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Geometry with Two Screens and Computational Graphics

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Geometry with Two Screens and Computational Graphics Geometry with two screens and computational graphics R´emi Langevin, IMB, Universit´ede Bourgogne June 10, 2014 CAGIR,L´od´z(Poland), July 1st–July 12th, 2013 Dijon (France), July 6th–July 19th 2014. Let us first quote Gaspard Monge ([Mo] page 62): Il faut donc que l’´el`eve s’accoutume de bonne heure `asentir la correspondance qu’ont entre elles les op´erations de l’analyse et celles de la g´eom´etrie ; il faut qu’il se mette en ´etat, d’une part, de pouvoir ´ecrire en analyse tous les mouvements qu’il peut concevoir dans l’espace, et, de l’autre, de se repr´esenter perp´etuellement dans l’espace le spectacle mouvant dont cha- cune des op´erations analytiques est l’´ecriture. And Ren´eMagritte Je n’ai pas eu d’id´ee, je n’ai pens´equ’`aune image. 1 1 Introduction Here will will concentrate on the extrinsic viewpoint: curves in the plane or in Euclidean space (sometimes in spheres) and surfaces in the Euclidean space (sometimes in S3). With two screens, we can understand better surfaces in R3 or S3. The first screen is what you see in space or on the screen of your computer. The choice of the second will depend on the particular geometrical problem you are studying. Gauss was looking simultaneously at a surface and at its normal vectors as points of a sphere. The set of lines helps to understand ruled surfaces and quadrics. In conformal geometry, the set of spheres is useful. With two screens, we can understand better Dupin cyclides and canal surfaces. Finally, we will conclude with some old and new examples using the set of circles. The goal of this course is twofold. - Present some notions, which, although apparently Euclidean, make sense in confor- mal geometry. Figure 1: The three types of tori of revolution (figure J-C. Sifre), POVray Figure 2: A Dupin cyclide and its two families of characteristic circles (figure G. Solanes) The Dupin cyclide of Figure 2 is the image of the regular torus of revolution on the right of Figure 1. The fact that visually some remembrance of the initial shape survive 2 after a conformal transformation of R3 indicates there is a good hope to find geometrical notions, which, although apparently Euclidean, are in fact conformal. - Propose ways to generate surfaces sweeping them by lines or circles or as envelopes. We will be particularly interested by generating surfaces using conformally defined objects. 2 Euclidean or affine versus conformal geometry The second picture (2 is conformally equivalent to the torus of revolution on the right of the first picture (1). Euclidean geometry in the plane or in space relies on the measure of distances. The isomorphisms are isometries, transformations which preserve distances. Isometries of the Euclidean space form a 6-dimensional group. Angles between curves, in particular lines can be obtained using distances. The interested reader can consult [Kl2] for a systematic approach. Affine geometry, that we will not consider in this set of lectures, considers properties of pictures invariant by invertible affine maps. Important objects for this geometry are then lines and planes. Another geometry where lines and planes make sense is projective geometry. Examples of Euclidean objects: Orthogonal lines in the plane, segments of given length, Perpendicular planes of R3, spheres of radius r, rectangles of length a and width b, tori of revolution with circles contined in planes of symmetry of radii a, b and b + 2a. Conformal geometry does not measure distances between points, but only angles be- tween curves or surfaces at a point of intersection. In dimension 3, this correspond to the fact that the group of isomorphism for this new viewpoint is larger: it is the M¨obius group, of dimension 10. Using differential geometry, one can define the M¨obius group as a subgroup of the group of (orientation preserving) diffeomorphisms Mob(R3)= {g ∈ Diff +(R3)|∀x ∈ R3, Df(x)(v)= λ(x)|v|} In dimension 2 the situation is more... complex. Conformal local diffeomorphisms are holomorphic (or antiholomorphic) maps. Still, one can define a M¨obius group in dimension 2: in terms of a complex variable it is given by the transformations of the form az + b az¯ + b z 7→ or z 7→ cz + d cz¯ + d These transformations map circle-or-line to circle-or-line. The group is of dimension 6 (4 complex parameters, but we get the same map multiplying all the coefficients