CONSTRUCTION AND ANALYSIS OF INVERSIONS IN S 2 AND H 2. Arunima Ray. Final Paper, MATH 399. Spring 2008 ABSTRACT

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1 CONSTUCTION AN ANALYSIS OF INVESIONS IN S AN H Arunima ay Final Paper, MATH 399 Spring 008 ASTACT The construction use to otain inversions in two-imensional Eucliean space was moifie an applie to otain inversions on susets of the two-imensional spherical an hyperolic spaces. The functions otaine showe remarkale symmetry with the Eucliean formula, an have similar properties. We investigate the conformality of the functions in the spherical an hyperolic geometries, ut have no conclusive results at this time. Introuction We otain the formulae for inversions as shown elow: Spherical Geometry: EuclieanGeometry: HyperolicGeometry: tan tan tan tanh tanh tanh where is the raius of the inverting circle, is the istance of the original point from the center of the inverting circle, an is the istance of the inverte point from the center of the inverting circle. Clearly, these functions are inverses for themselves an show consierale similarity to one another. In aition, we iscovere that an inverting circle of raius of π/4 yiels a conformal function in spherical geometry. a. Inversions ackgroun In Eucliean Geometry, an inversion can e consiere to e the reflection of points on a circle, such that points in the interior the circle are mappe to the exterior, points in the exterior are

2 mappe to the interior an points on the circle are mappe to themselves. Inversions are their own inverses, namely, the composition with itself is the ientity function. Figure : Constructing the Eucliean Inversion. (i) Points insie the inverting circle. (ii) Points outsie the inverting circle.. Constructing the Eucliean Inversion In Eucliean geometry, the inversion can e geometrically constructe (Figure ). Consier a point P insie a circle I (to e inverte across) with center O. We construct the raius of the circle through P, an then the chor perpenicular to this raius at P. The chor intersects the circle I at the points A an. The tangents to I at A an intersect at the point P. If the point P to e inverte is outsie the circle I, we construct the line segment from the center O to P, an otain the mipoint M. We construct the circle with center M an raius P. This circle intersects the circle I at the points A an. The line segment joining A an intersects OP at the point P. c. Testing the Eucliean Inversion Let the raius of the inverting circle e r, OP =, an OP =. In the first case, where we invert a point in the interior of the circle of inversion, ΔPOA ~ Δ AOP OP OA Thus. OA OP r or, r r or,

3 Similarly, in the secon case, where P is in the exterior of I, ΔP OA ~ Δ AOP OA OP Thus. OP OA r or, r r or, Notaly, this satisfies the conitions of eing an inversion. Thus,. Examples Figure : Ojects eing inverte across a circle, on the Eucliean plane. (i) The green circle is inverte to the lue circle. (ii) The green line is inverte to the lue circle. In the first example, the portion of the green circle insie the inversion circle is mappe to the portion of the lue circle in the exterior of the inversion circle. The part of the green circle outsie the inversion circle is mappe to the part of the lue circle in the interior of the inversion circle. It is clear that istances are not preserve. The lue an green curves intersect on the inversion circle, ce points of the green circle on the inversion circle get mappe to themselves, y the characteristic property of inversions. In the secon example, the green line is mappe to the lue circle. Again, the two curves intersect on the inversion circle. Similarly, insie of green correspons to outsie of lue, vice versa. Certainly, shapes are not preserve uner Eucliean inversions. In oth these examples, it serves to keep in min that the lue circles woul also get inverte to the green circle an line respectively, ue to the inversive nature of the function.

4 e. Conformality Figure 3: The angles efore an after inversion are equal. espite not conserving istances, inversions o preserve angles etween curves, i.e. they are conformal. We know that if f: X n is a function an f (x) is the matrix of partial erivatives, f is conformal if f (x) is an orthogonal matrix. A matrix orthogonal if its transpose is its inverse, therefore, the eterminant of an orthogonal matrix is always ±. This gives us a metho of etermining whether functions are conformal. f. A survey of -imensional Eucliean an Non-Eucliean geometries. i. Eucliean Geometry In -imensional Eucliean geometry, or E, the shortest istance etween two points is along a straight line. There is a unique straight line etween any two given points, an the angles of a triangle a up to 80. The inversions iscusse so far are all in E. ii. Spherical Geometry Figure 3: A triangle on a sphere with angle sum 70 The stanar moel for -imensional spherical geometry is the unit sphere S in 3. The spherical istance etween two points x an y is given y the Eucliean angle etween their position vectors. On the unit sphere, therefore, this is equal to the length of the arc of the

