Foundations for Geometry

JWR April 26, 2002

§1 Deﬁnition. A geometry is a pair (M,G) where M is a set and G is a group of transformations of M. (A transformation of M is a one-one onto map from M to itself; a group of transformations is a set of transformations which contains the identity transformation, contains the composition of any two of its members, and contains the inverse of each of its members.) Here is a list of some key examples (deﬁnitions follow):

• The Aﬃne Plane: M = R2, G = the group of plane aﬃne transformations.

• The Euclidean Plane: M = R2, G = the group of rigid motions. • The Projective Line: M = R ∪ {∞}, G = the group of projective transformations.

• The Inversive Plane: M = C ∪ {∞}, G = the group of inversive transformations.

§2 A line in R2 is a set of form 2 L = {(x, y) ∈ R : ax + by + c = 0} where either a or b (or both) is not zero. An affine transformation of the plane is a transformation of R2 which maps lines to lines. §3 (Fundamental Theorem of Plane Affine Geometry.) A transformation T is affine if and only if there are numbers a, b, c, d, p, q such that the equation (x0, y0) = T (x, y) takes the form x0 = ax + by + p (1) y0 = cx + dy + q.

1 §4 Remark. Equations (1) define a transformation if and only if for every (x0, y0) there is a unique solution (x, y). By linear algebra this is so if and only if the determinant of coefficients is nonzero, i.e. ad − bc 6= 0. An affine transformation is called direct (also called orientation preserving) if ad − bc > 0. 2 §5 The distance between two points P1 = (x1, y1) and P2 = (x2, y2) of R is denoted |P1 − P2| and defined by

p 2 2 |P1 − P2| = (x1 − x2) + (y1 − y2) .

A rigid motion is a transformation T of R2 which preserves distance, i.e.

0 0 |P1 − P2| = |P1 − P2|

0 0 whenever P1 = T (P1) and P2 = T (P2) §6 (Fundamental Theorem of Plane Euclidean Geometry.) A rigid motion is an aﬃne transformation. A transformation T as in Equation (1) of Theorem 3 is a rigid motion if and only if the coeﬃcients a, b, c, d satisfy either a b cos θ − sin θ a b cos θ sin θ = or = c d sin θ cos θ c d sin θ − cos θ for some θ. (In the former case ad − bc = 1 so the transformation is direct; in the latter case ad − bc = −1.) §7 (SSS Theorem) Let A, B, C, A0,B0,C0 ∈ R2. Then there is a rigid motion T such that

A0 = T (A),B0 = T (B),C0 = T (C) if and only if

|A0 − B0| = |A − B|, |B0 − C0| = |B − C|, |C0 − A0| = |C − A|.

§8 For three distinct points A, B, C ∈ R2 deﬁne the quantities (A − B) · (C − B) cos ABC = ∠ |A − B| |C − B|

2 (A − B) × (C − B) sin ABC = . ∠ |A − B| |C − B| Here · and × denote the familiar operations of dot product and cross product from elementary calculus, i.e.

(u1, v1) · (u2, v2) = u1v1 + u2v2 and (u1, v1) × (u2, v2) = u1v2 − u2v1.

§9 (SAS Theorem) Let A, B, C, A0,B0,C0 ∈ R2. Then there is a rigid motion T such that A0 = T (A),B0 = T (B),C0 = T (C) if and only if |A0 − B0| = |A − B|, |B0 − C0| = |B − C|, and 0 0 0 cos ∠A B C = cos ∠ABC. 0 0 0 When such a rigid motion exists we have sin ∠A B C = ± sin ∠ABC and 0 0 0 the motion is direct if and only if sin ∠A B C = sin ∠ABC. §10 The cross ratio of four distinct points x1, x2, x3, x4 ∈ R∪{∞} is deﬁned by x1 − x3 x2 − x4 (x1, x2; x3, x4) := · x1 − x4 x2 − x3 If one of these points is ∞ then the cross ratio is deﬁned as the corresponding limit, e.g. x2 − x4 (∞, x2; x3, x4) = . x2 − x3 A projective transformation of R ∪ {∞} is a transformation T which preserves cross ratio, i.e.

0 0 0 0 (x1, x2; x3, x4) = (x1, x2; x3, x4) 0 for any four distinct points x1, x2, x3, x4 ∈ R ∪ {∞} where xi = T (xi). §11 (Fundamental Theorem of Projective Geometry.) A transformation T of R ∪ {∞} is projective if and only if it has the form ax + b T (x) = cx + d

3 where a, b, c, d ∈ R. The value T (∞) is deﬁned as the limit ax + b T (∞) = lim x→∞ cx + d (so T (∞) = a/c if c 6= 0 and T (∞) = ∞ if c = 0) and T (x) is deﬁned to be ∞ if the denominator vanishes, i.e. cx + d = 0. In contrast to the practice in elementary calculus we do not distinguish +∞ and −∞.

