Circular reflection

The reflection on the circle or circle reflection is a representation of the planar geometry , which interchanges the inside and the outside of a given circle .

The mapping is conformal and is one of the special conformal transformations .

A circular reflection is the plane case of a (geometric) inversion . An inversion in space is the reflection on a sphere , or spherical reflection for short, with properties similar to those of circular reflection.

To define the reflection on a circle

definition

For the mirroring of a circle on a circle with a center point and radius , the image point of a point is determined by the fact that they lie on a line or on a half- line and the condition ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle P'}$ ${\displaystyle P}$ ${\displaystyle P'}$ ${\displaystyle |{\overline {MP}}|}$ ${\displaystyle [MP}$

${\displaystyle |{\overline {MP'}}|={\frac {R^{2}}{|{\overline {MP}}|}}}$

must meet. The original point must not coincide with the center point . Occasionally, this problem can be avoided by adding a new point to the plane and defining it as the pixel of . The image point of this new point is the center of the inversion circle. Often only the center is important, but not the radius , so that you can draw a circle with any radius (e.g. 1). ${\displaystyle P}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle M,}$ ${\displaystyle R}$

Analytical description

If the origin is in a Cartesian coordinate system , the reflection on the circle can be passed through ${\displaystyle M}$ ${\displaystyle x^{2}+y^{2}=R^{2}}$

• ${\displaystyle (x,y)\rightarrow {\frac {R^{2}\cdot (x,y)}{x^{2}+y^{2}}}}$

describe.

In planar polar coordinates , a mirroring of a circle has a particularly simple representation: ${\displaystyle r,\varphi }$

${\displaystyle (r,\varphi )\rightarrow \left({\frac {R^{2}}{r}},\varphi \right)}$.

The reflection on the unit circle is then

${\displaystyle (r,\varphi )\rightarrow \left({\frac {1}{r}},\varphi \right)}$

and justifies the term inversion.

In the complex analysis treating the inversions and their production of circle-preserving transformations in the best complex ( "Gaussian") number plane . An inversion on the unit circle is described by the figure . Therein denotes a complex number and the associated complex conjugate number. ${\displaystyle z\mapsto {\frac {1}{\overline {z}}}}$ ${\displaystyle z}$ ${\displaystyle {\overline {z}}}$

construction

With compass and ruler

Fig. 1: Construction of the image point mirrored on the inversion circle (red) with compass and ruler: the perpendicular to the line connecting the center of the circle to the original image point is formed at the point ; the two tangents at the two points of intersection with the circle meet in the image point ${\displaystyle P'}$ ${\displaystyle P}$ ${\displaystyle P}$ ${\displaystyle P'.}$

• If lies on the given circle, it is equal . ${\displaystyle P}$ ${\displaystyle P'}$ ${\displaystyle P}$

• If the point in the circle inside is located (Figure 1), to draw the to half-line vertical chord through and the two tangential circles at the ends of this chord. then results as the intersection of these tangents. ${\displaystyle P}$ ${\displaystyle [MP}$ ${\displaystyle P}$ ${\displaystyle P'}$

• If, on the other hand, the point is outside the circle, you begin with the two circle tangents using the Thales circle . Then you bring the connecting line of the two points of contact with the half-line to the intersection. The point of intersection is the image point that is sought . ${\displaystyle P}$ ${\displaystyle P}$ ${\displaystyle [MP}$ ${\displaystyle P'}$

The proof that the image point is obtained in this way follows directly from the set of cathets .

Alone with a circle

Image 2: The original image point is only mirrored with the help of a compass on the inversion circle (red), the image point results ${\displaystyle P}$ ${\displaystyle P'.}$

If the point is outside the inversion circle (Fig. 2), draw around a circle through the center of the inversion circle. This intersects the inversion circle in two points. Also around these points draw circles through the center. These two circles now intersect in the image point . ${\displaystyle P}$ ${\displaystyle P}$ ${\displaystyle P'}$

If lies on the inversion circle, no construction is necessary, it applies ${\displaystyle P}$ ${\displaystyle P'=P.}$

Lies within the inversion circle, z. B. by dividing the possible positions of the point into three areas (Fig. 3–5), a significant simplification of the construction effort for two areas can be achieved. For this one imagines, quasi-mentally, a circular area (light gray), the radius of which is equal to half the radius of the inversion circle. The circular area (light gray) is not required for the actual construction. The three areas of the possible position of the point , usually given as the distance to the center of the inverse circle, and the possible construction methods are: ${\displaystyle P}$ ${\displaystyle P}$ ${\displaystyle P}$ ${\displaystyle M}$

1. The distance from the point to (Fig. 3) is greater than half the radius of the inversion circle, i.e. H. ${\displaystyle P}$ ${\displaystyle M}$ ${\displaystyle |{\overline {MP}}|>{\frac {1}{2}}R.}$

The description for the construction can be taken from the construction above for If the point lies outside the inversion circle (Figure 2) ${\displaystyle P}$ .

