tractoide as precisely derived from a tractrix. This area shares with the field property any part of it exhibits the same curvature (we're talking about the curvature of a surface, not the curvature of a line), a constant curvature. While for the constant curvature sphere is positive for seudoesfera (tractoide) its constant curvature is negative. For this advantageous property, seudoesfera was precisely the area used by Beltrami to settle once and for all the issue of whether hyperbolic geometry discovered simultaneously by Janos Bolyai and Nicolai Lobachevsky geometry could be regarded as consistent as Euclidana geometry, classical (the word used is equiconsistencia ), which which failed to show. This does not mean that Beltrami has succeeded in demonstrating the overall consistency of both geometries, giving us the absolute guarantee that sooner or later no contradictions arise upon entering the demonstration of new theorems. What Beltrami showed that if the hyperbolic geometry is inconsistent, then so is Euclidean geometry, and vice versa.