Geometric group theory

In plane geometry, two lines are hyperbolic orthogonal when they are reflections of each other over the asymptote of a given hyperbola. Two particular hyperbolas are frequently used in the plane:

(A) x y = 1 with y = 0 as asymptote.

When reflected in the x-axis, a line y = mx becomes y = −mx.

In this case the lines are hyperbolic orthogonal if their slopes are additive inverses.

(B) x² − y² = 1 with y = x as asymptote.

For lines y = mx with −1 < m < 1, when x = 1/m, then y = 1.

The point (1/m , 1) on the line is reflected across y = x to (1, 1/m).

Therefore the reflected line has slope 1/m and the slopes of hyperbolic orthogonal lines are reciprocals of each other.

The relation of hyperbolic orthogonality actually applies to classes of parallel lines in the plane, where any particular line can represent the class. Thus, for a given hyperbola and asymptote A, a pair of lines (a,b) are hyperbolic orthogonal if there is a pair (c,d) such that $a\rVert c,b\rVert d$, and c is the reflection of d across A.

The property of the radius being orthogonal to the tangent at the curve, is extended from the circle to the hyperbola by the hyperbolic orthogonal concept. See Lewis & Wilson or Felsagen for the radius-tangent orthogonality.

Since Hermann Minkowski's foundation for spacetime study in 1908, the concept of points in a spacetime plane being hyperbolic-orthogonal to a timeline (tangent to a World line) has been used to define simultaneity of events relative to the timeline. In Minkowski's development the hyperbola of type (B) above is in use (see the wikisource link below). Two vectors $x,y,z,t\quad {\text{and}}\quad x_{1},y_{1},z_{1},t_{1}$ are normal (meaning hyperbolic orthogonal) when

$c^{2}t\ t_{1}-x\ x_{1}-y\ y_{1}-z\ z_{1}=0.$

When c = 1 and the y's and z's are zero, x≠0, t₁≠0, then ${\frac {t}{x}}={\frac {x_{1}}{t_{1}}}$.

Given a hyperbola with asymptote A, its reflection in A produces the conjugate hyperbola. Any diameter of the original hyperbola is reflected to a conjugate diameter. The directions indicated by conjugate diameters are taken for space and time axes in relativity. As E. T. Whittaker wrote in 1910, "[the] hyperbola is unaltered when any pair of conjugate diameters are taken as new axes, and a new unit of length is taken proportional to the length of either of these diameters." On this principle of relativity, he then wrote the Lorentz transformation in the modern form using rapidity.

Edwin Bidwell Wilson and Gilbert N. Lewis developed the concept within synthetic geometry in 1912. They note (p.415) "in our plane no pair of perpendicular [hyperbolic-orthogonal] lines is better suited to serve as coordinate axes than any other pair".

In the terminology of projective geometry, the operation of taking the hyperbolic orthogonal line is a hyperbolic involution. Suppose the slope of a vertical line is denoted ∞ so that all lines have a slope in the real projective line. Then whichever hyperbola (A) or (B) is used, the operation is an example of a hyperbolic involution.

References

• G. D. Birkhoff (1923) Relativity and Modern Physics, pages 62,3, Harvard University Press.
• Francesco Catoni, Dino Boccaletti, & Roberto Cannata (2008) Mathematics of Minkowski Space, Birkhäuser Verlag, Basel. See page 38, Pseudo-orthogonality.
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• Various English translations on Wikisource: Space and Time

• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010(see page 441)
• Bjørn Felsager (2004), Through the Looking Glass - A glimpse of Euclid’s twin geometry, the Minkowski geometry, ICME-10 Copenhagen; pages 6 & 7.
• Edwin B. Wilson & Gilbert N. Lewis (1912) "The Space-time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics" Proceedings of the American Academy of Arts and Sciences 48:387-507.