Polnomials. Math 4800/6080 Project Course 2. The Plane. Boss, boss, ze plane, ze plane! Tattoo, Fantas Island The points of the plane R 2 are ordered pairs (x, ) of real numbers. We ll also use vector notation to denote a point of the plane: x R 2 In high school geometr, we talk about congruences and similarities. These are examples of affine linear transformations of the plane. Definition. A general linear transformation is an invertible map: A : R 2 R 2 with the following two properties: ( ) ( ) ( ) x1 x2 x1 x2 A A A 1 2 1 2 for all pairs of vectors, and ( ) x A α ( x αa for all scalars α R. In particular, A fixes the origin (0, 0). A linear transformation is realized b matrix multiplication: ( ) x a b x A c d where the matrix is computed via: ( ) ( ) 1 a 0 b A and A 0 c 1 d Examples. The rotation counter-clockwise (around the origin) b the angle θ is given b: ( ) x cos(θ) sin(θ) x R sin(θ) cos(θ) A rotation is a rigid motion of the plane, meaning that it does not change the distances between two points (or the angles between two vectors). It also preserves orientation, meaning that a clock would measure clockwise in the same wa before and after a rotation. 1 )
2 A reflection is another rigid motion. The basic reflections are: 1 0 1 0 and 0 1 0 1 across the -axis and x-axis, respectivel. Reflecting across the x-axis, followed b rotating b θ gives rise to another reflection: cos(θ) sin(θ) 1 0 cos(θ) sin(θ) sin(θ) cos(θ) 0 1 sin(θ) cos(θ) this time across the line given (in polar coordinates!) b: r θ 2 A little exercise with trigonometr angle addition formulas gives: cos(θ2 ) sin(θ 2 ) sin(θ 2 ) cos(θ 2 ) cos(θ1 ) sin(θ 1 ) sin(θ 1 ) cos(θ 1 ) cos(θ1 θ 2 ) sin(θ 1 θ 2 ) sin(θ 1 θ 2 ) cos(θ 1 θ 2 ) from which we verif the (intuitivel obvious) fact that rotating b θ 1 and then b θ 2 results in a rotation b θ 1 θ 2. In particular, rotations commute with each other. On the other hand, reflecting across r θ 1 /2 followed b reflecting across r θ 2 /2 gives rise to: cos(θ2 ) sin(θ 2 ) sin(θ 2 ) cos(θ 2 ) cos(θ1 ) sin(θ 1 ) sin(θ 1 ) cos(θ 1 ) cos(θ2 θ 1 ) sin(θ 2 θ 1 ) sin(θ 2 θ 1 ) cos(θ 2 θ 1 ) which is a rotation b the difference of the two angles θ 2 θ 1. In particular, the two reflection matrices do not (usuall) commute since rotating b θ 1 θ 2 is the inverse of rotating b θ 2 θ 1. Definition. A set of linear transformations is a group (or class) if it is closed under compositions and taking inverses. In our examples, (a) The set of rotations (b an angle 0 θ < 2π) is a group, but (b) The set of reflections across the lines r θ/2 is not a group, since the composition of two reflections is a rotation, not a reflection. However, (c) The union of the sets of rotations and reflections is a group. Remark. To complete a verification of (c), ou would need to check that a rotation followed b a reflection is a reflection, and also that a reflection followed b a rotation is a reflection. I encourage ou to check these and also to verif the two matrix multiplications above.
