3 Analytic Geometry

3 Analytic Geometry 3.1 The Cartesian Co-ordinate System Pure Euclidean geometry in the style of Euclid and Hilbert is what we call synthetic: axiomatic, without co-ordinates or explicit formulæ for length, area, volume, etc. Nowadays, the practice of ele- mentary geometry is almost entirely analytic: reliant on algebra, co-ordinates, vectors, etc. The major breakthrough came courtesy of Rene´ Descartes (1596–1650) and Pierre de Fermat (1601/16071–1655), whose introduction of an axis, a fixed reference ruler against which objects could be measured using co-ordinates, allowed them to apply the Islamic invention of algebra to geometry, resulting in more efficient computations. The new geometry was revolutionary, so much so that Descartes felt the need to justify his argu- ments using synthetic geometry, lest no-one believe his work! This attitude persisted for some time: when Issac Newton published his groundbreaking Principia in 1687, his presentation was largely synthetic, even though he had used co-ordinates in his derivations. Synthetic geometry is not without its benefits—many results are much cleaner, and analytic geometry presents its own logical difficulties— but, as time has passed, its study has become something of a fringe activity: co-ordinates are simply too useful to ignore! Given that Cartesian geometry is the primary form we learn in grade-school, we merely sketch the familiar ideas of co-ordinates and vectors. • Assume everything necessary about on the real line. continuity y 3 • Perpendicular axes meet at the origin. P 2 • The Cartesian co-ordinates of a point P are measured by project- ing onto the axes: in the picture, P has co-ordinates (1, 2), often 1 written simply as P = (1, 2). • Curves are defined using equations. For instance, the Pythagorean 2 1 1 2 3 distance function says that the circle of radius 1 centered at the − − 1 x origin may be described by the equation x2 + y2 = 1. − 2 − In the 1600’s this was not considered a new axiomatic system, but rather a collection of computational tools built on top of Euclid. We may therefore assume anything from Euclid and mix strategies as appropriate. To see this at work, consider a simple result. B Lemma 3.1. Let O = (0, 0), A = (x, y) and B = (v, w) where O, A, B are non-collinear. If C = (x + v, y + w), then the quadri- C lateral OACB is a parallelogram. p Proof. Calculate distances: BC = x2 + y2 = OA , etc. j j j j Now use side-side-side to see that OAC = CBO. The usual 4 ∼ 4 discussion of parallel lines/alternate angles from Euclid forces O opposite sides to be parallel. A 1There is some argument over Fermat’s birth given that he possibly had a deceased older brother, also named Pierre. 1 Vector Geometry Vectors come to us courtesy of several mathematicians, most prominently William Rowan Hamilton (1805–1865), Oliver Heaviside (1850–1925) and J. Willard Gibbs (1839–1903). Hamilton had stumbled upon the algebra of quaternions when attempting to extend to three dimensions the contemporary use of complex numbers to describe planar geometry.2 Heaviside and Gibbs independently developed vector calculus. The revolution was rapid; by 1900 vector calculations were dominant in physics. Definition 3.2. A directed line segment −!AB is a segment together with an orientation.a The position vector of a point A is the directed line segment OA−! where O is the origin. A vector is an equivalence class of directed line segments where two segments are equivalent if and only if they are congruent and oriented in the same direction. aWe write −!AB for the directed line segment so as to distinguish it from the ray −!AB of Hilbert’s Euclidean geometry. A vector has length and direction, but no fixed location. All directed line segments with the same length and direction represent the same vector. The standard representation of a vector involves placing its tail at the origin: we can then describe the vector by giving the co-ordinates of its head. Example 3.3. In the picture, the standard representation of a vector y 3 v is shown in blue. The green arrows are other representations of the same vector. Various common notations include 2 2 v = ~v = v = OA−! = = 2, 1 = 2i + j 1 A 1 h i O The usual convenient abuse of terminology is at work: strictly OA−! v 1123 2 − 1 since v is an equivalence class, but no-one writes this. − x Addition and Scalar Multiplication are defined algebraically in the familiar manner: v w v + w lv v + w = 1 + 1 = 1 1 , lv := 1 v v2 w2 v2 + w2 lv2 w Vector addition can be visualized by placing representative segments nose- to-tail. In view of Lemma 3.1, the commutativity of addition v + w = v + w w w + v is often known as the parallelogram law. Indeed the Lemma may be rephrased in this language: the parallelogram OACB is spanned by O v x v OA−! = and OB−! = y w The scalar multiple lv may be viewed as stretching or shrinking v, and reversing its direction when l < 0. 