5 unique great circle (unit circles lying on the sphere) joining x an y. Spherical geometry is one of the two homogeneous examples of non-eucliean geometries, ce Eucli s fifth postulate oes not hol true on a sphere. Great circles are the geoesics on the sphere an it is clear that through two antipoal points there exist an infinite numer of great circles. Also, the angle sum of triangles is always greater than 80 (Figure 3). iii. Hyperolic Geometry Figure 4: A triangle on H -imensional hyperolic geometry (H ) can e consiere to e the geometry on the upper sheet of a unit hyperoloi in 3. Geoesics on the hyperoloi are unsurprigly hyperolas on its surface, an the istance etween two points may e consiere to e the arc length of the unique hyperolic segment etween them. In H, the angle sum of a triangle is always less than 80. Hyperolic geometry is also a non-eucliean geometry. Spherical an hyperolic geometries show an interesting uality, which egins with the nature of how Eucli s fifth postulate is negate in each case. Eucli s fifth postulate is equivalent to the moern parallel postulate, which states that through a point outsie a given straight line there is exactly one straight line parallel to the given line. In spherical geometry, where great circles may e consiere to e straight lines, through a point outsie a given great circle, there are no great circles parallel to it (all great circles must necessarily intersect one another). In hyperolic geometry, where hyperolas may e consiere to e straight lines, through a point outsie a given hyperola, there are infinitely many hyperolas parallel to it. Notaly, Eucliean geometry is frequently miway etween spherical an hyperolic geometries. For instance, the curvature of S is, that of the Eucliean plane is 0, an that of H is -.

6 Moifying the Construction for Spherical an Hyperolic Geometries a. Spherical Geometry Figure 5: Constructing the inversion on S. In the spherical case, straight lines are replace y great circles. The inverting circle (given in Figure 5 y ZYZ ) is some circle on the surface of the sphere, not necessarily a great circle. The center of the inverting circle is given y P, an we shall invert the point X. As in the Eucliean case, our first step is to take the raial line (in this case the great circle PXY) through X. We take the great circle perpenicular to PXY through X. This great circle intersects the inverting circle at Z an Z. We take the raii to Z an Z an construct the great circles perpenicular to them. These tangent great circles intersect at X. Let S (P,Y)=; S (P,Z)=; S (P,X)=; S (X,Z)=; S (P,X )= ; ÐPX Z=q; ÐXPZ=; ÐPZX=; Applying the spherical laws of es an coes, ( ) FromXX Z, ( ) FromPX Z,

7 Also, Also, PXZ, From or, tan cot cot or,cot cot or,cot or, ) ( tan tan or,tan

8 Notice, the certainly this function maintains the necessary inversive nature, namely the composition of the function with itself is the ientity function. Also, it is not possile to invert eyon the hemisphere, therefore, we restrict the function to the hemisphere. On the other han, it is clear that points in the interior of the inverting circle get mappe to the exterior an vice versa, an points on the circle are mappe to themselves. Therefore, on the restricte omain, this function is inee an inversion.. Hyperolic Geometry Figure 6: Constructing the inversion on H. Points on H are projecte own to the plane. We start with projecting points in H perpenicularly own to the plane. The inverting circle (shown in re in Figure 6) has center P, an X is the point to e inverte. We take the raius through X an the hyperola perpenicular to this raius at the point X. This chor hyperola intersects the inverting circle at Z an Z. Tangent hyperolas at these points intersect at X Let H (P,Y)=; H (P,Z)=; H (P,X)=; H (X,Z)=; H (P,X )= ; ÐPX Z=q; ÐXPZ=; ÐPZX=; Applying the hyperolic laws of es an coes, h( ) h FromXX Z, ( ) h h h FromPX Z, h

9 h h h h Also, h Also, h h h h PXZ, From h h h h h h or,h h h h h h h h h h h h tanh coth h h h h h h h h h h h h h h h h h coth or,coth h h h h h coth or,coth h h h h h hh or, h h h h ) h( tanh tanh or,tanh

10 We can see, once again, the inversive nature of this function. However, this function as it stans is not well-efine. Namely, a isc concentric to the inverting circle oes not get mappe anywhere. The raius of this isc can e foun as follows: tanh tanh tanh tanh tanh or, tanh (tanh ) Notaly, the raius of this circle increases with increase in the raius of the inverting circle. We efine the inversion on the remainer of H. On the other han, it is clear that points in the interior of the inverting circle get mappe to the exterior an vice versa, an points on the circle are mappe to themselves. Therefore, on the restricte omain this function is an inversion. esults a. Inversions The symmetry of the otaine graphs (Figure 6 an 7) aroun y=x verifies that these functions are their own inverses. The secon set of graphs show that there are iscs aroun the center of the inverting circle which o not get mappe uner the hyperolic inversion.. Investigation of conformality The matrix of partial erivatives with respect to spherical co-orinates for the inversion on the hemisphere is given elow: If x r,, thenf ( x) r,,cot cot cot

11 f ( x) cot ( cot 4 cot ( )cot cot ) Since, et(f (x))= for all if an only if =, we can infer that the inversion over a circle of raius is conformal on the hemisphere. Figure 7: Graphs for inverting on the hemisphere. The x an y axes represent the istances of the original an inverte point respectively from the center of the inverting circle.

12 Figure 7: Graphs for inverting on the hemisphere. The x an y axes represent the istances of the original an inverte point respectively from the center of the inverting circle. Future irections We woul like to investigate the ehavior of angles uner these inversions in greater etail. A metho we have consiere ug involves taking conformal projections of curves efore an after inversion, onto the Eucliean plane. If angles are preserve on the Eucliean plane we woul e ale to infer conformality on the original surface. We woul also like to etermine how lines an circles are change uner the inversions.