3 Definition. A translation of R 2 is a mapping: given b: T : R 2 R 2 ( ) x x T x0 0 Important Remark. A translation is not a linear transformation and cannot be described b matrix multiplication. However, a translation is a rigid motion, since it clearl does not change the lengths of vectors or the angles between them. It also, like rotations (and unlike reflections) preserves orientation. Congruences are rigid motions of the plane that preserve orientation. Each of these can be expressed as a rotation followed b a translation. ( ) x cos(θ) sin(θ) x x0 C sin(θ) cos(θ) 0 But a congruence ma be more intuitive if the translation is done first, followed b the rotation. Linear algebra comes to our rescue: ( ) cos(θ) sin(θ) x x0 sin(θ) cos(θ) 0 cos(θ) sin(θ) x cos(θ) sin(θ) x0 sin(θ) cos(θ) sin(θ) cos(θ) 0 is also a rotation followed b a translation (but not b the same vector!) One can check that a composition of congruences is a congruence and that the inverse of a congruence is a congruence. The are a group. Debate. Some might argue that a planar shape is congruent to its mirror image under a reflection. For this to be so, we need to enlarge our group of congruences to include reflections followed b translations. Similarities are like congruences, except instead of fixing lengths of line segments, all segments are magnified b a fixed scaling factor α 0. This is accomplished b multipling the rotation matrix b a scaling matrix: ( ) x α 0 cos(θ) sin(θ) x x0 S 0 α sin(θ) cos(θ) 0 A similarit that fixes the origin might be called a rotation with scaling: ( ) x α 0 cos(θ) sin(θ) x S 0 α sin(θ) cos(θ)
4 Since a composition of similarities will scale lengths b the product of the α s, it follows that similarities with a fixed scaling factor are not a group (unless the are congruences). The similarities of all possible (positive) scaling factors do, however, form a group. General Linear Transformations scale areas, rather than lengths: ( ) x a b x A c d has an area scaling factor equal to the absolute value of the determinant a b det ad bc c d (the sign of the determinant determines whether or not orientation is preserved b the transformation). Thus, it is in particular required that the determinant not be zero. These form a group. It is important to keep in mind that a linear transformation ma not preserve angles, so circles ma turn into ellipses, angles of triangles ma change, etc. A general linear transformation followed b a translation is an Affine (General) Linear Transformation: ( ) x a b x x0 F c d 0 These are, for now, our most general transformations of the plane. As a model for all the other claims we ve made, let s prove: Proposition. The set of Affine Linear Transformations is a group. Proof. Given two affine linear transformations: ( ) x a1 b F 1 x x1 d 1 1 c 1 c 1 and ( ) x a2 b G 2 x x2 c 2 d 2 2 The composition: ( ) ( x a2 b G F 2 a1 b 1 x c 2 d 2 c 1 d 1 ( ) ( a2 b 2 a1 b 1 x a2 b 2 x1 c 2 d 2 c 1 d 1 c 2 d 2 1 is again an affine linear transformation, and the inverse of F is: ( ) 1 1 x F 1 a1 b 1 x a1 b 1 x1 d 1 d 1 1 c 1 ) x1 x2 1 2 x2 2 )
Examples of interesting general linear transformations: 1 s x x s 0 1 is a shear. It fixes points on the x-axis, and fixes the -coordinates of an point, but it shoves the points of a given -coordinate to the right (or left) b the fixed multiple s of the -coordinate. A distortion has different scaling factors in the x and directions: u 0 x ux 0 v v It is not hard to see that ever linear transformation with positive determinant (preserving orientation) can be decomposed into a rotation followed b a shear followed b a distortion. The Projective Plane is a wa of completing the plane with points at infinit. Projective transformations of the projective plane will be an even larger group of transformations that will be our most general transformations. We will introduce both the projective line and the projective plane simultaneousl (to contrast them). The astute reader will notice that there is a pattern here that can be extended to give projective spaces of all dimensions. Definition. (i) The projective line RP 1 is the set of all the lines through the origin in the plane R 2. (ii) The projective plane RP 2 is the set of all the lines through the origin in space R 3. First Remark. Ever line through the origin intersects the unit sphere in exactl two points. The unit sphere in the plane is more commonl referred to as the unit circle: S 1 : {(x, ) x 2 2 1} whereas in three space it is what we usuall think of as a sphere: S 2 : {(x,, z) x 2 2 z 2 1} The fact that ever line meets the sphere in two (antipodal) points means that there are surjective maps: a 1 : S 1 RP 1 and a 2 : S 2 RP 2 with the propert that the inverse image of an point consists of the two poles of the intersection of the line with the sphere. These maps make RP 1 and RP 2 into topological manifolds. 5
6 The topological manifold RP 1 is eas to describe. It is also a circle. In fact, the circle is the onl connected, one-dimensional closed manifold. In this case, walking along RP 1 is just the same as walking along the unit circle, except that ou return to the start after π radians, rather than after 2π radians. Interestingl, the manifold RP 2 is not a sphere, as we will see. Second Remark: Ratios give coordinates of points of RP 1 and RP 2. For RP 1, these are ordinar ratios: (x : ) (0 : 0) with (x : ) (αx : α) for all α 0 Recall that ratios are almost like fractions, except that ratios (x : 0) are allowed, while fractions are not allowed a 0 in the denominator. Onl the ratio (0 : 0) is not permitted. In the case of RP 2, the ratios are triple ratios: (x : : z) (0 : 0 : 0) with (x : : z) (αx : α : αz) Special Ratios. The ratios (x : 1) and (1 : ) determine two embeddings of the real line in RP 1 ; f : R RP 1 ; f(x) (x : 1) and g : R RP 1 ; g() (1 : ) since two ratios (x : 1) and (x : 1) are different if x x. These two embeddings cover RP 1, with each one missing a different point. Similarl, the ratios: (x : : 1), (x : 1 : z) and (1 : : z) determine three embeddings of R 2 that cover the projective plane: f(x, ) (x : : 1), g(x, z) (x : 1 : z) and h(, z) (1 : : z) though each of these embeddings misses a cop of RP 1. For example: ( ) RP 2 {(x : : 1)} {(x : : 0)} and the latter set is identified in the obvious wa with RP 1. All these embeddings ma be realized geometricall. In the case of RP 1, a line through the origin in R 2 intersects the line 1 at a point (x, 1) (unless it is the x-axis). Similarl, it intersects x 1 at a point (1, ) unless it is the -axis. In the case of RP 2, the lines through the origin intersect the planes z 1, 1 and x 1, respectivel in points (x,, 1), (x, 1, z) and (1,, z) unless the lines are contained in the x-plane, xz-plane and z-plane, respectivel.