2Quaternions are objects of the form a + bi + cj + dk where a, b, c, d R, i2 = j2 = k2 = 1 and i, j, k multiply as if 2 − using the cross-product (ij = k = ji, etc.). Hamilton couldn’t make the three-dimensional part (a = 0) into a suitable − algebra, but a fourth dimension fixed things. Hamilton eventually realized that if he dropped his requirement of having a well-defined multiplication, he could apply his vector approach to the study of geometry in any dimension. 2 We finish with a famous result showing how easy it can be to work in analytic geometry. Theorem 3.4. The medians of a triangle meet at a point 1/3 of the way along each median. Proof. Given OAB, let a = OA−! and b = OB−!. 4 1 A If M is the midpoint of AB, then OM−−! = 2 (a + b). 2 Let G lie 3 of the distance along OM−−!: that is 2 1 1 OG−! = (a + b) = (a + b) 3 · 2 3 The point 1 of the way along the median through has position M 3 A a vector 1 1 1 1 OB−! + OA−! OB−! = (a + b) G 2 3 − 2 3 Similarly, the point 1 of the way along the median through B has 3 1 ( + ) position vector 3 a b O b B 1 1 1 1 OA−! + OB−! OA−! = (a + b) 2 3 − 2 3 All three points have the same position vector, and are therefore the same point G! Compare this to the argument/exercise using Ceva’s Theorem at the end of our discussion of Eu- clidean Geometry! The proof shows another important aspect of analytic geometry. The standard approach is to place the origin and orient a figure in a manner which makes calculations simple. This is essentially Eu- clid’s (sketchy) superposition principle, or Hilbert’s congruence, but can be more rigorously grounded in a full discussion of isometries. Exercises. 3.1.1. Given a quadrilateral ABCD, let W, X, Y, Z be the midpoints of the sides AB, BC, CD and DA respectively. Use vectors to prove that WXYZ is a parallelogram. 3.1.2. (a) Let A = (a1, a2) and B = (b1, b2) be given. Show that any point on the line segment joining A and B has co-ordinates ((1 t)a1 + tb1, (1 t)a2 + tb2) where 0 t 1. − − 1 1 ≤ ≤ (b) Show that the midpoint of A and B has co-ordinates 2 (a1 + b1), 2 (a2 + b2) . (You’ll probably find it easiest to think in terms of vectors for both parts) 3.1.3. (a) Perform a pure co-ordinate proof (no vectors) of Theorem 3.4. (b) Descartes and Fermat did not, in fact, have a fixed perpendicular second axis! Their approach was equivalent to choosing a a second axis whose angle to the first was chosen to make a problem as easy as possible. Given OAB, where B = (b, 0), choose the second 4 axis to point along OA−! so that A has co-ordinates (0, a). Now give an even simpler proof of the centroid theorem. 3 3.2 Angles in Planar Analytic Geometry We define angle measure a little differently: this time we use radians and extend to any angle. Definition 3.5. Suppose A, B, C are distinct and draw any circle centered at A. The radian-measure ]BAC is the ratio of the arc-length to the radius measured counter-clockwise from −!AB to −!AC. Being ratios, radians are naturally unitless. θ − In analytic geometry it is common to label angles using their C θ radian measure. In part this is since. B Angles are congruent if and only if their radian measures are equal or negative each other modulo 2p. A In the picture, q = BAC and CAB = 2p q. This last is 2π θ ] ] − often taken instead to be q. − − Definition 3.6. Let P = (x, y) lie on a circle of radius r such that the P segment OP−! makes angle q radians measured counter-clockwise from the positive x-axis. The cosine and sine of q are defined by r y θ x = r cos q, y = r sin q O x The AAA theorem for similar triangles says we can view these as well- defined functions of radian-measure, not merely angle! p 3p The definitions work for any angles: if, say, 2 < q < 2 , simply take x < 0.

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