True-circle illustration

A geometric or mathematical picture of the level or the number of ball to be called circular faithful or circle used when the image of an arbitrary circle always turn is a circle.

This particular property has for example the similarity maps and the stereographic projection . The orientation-preserving circle preserving bijections the Riemann sphere are exactly the Möbius transformations .

definition

A mapping of the plane on itself is called true to a circle if circles are mapped onto circles. The circle fidelity only refers to the circle line. The image point of the center of the original circle is generally not identical to the center of the image circle. Examples of true-to-circle images are similarity images such as parallel shifts , rotations , axis and point reflections or centric stretching . ${\ displaystyle \ mathbb {R} ^ {2}}$

A mapping of the Riemann number ball on itself is called true to a circle, if circles are mapped onto circles, whereby straight lines in a plane count as circles through the infinitely distant point. In addition to the similarity images, one also has the stereographic projection and generally all furniture transformations as true-to-circle images. ${\ displaystyle \ mathbb {R} ^ {2} \ cup \ left \ {\ infty \ right \}}$

More generally, a mapping of the -dimensional Euclidean space or the -dimensional sphere on itself is called a true-to-circle mapping when it maps circles onto circles. ${\ displaystyle n}$ ${\ displaystyle n}$

Stereographic projection

The stereographic projection depicts a spherical surface with the help of a central projection onto a plane, with the projection center lying on the spherical surface. The image of the center of projection is an infinitely distant point that is added to the plane. The plane can be understood as a complex number plane , which is expanded by the infinitely distant point, and the sphere as a Riemannian number sphere . The stereographic projection maps both surfaces bijectively onto one another.

The principle of stereographic projection was already known in ancient times. Its property as a true-to-circle image of the celestial sphere on a plane is said to be around 130 BC Was used by Hipparchus to build an astrolabe. In the 2nd century AD, this figure was described in detail by Ptolemy and it was geometrically proven to be true to the circle. Because they are true to the circle, circular orbits of the celestial bodies are also shown in a circle on flat maps. This property enabled the simple construction of star maps , navigation maps or the dials of astronomical clocks. The circular star paths in the sky could be drawn with compasses on flat disks. For the cartographic projection of the earth's surface onto a map, the principle was first used around 1500 and promoted particularly by the Nuremberg astronomer and mathematician Johannes Werner .

Illustration of the true-to-circle accuracy of the stereographic projection

The illustration opposite shows the section through a sphere. This section contains the projection center of the stereographic projection, the contact point of the image plane and the center of a circle to be mapped. The points and are the two points of the primordial circle on the shown meridian, and their image points. The angular center of the sphere , according to the circular angle set twice as large as the corresponding angle . The angles and are the same size, as their legs are perpendicular to each other in pairs . From consideration of the sum of the angles in the triangles , and finally follows that the angles shown in red are equal. The projection rays from through the original image circle form an elliptical cone. The archetype circle through and as well as its image through and intersect the cone at the same angle. Therefore the image of the primordial circle must also be a circle. ${\ displaystyle P}$ ${\ displaystyle T}$ ${\ displaystyle C}$ ${\ displaystyle A}$ ${\ displaystyle B}$ ${\ displaystyle A '}$ ${\ displaystyle B '}$ ${\ displaystyle M}$ ${\ displaystyle P}$ ${\ displaystyle \ angle CMT}$ ${\ displaystyle \ angle CSA '}$ ${\ displaystyle PB'T}$ ${\ displaystyle PB'A '}$ ${\ displaystyle BB'S}$ ${\ displaystyle P}$ ${\ displaystyle A}$ ${\ displaystyle B}$ ${\ displaystyle A '}$ ${\ displaystyle B '}$

Furniture transformation

Möbius transformations form the complex numbers, expanded by the infinitely distant point, onto themselves. Their general formula is given by

{\ displaystyle \ phi: z \ mapsto {\ frac {az + b} {cz + d}}}

,

where are complex numbers that satisfy. This condition ensures that the image is not imaged onto a fixed pixel and that the image is bijective . ${\ displaystyle a, b, c, d}$ ${\ displaystyle ad-bc \ neq 0}$ ${\ displaystyle z}$

They are named after August Ferdinand Möbius , who examined them in a purely geometric representation in his work The Theory of Circular Relationship in 1855 and who described their group characteristics .

