**Excursion through**

Inversion

Inversion

**Inversion**

“An inversion in a circle, informally, is a transformation of a plane that flips the circle inside-out. That is, points outside the circle get mapped to points inside the circle and points inside the circle get mapped outside the circle.” -Kozai and Libeskind

Inversion occurs with respect to a specified circle of inversion

C

with center

O

and radius

r

The image of a point

P

is

P'

lying on the ray OP such that OP*OP' =

r

^2

If we chose

r

= 1, OP = 1/(OP') = (OP)^-1

Inversion in the Standards

“Dynamic geometry environments provide students with experimental and modeling tools that allow them to investigate geometric phenomena in much the same way as computer algebra systems allow them to experiment with algebraic phenomena.” -Common Core Standards

Properties of Inversion

A line through O maps to itself

A line not through O maps to a circle through O

A circle not through O maps to a circle not through O

A point P on the circle of inversion maps to itself

History

Inversion dates back to more than 2000 years ago. Apollonius of Perga (225 B.C. - 190 B.C.) investigated one particular family of curves (the Apollonius curve), which is significant in inverse geometry. He defined the Apollonian

curve C sub k(A,B) to be the locus of points P such that PA = kPB, where A and B are points in the Euclidean plane, and k is a positive constant. He gave the following theorem:

1) C sub k(A,B) is a line if k = 1 and a circle for all other positive k.

2) if C sub k(A,B) is a circle with center O and radius r, then

a) O,A,B are collinear with A and B on the same side of O; b) OA.OB = r^2; and c) If A,B, and the given circle satisfy (a) and (b) then the circle is an Apollonian curve.

He also proposed the problem to construct a circle that is tangent to three circles. This problem can be solved using inversion. Dandelin solved this problem using inversion in a memoir in 1825.

Applications of Inversion

There are a variety of applications associated with Inversion in geometry.

The main applications include:

Apollonius' Problem

Steiner Chain

Peaucellier Inversor (linkage mechanism)

Other uses include:

Hart's Linkage

Pappus Chain

Magnetism

Poincaré Disk

Apollonius' Problem

Given three circles (non-intersecting), construct a circle tangent to all three circles.

The problem can be solved by using inversion and will be displayed in GSP.

Inversion with Conics

Inversion with Triangles

Make Geometric Constructions

12. Make formal geometric constructions with a variety of tools and methods.

→Using tools such as GSP allows students to explore Inversion Geometry by making formal constructions. After making a specific tool that inverts, it is easier to explore properties of inversion with different shapes, centers of inversion, etc.

Example:

Constructing P’ (inside and outside the circle of inversion).

Making Geometric Constructions

13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

→Use GSP to make these constructions, and then invert them using the circumcircle as the circle of inversion.

Understand and Apply Theorems about Circles

3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

→After constructing and inverting in GSP, have students make observations, especially about what happens to the angles in the inversion image.

Connections

Since inversion is not mentioned in the standards explicitly, we have to take bits and pieces from the Geometry section and expand on them to include inversion.

Teaching inversion is useful for challenging students and motivating them to work on more in-depth and difficult problems.

Steiner Chain

Peaucellier's Linkage

Named after Charles-Nicolas Peaucellier (1832-1913), a French engineer, who discovered it in 1864

The purpose of this tool is that it converts circular motion to linear motion

By using this device, we are able to draw the inverse of the curve/shape

In 330 A.D. Pappus of Alexandria (290-350) was the first to mention that this transformation carried a line or circle into a line or circle. A more in depth study of inversion did not happen until the 19th century.

Steiner (1769-1863) wrote many papers on the subject of inversion, but he never formally published them before he died. He explicitly states that the inverse of a parabola is a closed curve passing through the center of inversion C; that the inverse of an hyperbola always

goes through C and the asymptotes transform into two asymptotic circles. However, he does not provide any proof as to how he arrived at these conclusions.

The center of a circle is not preserved in inversion, however, the inverted center does remain on the same ray as the original center.

Illustration of linkage in GSP

The quasi-transformation, “Given a circle w with center O and radius k and a point P different from O, we define the inverse of P to be the point P’, on the ray OP, whose distance from O satisfies the equationOPOP'=k2” was invented by Magnus in 1831 (Coxeter and Greitzer, p. 108). Ludwig Immanuel Magnus (1790-1861) was a German Jewish mathematician.

Bellavitis (1803-1880) discusses inversion nearly completely in a paper in 1836, but it is clear he does acknowledge that he is not the first to study the material.

Angle Preservation

Angles are preserved under inversion. If two curves intersect at an angle measurement of

b

, their inverses will also intersect at an angle measurement of

b

, but in a reversed direction of sweeping.

Here we have four circles tangent at and passing though

O

and a fifth circle passing through

O

, but not tangent to the other circles. If we take the inverse of all these circles, we get five lines: four that are parallel and one that is intersecting the parallel lines (and is thus a transversal). Under inversion, the angles made by the parallel lines and the transversal have the same measurement as the angles made by the intersection of the original circles.

As B traces the circle, D traces a line (circular to linear motion)

The blue line is the inverse of the red circle, and O is the center of inversion

The two segments from O have the same specified length and the four smaller segments also have the same specified length. These lengths do not change as B is moved.

How Does It Work?

Two circles are drawn/given

One is within the other

They do not intersect at any points, and they do not necessarily have to be concentric

Draw a circle tangent to the two circles

Then draw a circle tangent to those three circles, and continue to do this until the chain is complete

Either the last circle will be tangent to the first constructed circle or will intersect

Steiner Chain created by Inversion

How do you think inversion was used in this construction?

Can you figure out where the circle of inversion might be?

Open GSP document for the answers...

**By Lexi Stear, Mimi Tsui, and Kendyl Wade**