Notes 20: Afine Geometry

Notes 20: Aﬁne Geometry

Example 1. Let C be the curve in R2 deﬁned by x2 + 4xy + y2 + y2 − 1 = 0 = (x + 2y)2 + y2 − 1.

If we set u = x + 2y, v = y, we obtain

u2 + v2 − 1 = 0 which is a circle in the u, v plane. Let

2 2 F : R → R , (x, y) 7→ (u = 2x + y, v = y). We look at the geometry of F.

Angles: This transformation does not preserve angles. For example the line 2x + y = 0 is mapped to the line u = 0, and the line y = 0 is mapped to v = 0. The two lines in the x − y plane are not orthogonal, while the lines in the u − v plane are orthogonal.

Distances: This transformation does not preserve distance. The point A = (−1/2, 1) is mapped to a = (0, 1) and the point B = (0, 0) is mapped to b = (0, 0). the distance AB is not equal to the distance ab.

Circles: The curve C is an ellipse with axis along the lines 2x + y = 0 and x = 0. It is mapped to the unit circle in the u − v plane.

Lines: The map is a one-to-one onto map that maps lines to lines.

What is a Geometry? We can think of a geometry as having three parts: A set of points, a special set of subsets called lines, and a group of transformations of the set of points that maps lines to lines. In the case of Euclidean geometry, the set of points is the familiar plane R2. The special subsets are what we ordinarily call lines. The transformations are rotations, translations and reﬂections. The transformations that preserve distance. In hyperbolic geometry the set of points is the upper half plane, the special subsets are portions of circles in the upper half plane that meet the real axis orthogonally, and the transformations are the fractional linear transformations with real entries. In aﬃne geometry the set of points is R2. The special subsets are lines. The transformations are the maps

2 2 R → R , (x, y) 7→ (u = ax + by + e, v = cx + dy + f) with ad − bc 6= 0. The last condition insures that the map is one-to-one and onto. We call these aﬃne transformations.

1 In order to understand affine geometry we work out a problem: Let P1,Q1 be two points in one of our geometries and let P2,,Q2 be another set of two points. When is the set P1,Q1 the ’same” as the other set P2,Q2? What do we mean when we say ’the same as’? We mean that there exists a transformation that takes on pair to the other pair. In the case of Euclidean geometry this means that the distance between P1,Q1 is the same as the distance between P2,Q2. The answer in hyperbolic geometry is the same. In affine geometry thee is always an affine transformation taking P1,Q1 to P2,Q2 for arbitrary pairs of distinct points. The problem of classifying three points in Euclidean geometry is the problem of deciding when two triangles are congruent. Question Are all triples of three non-collinear points in affine geometry the ’same’? Another way of asking this is: Let P, Q, R be three points in the affine plane. Is there an affine transformation taking this set of three points to the set (0, 0), (1, 0), (0, 1)? We interpret homework 8 from this point of view. We have shown that every degree 2 curve in R2 is affine equivalent to one of the following: 1. A circle (x2 + y2 − 1 = 0).

2. A parabola (x2 + y = 0).

3. A hyperbola (x2 − y2 − 1 = 0).

4. A pair of distinct lines (x2 − y2 = 0) that intersect.

5. A pair of parallel lines (x2 − 1 = 0).

6. A line counted twice. This is the case where the equation can be changed into the equation x2 = 0.

7. The empty set (x2 + y2 + 1 = 0) or x2 + 1 = 0.

We give one more example to illustrate the difference between affine and Euclidean geometry. We define (some of) the curves of degree two in Euclidean geometry using distance as follows:

• A circle is the locus of points equidistant to a ﬁxed point.

• A parabola is the set of point equidistant to a line and a point on on the line.

• An ellipse is the set of points so that the sum of the distances to two ﬁxed points is a ﬁxed constant.

• A hyperbola is the set of points so that the difference of the distances to two fixed points is a fixed constant.

2 How do you construct these curves using aﬃne geometry and without resorting to coordinates?

Answer: We construct a mechanism that draws conic sections. Fix two points P1,P2 and two lines M1,M2 so that neither P1 nor P2 is on M1 ∪ M2. Now let Q be third ﬁxed point and for each line L through Q we get a pair of points, namely S1 = L ∩ M1,S2 = L ∩ M2. Let Li be the line through the points Pi,Si, i = 1, 2. Let P = L1 ∩ L2. As L moves among the lines through Q, the points P sweep out a conic section. Note that we do not use a notion of distance in this construction. It just depends on intersections of lines. This brings up numerous questions:

Question: Can you show, without resorting to coordinates, that the locus I have con- structed intersects a many lines in two points and no line in more than two points?

Question: Using coordinates, can we ﬁnd the equation of the locus we have described?

This course is NOT a course in affine geometry. We have introduced objects, ideas, questions that go beyond affine geometry. Affine geometry is a somewhat artificial construct. We really should be talking about projective geometry, but we have not gotten that far yet. Projective geometry was introduced during the renaissance along with notions of perspective in drawing. We give an example of a transformation that appears in algebraic geometry that has features far from those of affine geometry. Consider the map 2 2 R → R , (x, y) 7→ (u = xy, v = y).

• This is a one-to-one map from R2 − {y = 0} to R2 − {v = 0}. • The image of the line y = 0 is the point (0, 0).

• We calculate the inverse image of the line u = mv in the u − v plane. We have

u = mv =⇒ y = mxy =⇒ y(1 − mx) = 0

is the equation of the inverse image. This is the union of the line y = 0 and the line x = 1/m. We see that the inverse image of point (u = 0, v = 0) in the x − y plane is the set of tangent directions to the point (u = 0, v = 0).

3 Figure 1: An Aﬃne Transformation

4 Figure 2: Blowing Up