By C. G. Gibson

ISBN-10: 0521834481

ISBN-13: 9780521834483

This advent to the geometry of traces and conics within the Euclidean aircraft is example-based and self-contained, assuming just a simple grounding in linear algebra. together with quite a few illustrations and a number of other hundred labored examples and workouts, the ebook is perfect to be used as a direction textual content for undergraduates in arithmetic, or for postgraduates within the engineering and actual sciences.

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Extra resources for Elementary Euclidean geometry: An undergraduate introduction

Example text

For instance, the three quadratic functions below all define the same conic x 2 + y 2 − 1, 2x 2 + 2y 2 − 2, −x 2 − y 2 + 1. The conic arising from a quadratic function Q will be denoted by the same symbol: on the occasions when we do need to draw a distinction we will deliberately use the term ‘quadratic function’. Associated to any quadratic function Q is its zero set {(x, y) ∈ R2 : Q(x, y) = 0}. Note that any scalar multiple of Q has the same zero set, so the concept makes perfect sense for conics.

When the following relation holds, determining a unique value of t, hence a unique point Q 0 = T • (R − P − t T ) = T • (R − P) − t (T • T ). Moreover, it is a strict minimum of the function, since the second derivative f (t) = 2(T • T ) is positive. For that value Q = P + t T = R + s N for some constant s, so R − P = −s N + t T . Taking the scalar product of both sides with N , we obtain −s = L(x, y) N • (R − P) = 2 . 3) by observing that the minimum value d is given by d 2 = (R − Q) • (R − Q) = −s N • −s N = s 2 (a 2 + b2 ).

9 31 Find the circles touching the lines x = 0, y = 0, x = 2a, where a is a positive constant. Find the circles touching the lines x = 2, y = 5, 3x − 4y = 10. Find the circles which touch both the coordinate axes and pass through the point (6, 3). Show that the three radical axes associated to three circles with distinct centres are concurrent or parallel. Show that if two of the radical axes associated to three circles with distinct centres coincide, then all three coincide. Let C, D be circles with distinct centres and radical axis the line L.

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Elementary Euclidean geometry: An undergraduate introduction by C. G. Gibson


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