Every Möbiust transformation can be achieved by linking the three elementary types

Translation ( ), ${\ displaystyle \ phi: z \ mapsto z + b}$
Twist stretch ( ) and ${\ displaystyle \ phi: z \ mapsto az}$
Inversion ( ) ${\ displaystyle \ phi: z \ mapsto {\ tfrac {1} {z}}}$

to be discribed. The first two figures are similarity figures and therefore obviously true to the circle. Since the inversion is also true to the circle (see next section ), concatenations of these images and thus every Möbius transformation are true to the circle. Even when combined with a stereographic projection onto a sphere, rotating the sphere, changing the projection center and projecting back onto the plane, the image remains true to the circle.

The circling and orientation-preserving images of the complex plane of numbers (including the infinite distant point) on themselves are exactly the Möbius transformations. They are also conformal (conformal). Of all conforming images, only they map this level of numbers bijectively onto themselves. Therefore, conforming mappings can be handled very effectively using the complex numbers.

Other circled images

Other true-to-circle images are the axis reflection and the circle reflection . The axis reflection is a congruence mapping , with the circle reflection the original image point and the image point lie on a half-line through the center of the circle. Both images are true to the angles, but the orientation of the angles is reversed - unlike in the case of the orientation-preserving furniture transformations.

These mappings and their linkage with orientation-preserving Möbius transformations can be described with the help of the conjugate complex number by: ${\ displaystyle {\ bar {z}}}$

{\ displaystyle \ phi: z \ mapsto {\ frac {a {\ bar {z}} + b} {c {\ bar {z}} + d}}, \ qquad ad-bc \ neq 0}

.

These figures describe precisely all figures of the Riemann number sphere that are true to a circle.

An inversion is composed of an axis and a circular reflection, so that the orientation is retained. ${\ displaystyle z \ mapsto {\ tfrac {1} {z}}}$

Tissot indicatrix

In map network designs , the local distortion is illustrated by a Tissot indicatrix , which represents the image of a circle as a distortion ellipse. This makes direction-dependent route distortion visible in the point under consideration. In conformal maps, all distortion ellipses are circles. This “circle loyalty” generally only applies locally and not for circles of any size.

True-to-circle images in higher dimensions

A mapping is called true to circle if it maps circles into circles. ${\ displaystyle f \ colon \ mathbb {R} ^ {n} \ cup \ left \ {\ infty \ right \} \ to \ mathbb {R} ^ {n} \ cup \ left \ {\ infty \ right \} }$

This is precisely the case when it can be represented as a series of a finite number of similarity images and reflections in hyperplanes and / or spheres.

In particular, the orientation-preserving circular images are exactly the higher-dimensional furniture transformations .

Individual evidence

↑ Eberhard Schröder: Map drafts of the earth. Harri Deutsch, Thun and Frankfurt / Main, 1988, p. 32f.
^ Klaus Fritzsche : Möbius Transformations. (PDF) Retrieved on March 19, 2016 (part of a lecture notes).
^ ^A ^b Günter M. Ziegler : Geometry. (PDF) July 6, 2012, pp. 67–72 , accessed on March 19, 2016 (preliminary lecture notes).
^ Ziegler, ibid.

[1] Eberhard Schröder: Map drafts of the earth. Harri Deutsch, Thun and Frankfurt / Main, 1988, p. 32f.

[Fritzsche-2] Klaus Fritzsche : Möbius Transformations. (PDF) Retrieved on March 19, 2016 (part of a lecture notes).

[Ziegler-3] A ^b Günter M. Ziegler : Geometry. (PDF) July 6, 2012, pp. 67–72 , accessed on March 19, 2016 (preliminary lecture notes).

[4] Ziegler, ibid.