We will not prove the theorem for all cases but sketch out the proof for the case where the angle is formed by two circles.
This is shown by using the same steps as in Exercise 4. Other cases of angles between circles or lines can be proved in a similar way.
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Note that when two circles are reflected in a circle, the angles are preserved but since the orientation of points is reversed the orientation of angles is also reversed. When forming the angles between two circles you can always do it so that exactly one of the supplementary angles is inside both circles. After inversion the two angles are reversed with regards to which angle that is inside both circles.
As a consequence of angle preservation, objects that are tangent to each other, are inverted to objects that are also tangent to each other. A geodesic is the hyperbolic equivalence of a Euclidean line. The reason why perpendicular generalized arcs are used as hyperbolic lines to model hyperbolic geometry, is because of the properties of perpendicular circles when doing circle inversion.
For this part of the proof we will use that a tangent to a circle through a point is perpendicular to the radius through the same point. We have that. When inverting in a circle, angles are preserved but distances are not. There is, however, a relation between distances that is preserved even after a circle inversion. In order for the hyperbolic distance to have the same functionality as Euclidean distance, it must have a number of properties. Thales theorem states that if two vertices of a triangle are the endpoints of a diameter of a circle, and if the third vertex also lies on the circle, then the angle at the third vertex is a right angle.
You don't have to do this. Anton Petrunin - Euclidean and Hyperbolic Planes, A minimalistic introduction with metric approach pdf.
Inversion in Circle. Non-Euclidean Geometry Inversion in Circle. Proof The proof is the task of Exercise 1. Proof We must prove the theorem for each of the three cases. The proof is the task of Exercise 3. The task is given with hints. The proof is the task of Exercise 4. Most Simple Science Theory of Reality. According to the general theory of relativity space without ether is unthinkable; for in such space there not only would be no propagation of light, but also no possibility of existence for standards of space and time measuring-rods and clocks , nor therefore any space-time intervals in the physical sense.
But this ether may not be thought of as endowed with the quality characteristic of ponderable media, as consisting of parts which may be tracked through time. The idea of motion may not be applied to it. They may not be thought of as consisting of particles which allow themselves to be separately tracked through time. Physical objects are not in space, but these objects are spatially extended. The four dimensional space of the special theory of relativity is just as rigid and absolute as Newton's space. Thus he agrees that the 'ether' or Space physically exists but that Matter, Space, and Time are intimately interconnected rigidly interconnected by Space , which is true.
Euclidean and Non-Euclidean Geometries, 4th Edition | Macmillan Learning for Instructors
As it turns out though, the Electromagnetic Fields Albert Einstein used were continuous and vector, whereas the true Wave Motion is scalar described by Wave-Amplitude only, see Point vi below and form discrete standing Waves, thus his task was ultimately impossible which of course explains his failure. In fact Lorentz, whose work founded Albert Einstein's Relativity, though he was also misled by the electromagnetic field and thus argued for an 'ether' for the vector field Electromagnetic Waves to exist in was closest to the truth when he wrote;. We shall add the hypothesis that, though the particles may move, the ether always remains at rest.
I cannot but regard the ether, which can be the seat of an electromagnetic field with its energy and its vibrations, as endowed with a certain degree of substantiality, however different it may be from all ordinary matter.
Now it seems that many people do not correctly understand the meaning of the four dimensional space-time continuum, but it simply means that three spatial dimensions and a time dimension are required to define the motion of bodies and the path of light in three dimensional Space. And yet there is no more common-place statement than that the world in which we live is a four-dimensional space-time continuum.
Space is a three-dimensional continuum. Similarly, the world of physical phenomena which was briefly called 'world' by Minkowski is naturally four dimensional in the space-time sense. For it is composed of individual events, each of which is described by four numbers, namely, three space co-ordinates x, y, z, and the time co-ordinate t. Thus we could number the Spherical In-Waves 1,2,3, Thus we see how Time is caused by Wave Motion.
For Matter moving past a massive body in Three Dimensional Space e. This is what we mean when we talk of the 'curvature of space'. The fundamental concepts of the 'straight line', the 'plane', etc. In the general theory of relativity the doctrine of space and time, or kinematics, no longer figures as a fundamental independent of the rest of physics.
The geometrical behavior of bodies and the motion of clocks rather depend on gravitational fields which in their turn are produced by matter. Now this is very important, for it is this 'curvature' that largely led to Albert Einstein's early fame. As we have explained, the measurable properties affected by the presence of spherically spatially extended Matter in Space is that the path of nearby light and matter is caused to be curved. It was the prediction by Albert Einstein that light curved as it grazed the sun subsequently confirmed by observation during a solar eclipse on the 29th May that resulted in his General Theory of Relativity becoming widely accepted and very famous!
Just like an asteroid, the curved path of light is simply caused by the higher mass-energy density of space near Matter and the resultant slower Wave-Velocity. As Albert Einstein rightly says;. His general principle is correct though, matter does determine the geometric properties of Space;.
Thus very tiny differences of Wave-Medium Density occur near very large masses such as the Sun due to this cumulative effect of many trillions of high mass-energy density of space Wave-Centers. This increasing mass-energy density of Space slows the velocity of the wave fronts and causes them to curve slightly when passing massive bodies like our sun.
Thus he describes matter in Space as a 4D space-time continuum using the changing vector force field direction and quantity of Force.
The continuous 'force field' is actually an approximation of many discrete standing Wave interactions and ultimately any force and resultant acceleration on a 'Particle' is caused by the change in velocity of their In-Waves. The vector directional component is not fundamental like in Albert Einstein's force field but is caused by the slowing down i.
Our In-Waves coming in through the Space of the Earth. If we are to be precise with our language, but it is not really as complicated as it sounds! But it cannot be claimed that those parts of the general relativity theory which can today be regarded as final have furnished physics with a complete and satisfactory foundation. In the first place, the total field appears in it to be composed of two logically unconnected parts, the gravitational and the electromagnetic. And in the second place, this theory, like the earlier field theories, has not up till now supplied an explanation of the atomistic structure of mater.
This failure has probably some connection with the fact that so far it has contributed nothing to the understanding of quantum phenomena. Rather, they start this, displaying it to the senses, This sort of procedure is inductive and it is as plain as a pikestaff that it does not amount to a demonstration of essence or of what it is to be a thing. Aristotle, Metaphysics. The Electric and Magnetic Force Fields were first founded on repeated observations of how many trillions of Electrons and Protons behaved how magnets and electrically charged objects attracted and repelled one another, and how moving magnets produced electrical currents in wires, and vice versa This explains why the fields were continuous, as many trillions of discrete standing wave interactions blend together into a continuous force.
It was only around Planck that Physicists discovered that energy exchanges were in discrete amounts, and by then the continuous field theory was well established by Faraday, Maxwell and Lorentz and soon by Albert Einstein!
Foundation of Euclidean and Non-Euclidean Geometries according to F. Klein, Volume 97
Thus Albert Einstein did not know what Matter was, he just knew certain Inductive Laws from repeated observation about how it behaves by measuring the forces that matter has on the matter around it. The theory of relativity may indeed be said to have put a sort of finishing touch to the mighty intellectual edifice of Maxwell and Lorentz, inasmuch as it seeks to extend field physics to all phenomena, gravitation included.
The theory of relativity leads to the same law of motion without requiring any special hypothesis whatsoever as to the structure and behavior of the electron.
I wished to show that space time is not necessarily something to which one can ascribe to a separate existence, independently of the actual objects of physical reality.