the location of 4 points

this activity is a page of geogebra-book elliptic functions & bicircular quartics & . . .(06.02.2023)

4 different complex points

are characterised möbius-geometrically by their complex cross product

2 quadruples of points with identical cross productcan be mapped by a Möbius transformation

: one onto the other, but depending on the order. Independent of the order is the absolute invariant of four points:

For an elliptic function with the differential equation

and different focal points

its absolute invariant is also called the absolute invariant of the elliptic function.

In the applet above on the right, a Möbiustransformation has been mapped

to 0,

and

. special positions: I.: The circles

through

and

through

are orthogonal exactly,, if the double ratio

is purely imaginary. In the picture on the right, this is the case exactly when the straight lines

and

are orthogonal, i.e. exactly then, when

on the THALES circle lies over

! The circles

through

and

through

are then also orthogonal. The absolute invariant

does not reveal any special feature. II.: The pairs of points

and

lie mirror-inverted on 2 orthogonal circles exactly then, when

; this is the case exactly if

holds; and this is the case exactly if the absolute invariant

is real and not positiv:

and

. reason:

is the theorem of THALES, see below.A circle to which the points

are mirror-images, must be a circle from the hyperbolic pencil of circles around

and

(the inversion at this circle interchanges the two points!) Exactly one circle each from this pencil of circles goes through

and through

and

can therefore only lie mirror-inverted to a circle through

, if the two circles of the pencil are identical. In the applet one obtains on the right

is therefore exactly the case if

lie around 0 on the same concentric circle. The inversion at this circle interchanges 0 and

, the angle bisectors of the straight lines

and

are orthogonal to the concentric circle, mirrored on them

and

are interchanged. We explain the connection with the absolute invariant

below!

III.: 4 points lie on a circle exactly when their double ratio is real; this is the case exactly if the absolute invariant of the 4 points is real and non-negative! reason: According to the Peripheral Angle Theorem, the double ratio is real exactly if, if the points lie on a circle. This property is independent of the order of the points. For real

real and non-negative!

Under which condition is the absolute invariant

real? The set of 6 relative invariants

and

is invariant under the finite group of involutory Möbius transformations

with

. For each of the 6 relative invariants holds

This cubic equation in

has only real coefficients.

is true if and only if

is real, i.e. if the points are concyclic.

is true if and only if

for any of the six relative invariants. By changing the order of the points can be achieved

i.e. mirror-image position on 2 orthogonal circles.

2 very special positions: If the 4 points are different and is

, then both are valid: the points are concyclic and lie mirror-image on 2 orthogonal circles. Example: the intersections of the 1st and 2nd bisectors with the unit circle.

is present for 4 points whose double ratio

is independent of the order. If one projects the points stereographically onto the unit sphere, one obtains a regular tetrahedron on the sphere.

the location of 4 points

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