by a complex constant). Examples of conformal objects Circles-or-line in R2, Sphere-or-planes in R3, Dupin cyclides, canal surfaces These examples can also be defined in the frame of Euclidean geometry Examples of relations defined up to diffeomorphism - Two curves are tangent at a point 3 Examples of relations defined up to conformal transformations - A curve is tangent to a circle-or-line at a point - A circle is the osculating circle to a curve at a point - A sphere is the osculating sphere to a curve at a point Examples of affine relations - A curve is tangent to a line at a point - A plane curve has an inflection point. Examples of Euclidean relations - A point m is equidistant from three points - A plane curve has at a point an osculating circle of radius 2. - A surface and a plane are perpendicular (The notion of plane does not make sense in conformal geometry, the perpendicularity does not make sense in affine geometry) Exercise 2.0.1. To which geometry belongs the following sentence? - Two lines intersect. - Two lines are parallel - A point is a vertex of a plane curve, that is, at the point the contact with the osculating circle is strictly better that the usual contact - A curve is tangent to a surface - A curve is contained in a surface - The quadrilateral is a square of side of length 1 2.1 Contact order Let us start by comparing, near the origin, a curve C given by an equation of the form y = f(x), f(0) = 0, f 0(0) = 0 (It goes through the origin and has an horizontal tangent at the origin) and affine lines of the plane. D D δ δ m m m δ O O x O x D Figure 3: Contact order of a line with a curve An affine line D can (see Figure 11): 1. Avoid the origin. The distance δ between a point of C close to the origin and a point of the line is “of the order of 1” that is does not have limit 0 when the point on the curves goes to the origin. Bad news: the previous sentence does not fit with the usual terminology in contact theory. 2. Go through the origin without being tangent to the curve. The distance δ between a point of C close to the origin and a point of the line is of the same order as the 4 distance from the point to the origin (and also of the same order as the coordinate x of the point and of the same order as the length of the arc of the curve joining the point to the origin. We can say that the contact is of order 0. 3. Go through the origin and be tangent to the curve (D is the tangent to the curve at O = (0, 0)). The distance δ between m ∈ C and the line D is negligible compared to |x|, and, if the curve is the graph of a function f of class C2, of the order of x2 in general; this distance may exceptionally be negligible compared to x2, for example if the curve C crosses its tangent at the origin. We can say that the contact is of order 1 if the vertical difference is exactly of the order of x2, and not of smaller order1. 2.2 Osculating circles We suppose in this paragraph that the curve C is smooth enough, and is, near the origin, the graph of a function f satisfying f(0) = f 0(0) = 0. We can write a Taylor expansion of f near 0 f”(0) f(x)= x2 + o(x2). 2 (recall that o(x2) designs a term negligible compared to x2 , in general this term is of the order of |x3|; it may sometimes be even negligible compared to |x3|). Figure 4: Pencil of tangent circles We can repeat the previous discussion replacing lines by parabolae or circles. As we are interested in conformally defined notions, let us compare the curve with a conformally defined family of curves: the circles-or-lines. Let us study contact with circles-or-line. When the circles-or-lines does not go through the origin or is not tangent to the curve at the origin, nothing new. 1Is this notion Euclidean, conformal, or defined up to diffeomorphism? 5 Figure 5: Family of parabolae tangent to a curve at a point We now have to estimate the distance of a point m ∈ C close from the origin when the circle is tangent to the curve at the origin. For that, let us consider all the circles-or-lines tangents to the curve at the origin; they form a pencil of circles (voir figure 4, see also below). In order to estimate the distance between a point m ∈ C and the circle ΓR of radius R tangent to C at the origin O, it is enough to compute the order of the “vertical distance” between m and ΓR. Locally, at a neighborhood of the point m, the curve C has the equation y = f(x) where x is an Euclidean coordinate on the vectorial line TmC and y an Euclidean coordinate on the line `N (m) normal at m to the curve.
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