Figure 4: The distance from the point to is equal to half the radius of the inversion circle (red),  ${\displaystyle P}$ ${\displaystyle M}$ ${\displaystyle |{\overline {MP}}|={\frac {1}{2}}R.}$

Image 3: The distance from the point to is greater than half the radius of the inversion circle (red),  ${\displaystyle P}$ ${\displaystyle M}$ ${\displaystyle |{\overline {MP}}|>{\frac {1}{2}}R.}$

2. The distance from the point to (Fig. 4) is equal to half the radius of the inversion circle, i.e. H. ${\displaystyle P}$ ${\displaystyle M}$ ${\displaystyle |{\overline {MP}}|={\frac {1}{2}}R.}$

First a circle with a radius is drawn around the point and then, by subtracting this radius three times from the point , its diameter is determined. Next, the last circle with the radius is drawn around the point . Finally, this radius needs to be removed twice, starting with the intersection point just created, in order to obtain the image point . ${\displaystyle P}$ ${\displaystyle |{\overline {MP}}|}$ ${\displaystyle M}$ ${\displaystyle |{\overline {MC}}|}$ ${\displaystyle |{\overline {MC}}|}$ ${\displaystyle C}$ ${\displaystyle D,}$ ${\displaystyle P'}$

3. The distance from the point to (Fig. 5) is less than half, but greater than one eighth of the radius of the inversion circle, i.e. H. ${\displaystyle P}$ ${\displaystyle M}$ ${\displaystyle {\frac {1}{8}}R<|{\overline {MP}}|<{\frac {1}{2}}R.}$

Image 5: The distance from the point to is less than half, but greater than an eighth of the radius of the inversion circle (red), ${\displaystyle P}$ ${\displaystyle M}$ ${\displaystyle {\frac {1}{8}}R<|{\overline {MP}}|<{\frac {1}{2}}R.}$

In the adjacent Figure 5, the small circular area (pink) illustrates one eighth of the radius of the inversion circle. The circular area (pink) is not required for the actual construction. This also applies to the dotted lines drawn; they are only intended to make a comparison with the construction with compass and ruler clear.

First a circle with a radius is drawn around the point and then its diameter is determined by removing this radius three times . This is followed by an arc of a circle with a radius on which the diameter is generated in the same way as before . Now an arc is drawn around with a radius that intersects the inversion circle in and . A circular arc around and with the radii or connect and intersect in Um , a circular arc with a radius is drawn on which the diameter is generated , analogously before . Next, the last circle with the radius is drawn around the point . Finally, this radius needs to be removed three times from the point to obtain the image point . ${\displaystyle P}$ ${\displaystyle |{\overline {MP}}|}$ ${\displaystyle |{\overline {MC}}|}$ ${\displaystyle C}$ ${\displaystyle |{\overline {MC}}|,}$ ${\displaystyle |{\overline {MF}}|}$ ${\displaystyle F}$ ${\displaystyle |{\overline {MF}}|}$ ${\displaystyle G}$ ${\displaystyle H}$ ${\displaystyle G}$ ${\displaystyle H}$ ${\displaystyle |{\overline {MG}}|}$ ${\displaystyle |{\overline {MH}}|}$ ${\displaystyle I.}$ ${\displaystyle I}$ ${\displaystyle |{\overline {MI}}|}$ ${\displaystyle |{\overline {ML}}|}$ ${\displaystyle {\overline {ML}}}$ ${\displaystyle L}$ ${\displaystyle M}$ ${\displaystyle P'}$

• Universal method for lies ${\displaystyle P}$ within the inversion circle :

First, halve the radius of the inversion circle until you get a new circle that no longer contains the point . (This is only possible with a pair of compasses.) Then, as above (Fig. 2), construct the image point of , whereby the inversion is carried out on the new circle. Finally, you double the distance between the image point twice as often as you halved the radius. (This is also possible with a compass alone.) This point is the image point we are looking for. ${\displaystyle P}$ ${\displaystyle P}$

Due to the complexity of this procedure, the construction will hardly be carried out, but it offers a possibility to prove the Mohr-Mascheroni theorem, which says that with compasses alone you can carry out all constructions that are possible with compasses and ruler .

With other aids

There are mechanical devices specially designed for inversion on a circle, for example Peaucellier's Inversor .

properties

• The illustration swaps the inside and outside of the inversion circle, the points on the edge are fixed points .

• If the inversion is applied twice, the initial situation is obtained again, the inversion is therefore an involution .

• The inversion is a conformal mapping ; i.e., it is conformal. In particular, objects that touch one another are also mapped onto such objects.

• The inversion reverses the orientation like the straight line reflection .

• Straight lines that run through the center of the inversion circle are mapped onto themselves.

• Straight lines that do not pass through the center point are mapped to circles that pass through the center point.

• Circles that run through the center point are mapped on straight lines that do not go through the center point.

• Circles that do not run through the center of the inversion circle are mapped onto such circles again. However, the inversion does not map the center of the original circle to the center of the image circle.

• In particular, circles that intersect the inversion circle at right angles are mapped onto themselves.

Since the inversion is not true to the line, it is not a congruence mapping in contrast to point, axis or plane reflection .

literature

• Coxeter, HSM , and SL Greitzer: Timeless Geometry, Klett Stuttgart 1983
• Roger A. Johnson: Advanced Euclidean Geometry . Dover 2007, ISBN 978-0-486-46237-0 , pp. 121-127 (first published in 1929 by the Houghton Mifflin Company (Boston) under the title Modern Geometry ), pp. 43-57

Web links

• Vladimir S. Matveev: Inversion on a circle (mirroring of a circle) . Part of a script on linear algebra at the University of Jena (PDF; 828 kB).
• Inversion to cut-the-knot
• Eric W. Weisstein : Inversion . In: MathWorld (English).

Individual evidence

1. ↑ Coxeter, HSM ; Greitzer, SL: Geometry Revisited . Washington, DC: Math. Assoc. Amer. 1967, p. 108 5.3 Inversion ( excerpt (Google) ) - English original edition of Timeless Geometry .
2. ↑ David A. Brannan, Matthew F. Esplen, Jeremy J. Gray: Geometry . Cambridge University Press 1999, 2nd edition 2011, ISBN 978-1-107-64783-1 , pp. 281–283 ( excerpt (Google) )