Hyperbolic orthogonality | Wikipedia audio article
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Hyperbolic orthogonality
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"The only true wisdom is in knowing you know nothing."
Socrates
SUMMARY
=======
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) xy = 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
‖
c
,
b
‖
d
{\displaystyle 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.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. Two vectors
x
,
y
,
z
,
t
and
x
1
,
y
1
,
z
1
,
t
1
{\displaystyle 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.
{\displaystyle 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
t
x
=
x
1
t
1
{\displaystyle {\frac {t}{x}}={\frac {x_{1}}{t_{1}}}}
.
A bilinear form is used to describe orthogonality in analytic geometry, with two elements orthogonal when their bilinear form vanishes. In the plane of complex numbers
z
1
=
u
+
i
v
,
z
2
=
x
+
i
y
{\displaystyle z_{1}=u+iv,\quad z_{2}=x+iy}
, the bilinear form is
x
u
+
y
v
{\displaystyle xu+yv}
, while in the plane of hyperbolic numbers
...