Focusing on the union ( ) above, we ma consider the closure S RP 2 of a subset S R 2 b adding points from the RP 1 at infinit. Example: (a) The closure of the line mx b is the single point: (x : mx : 0) (1 : m : 0) at infinit. This is because the projective coordinates of the finite points of the line (with the exception of the -intercept) are: (x : mx b : 1) (1 : m b/x : 1/x) and the limit as x ± is the point (1 : m : 0). This means that following the line in one direction to infinit will flip to the opposite end of the line. The closure of the line is a cop of RP 1! (b) Consider the region bounded above b: and bounded below b max{ x, 1, x} min{ x, 1, x} The points to add to this bow-tie shaped region to obtain its closure in RP 2 is an interval of points at infinit: {(1 : m : 0) 1 m 1} If ou were to walk along the positive x-axis (growing linearl taller and moving exponentiall faster in proportion to our distance from the origin), ou would reach infinit on the right and then flip upside down as ou return from infinit on the left. This means that the closure of the bow-tie is a cop of the Möbius strip, which is a one-sided surface that cannot be embedded in a sphere. Moreover, the complementar region to the bow-tie closure is a topological disk, realizing the projective plane as a capped Möbius strip (the edge of the Möbius strip is a circle!) which cannot be placed into R 3. Thus RP 2, unlike RP 1, is not another sphere, but rather a (non-orientable) surface of a different topological tpe. Finall, we consider the projective linear transformations. As with general linear transformations, these are matrices, but of size one larger than the dimension of the projective space: Projective Transformations of RP 1 are 2 2 Invertible Matrices: ( ) x a b x A : RP 1 RP 1 ; A c d 7
8 This is well-defined because matrix multiplication takes lines through the origin to lines through the origin. It is different from a general linear transformation of the plane because the vectors of the matrix multiplication are ratios: α 0 a b x a b αx a b x 0 α c d c d α c d So scaled transformations are the same projective transformations. Recall that an affine linear transformation of the real line is: F (x) ax b where b is the translation, and a is the scaling factor. Consider the projective transformation of the projective line: ( ) x a b x F 0 1 For points of R (with coordinates (x : 1)) this is the same as F : ( ) x a b x ax b F 1 0 1 1 1 and the additional point at infinit is fixed: ( ) 1 a b 1 a F 0 0 1 0 In other words, the affine transformations of the line are a subgroup of the projective transformations of the projective line. This is also the case with the projective plane, except that something interesting happens there at infinit: Projective Transformations of RP 2 are 3 3 Invertible Matrices: A : RP 2 RP 2 ; A x a 1,1 a 1,2 a 1,3 a 2,1 a 2,2 a 2,3 x z a 3,1 a 3,2 a 3,3 z As with the projective line, a matrix and a scalar multiple of it define the same projective linear transformation, and also as with the projective line, an affine linear transformation: ( ) x a1 b F 1 x d 1 c 1 x1 1
can be extended to the projective linear transformation of RP 2 : F x a 1 b 1 x 1 c 1 d 1 1 x z z This transformation is just F on the points of R 2 : F x a 1 b 1 x 1 c 1 d 1 1 x a 1x b 1 x 1 c 1 x d 1 1 1 1 1 but interestingl, unlike the previous case, this extended transformation moves around the points at infinit via a projective transformation of the projective line of points at infinit: F x 0 a 1 b 1 x 1 c 1 d 1 1 x 0 a 1x b 1 c 1 x d 1 0 Permutations are another important group of transformations. In the projective line case, these are: 0 1 1 0 r and e 1 0 0 1 with r 2 e The transformation r switches the x and coordinates, hence: ( ) ( ) x 1 r 1 x so in particular, r switches the two embeddings of R in RP 1. In the projective plane case, there are six permutations: e 1 0 0 0 1 0, c 1 0 0, c 2 0 1 0 0 1 0 1 0 0 r x 0 1 0 1 0 0, r xz 0 1 0 1 0 0, r z 1 0 0 0 1 0 which permute the three embeddings of R 2 in the projective plane. An Application. We will use the permutation transformations to swing points in from infinit. In other words, given a subset S RP 2, we ma appl permutation transformations to S to move points at infinit into the plane R 2, and then examine them. 9