Euclidean Geometry
Affine real spaces can be provided with an additional “scalar product”, which yields corresponding notions of distance, angle, perpendicularity. And of course various additional geometric notions can now be studied: squares, rectangles, rotations, orthogo
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Euclidean Geometry
The next step in our study of geometry, using the methods of linear algebra, is to introduce the notions of distance and angle. This will be achieved by adding a scalar product to a real affine space. We obtain what is called a Euclidean space. We provide various examples and applications, study the metric properties of triangles, the orthogonal projections and orthogonal symmetries. We pay special attention to the isometries: the affine transformations which respect angles and distances, but also to the similarities, those which only respect angles.
4.1 Metric Geometry Geometry, as the name indicates, is the art of measuring the Earth. The emphasis here is on the act of measuring—the physical nature of what we are measuring, if it has a physical nature, is besides the point. The question is thus: can we measure and compare lengths, angles, surfaces, volumes, in an affine space? For example in an affine plane, can we speak of a square: a figure with four sides of equal lengths and four “right” angles? Can we define the perimeter or the surface of such a figure? In Definition 3.1.1 we have introduced the notion of a segment in a real affine space. But what about the length of such a segment? Of course in an affine space (E, V ) over a field K, when for A, B, C, D ∈ E and k ∈ K −→ −→ AB = k CD, −→ −→ we are tempted to say that AB is k times as long as CD. This is essentially what we have done in Definition 2.10.3. In this spirit, a length should be an element k ∈ K, an element that we probably want to be positive in the real case. However, the argument above does not take us very far. What about the case −→ −→ where the vectors AB and CD are not proportional? In any case, if we want lengths to be positive numbers, we should once more restrict our attention to “ordered fields”. We have already observed at the end of F. Borceux, An Algebraic Approach to Geometry, DOI 10.1007/978-3-319-01733-4_4, © Springer International Publishing Switzerland 2014
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Euclidean Geometry
Fig. 4.1
Sect. 3.1 that the field Q of rational numbers does not seem to be adequate to generalize “classical geometry”. Let us give an additional reason. If a sensible metric geometry can √be developed, a square whose side has length 1 should have a diagonal with length 2, which is no longer a rational number! Finally, we seem to end up again with the single case: K = R! Once more this conclusion is too severe, but we shall not insist on the possible generalizations. If we fix K = R, can we confidently state that we now have sound notions of length and angle? Take for example the vector space C(R, R) of real continuous functions, regarded as a real affine space. What is the distance between a parabola and the “sin x” function (see Fig. 4.1), viewed as points of the affine space? What is the angle between these two functions as vectors? The answer is not at all clear. After all, even in the ordinary real plane, if you are French (and work with centimeters) or British (and work with inches), the measures that you will gi
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