§12 The cross ratio of four distinct points z1, z2, z3, z4 ∈ C∪{∞} is defined by z1 − z3 z2 − z4 (z1, z2; z3, z4) := · ; z1 − z4 z2 − z3 the formula is exactly the same as for R ∪ {∞} but now complex numbers are allowed. The proof of Theorem 11 goes through mutatis mutandis:A transformation T of C ∪ {∞} preserves cross ratios if and only if it has the form az + b T (z) = (3) cz + d where a, b, c, d ∈ C. A transformation of form (3) is called a Möbius transformation (A Möbius transformation is also called a fractional linear transformation). §13 An inversive circle is a subset of C ∪ {∞} which is either a circle in C = R2 or a set of form L ∪ {∞} where L is a line. An inversive transformation is a transformation of C∪{∞} which maps inversive circles to inversive circles. §14 (Fundamental Theorem of Inversive Geometry.) A transformation T of C ∪ {∞} is inversive if and only it has one of the forms T (z) = S(z) or T (z) = S(¯z) where S is a Möbius transformation.

§15 Exercise. (90 points) Prove the Fundamental Theorem of Plane Aﬃne Geometry. ‘If’ is straightforward, but if you want to try ‘only if’ see me for hints. (The proof of ‘if’ is worth 10 points.) §16 Exercise. (10 points.) Show that the transformations (x0, y0) = T (x, y) of form (1) are a group, i.e. the identity transformation I(x, y) = (x, y) has

4 this form, the composition T1 ◦ T2 of two transformations T1 and T2 of this form is again of this form, and the inverse T −1 of a transformation T of this form is again of this form. Show that the direct aﬃne transformations form a subgroup. §17 Exercise. (10 points.) Identify the set R2 of pairs of real numbers with the set C of complex numbers by making the pair (x, y) correspond to the complex number z = x + iy. Show that under this identiﬁcation a transformation z0 = T (z) has the form

x0 = (cos θ)x − (sin θ)y + p y0 = (sin θ)x + (cos θ)y + q if and only if it has the form

z0 = µz + w (4) where µ and w are complex numbers and |µ| = 1. Prove that the set of all such transformations is a group, that any transformation of one of the two forms z0 = µz + w or z0 = µz¯ + w (5) is a rigid motion, and that the set of all rigid motions of one of the two forms in (5) is a group. (Eventually we will prove that the transformations (5) are the rigid motions and the transformations (4) are the direct rigid motions.) §18 Exercise. (10 points) Using the deﬁnitions from §8 prove that

2 2 cos ∠ABC + sin ∠ABC = 1. Hint: This is a calculation in high school algebra. The calculation will be more transparent if you introduce the abbreviations

u1 = x1 − x2, v1 = y1 − y2, u2 = x3 − x2, v2 = y3 − y2.

§19 Exercise. (10 points) Prove the SSS Theorem. §20 Exercise. (10 points) Prove the SAS Theorem. §21 Exercise. (20 points) Prove the Fundamental Theorem of Plane Eu- clidean Geometry. (Hint for ‘only if’. First show that a rigid motion which

5 satisﬁes T (0, 0) = (0, 0), T (1, 0) = (1, 0), and T (0, 1) = (0, 1) is the identity. Then use Exercises 17 and 7. §22 Exercise. (20 points) Find a formula for reﬂection in the line

ax + by + c = 0.

§23 Exercise. (40 points) Show that a rigid motion which is not direct leaves some line invariant. Conclude that it is a reflection in this line followed by a translation along the line. Show that a transformation is a direct rigid motion if and only if it is the composition of two reflections. Hint: If an affine a b transformation is rigid, what are the eigenvalues of the matrix of c d coefficients? §24 Exercise. (30 points) Prove the Fundamental Theorem of Projective Geometry. §25 Exercise. (30 points) Prove that the cross ratio of any four distinct four points is real if and only if they lie on an inversive circle. §26 Exercise. (30 points) Prove the Fundamental Theorem of Inversive Geometry. (Hint: Use Exercise 25.) §27 Exercise. Find a formula for inversion in the circle of radius k centered at the point Q and conclude that inversion is an inversive transformation, but not a Möbiustransformation. (Recall that inversion in this circle is determines by the condition that R0 = T (P ) if and only in Q, P , and P 0 are collinear and |Q − P | |Q − P 0| = k2; also T (Q) = ∞ and T (∞) = Q.) §28 Exercise. (30 points) Using calculus prove that a Möbiustransforma- tion preserves oriented angles. (See me for hints. To make this statement meaningful we must define angles at infinity.) §29 Remark. A transformation from an open subset of C to another open subset of C which preserves oriented angles is called conformal. Liouville’s Theorem (normally taught in Math 623) says that a conformal transformation of C ∪ {∞} is a Möbius transformation.