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Geometrical Constructions 2 Geometrical Constructions 2 by Pál Ledneczki Ph.D. Table of contents 1) Pencils of circles, Apollonian problems 2) Conic sections 3) Approximate rectification of an arc 4) Roulettes 5) Methods of representation in 3D 6) 3D geometrical constructions 7) Regular and semi-regular polyhedrons 8) Geometrical calculations "When he established the heavens I was there: when he set a compass upon the face of the deep.“ (Proverbs, Chapter 8 par. 27 ) Geometria una et aeterna est in mente Dei refulgens: cuius consortium hominibus tributum inter causas est, cur homo sit imago Dei. Geometry is one and eternal shining in the mind of God. That share in it accorded to men is one of the reasons that Man is the image of God. (Kepler, 1571-1630) God the Geometer, Manuscript illustration. Geometrical Constructions 2 2 Pencil of Circles Intersecting (or „elliptic”) pencil Radical center C of coaxal circles radical axis P C All tangents drawn to the circles For three circles whose centers form of a coaxal pencil from a point a triangle, the three radical axes (of on the radical axis have the the circles taken in pair) concur in a same length. point called the radical center. Geometrical Constructions 2 Apollonian Problems 3 Apollonius of Perga (about 262 - about 190 BC) Apollonius of Perga was known as 'The Great Geometer'. Little is known of his life but his works have had a very great influence on the development of mathematics, in particular his famous book Conics introduced terms which are familiar to us today such as parabola, ellipse and hyperbola. Perga was a centre of culture at this time and it was the place of worship of Queen Artemis, a nature goddess. When he was a young man Apollonius went to Alexandria where he studied under the followers of Euclid and later he taught there. Apollonius visited Pergamum where a university and library similar to Alexandria had been built. Pergamum, today the town of Bergama in the province of Izmir in Turkey, was an ancient Greek city in Mysia. It was situated 25 km from the Aegean Sea on a hill on the northern side of the wide valley of the Caicus River (called the Bakir river today) http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Apollonius.html Geometrical Constructions 2 Apollonian Problems 4 Apollonius Circle DEFITION 1: the set of all points whose distances from two fixed points are in a constant ratio DEFITION 2: one of the eight circles that is simultaneously tangent to three given circles http://mathworld.wolfram.com/ApolloniusCircle.html Geometrical Constructions 2 Apollonian Problems 5 Apollonian Problems on Tangent Circles Combinatorial approach Options Circle C 1) (PPP) ; Point (circle of 0 radius) P 2) (PPL) ; Straight line (circle of infinite radius) L 3) (PPC) ; 4) (PLL) ; Apollonian tangent circle problem: choose three from the elements 5) (PLC) : of the set {P, L, C} (an element can be chosen repeatedly) and fid the circles tangent to or passing through the given elements. 6) (PCC) : (In combinatorics: third class combinations with repetitions of three 7) (LLL) ; elements.) 8) (LLC) : 9) (LCC) : ; subject of our course 10) (CCC) : : not subject of our course Geometrical Constructions 2 Apollonian Problems 6 1) (PPP) 2) (PPL) c P1 e d P c 1 i d g P2 P3 f P 2 h l Geometrical Constructions 2 Apollonian Problems 7 3) (PPC),4)(PLL) l c 1 d P 1 e c d c P h i h g f l2 P 2 f e g Geometrical Constructions 2 Apollonian Problems 8 5) (PLC) Hint: (PLC) can be reduced to (PPL). An additional point Q can be constructed as the point of intersection of A1P and the circle through B, A2 and P. In this way two circles, passing through P, tangent to the given circle c and to the given line l can be constructed. Change the points A1 and A2 to obtain two more solutions. A1 d c P c A2 g Q P e B f Q1 Q2 l Geometrical Constructions 2 Apollonian Problems 9 7) (LLL) 8) (LLC) l f l3 2 c d e l1 g d c l f c l 2 1 e Hint: the solutions are the incircle and Hint: the problem can be reduced to the the excircles of the triangle (PLL) by means of dilatation that means, formed by the lines. The centers draw parallels at the distance of the radius are the points of intersection of of the given circle. The circles passing the bisectors of the interior and through the center of the given circle and exterior angles. tangent to the parallel lines, have the same centers as the circles of solution. Geometrical Constructions 2 Apollonian Problems 10 Chapter Review Vocabulary Geometrical Constructions 2 Apollonian Problems 11 Conic Sections Ellipse Parabola Hyperbola Geometrical Constructions 2 Conic Sections 12 Ellipse t P Definition: r1 r Let two points F1, F2 2 Ellipse is the set of points, major foci and a distance 2a whose sum of distances axis F F be given. 1 2 from F1 and F2 is equal to the given distance. axis minor r1 + r2 = 2a. Dist(F1, F2 )=2c, a>c. r1 r2 2a The ellipse is symmetrical with respect to the straight line F1F2 , to the perpendicular bisector of F1F2 and for their point of intersection O. The tangent at a point is the bisector of the external angle of r1 and r2. Geometrical Constructions 2 Conic Sections 13 Properties of Ellipse Antipoint E: dist(P,E) = r , dist(F ,E) = 2a. director circle 2 1 Director circle: set of antipoints, circle about a focus with the radius of 2a. principal circle The director circle is the set of points (antipoints) that are reflections of a focus with respect to all tangents r1 r2 E of the ellipse. The ellipse is the set of centers of circles P passing through a focus and tangent to a circle, i.e. M the director circle about the other focus. r1 r2 2a O Principal circle: circle about O with the radius of a. r F F 2 1 2 The principal circle is the set of pedal points M of lines from F2 perpendicular to the tangents of the ellipse. Under the reflection in the ellipse, a ray emitted from a focus will pass through the other focus. A point of the ellipse (P), the center of the director circle (F1) and the antipoint (E) corresponding to the given point, are collinear. Geometrical Constructions 2 Conic Sections 14 Osculating circle of an Ellipse Osculating circle at C C Perpendicular to BC A O L1 B Osculating circle at B L2 Osculating circle of a curve Let three (different, non collinear) points P1, P2 and P3 tend to the point P0. The three points determine a sequence of circles. If the limiting circle exists, this P1 osculating circle is the best approximating P2 P3 circle of the curve at the point P0. P0 Geometrical Constructions 2 Conic Sections 15 Approximate Construction of Ellipse Approximate ellipse composed Approximate ellipse composed of two circular arcs of three circular arcs (Five-Center Method) A’ C C A” B K1 O L1 A K Perpendicular bisector of AA” K2 L2 Geometrical Constructions 2 Conic Sections 16 Parabola axis t Definition: Let a point F focus and a straight line d directrix Parabola is the set of be given. The line is not points equidistant from passing through the the focus and the dirctrix. point. P FP = dist(P,d) = PE F V d E The parabola is symmetrical with respect to the line, passing through the focus and perpendicular to the directrix. This line is the axis of the parabola. The tangent at a point is the bisector of the angle Ë FPE. Geometrical Constructions 2 Conic Sections 17 Properties of Parabola s i Antipoint E: pedal point of the line passing x a through P perpendicular to d. The set of antipoints is the directrix. The directrix is the set of points (antipoints) that are reflections of a focus with respect to all tangents of the parabola. The parabola is the set of centers of circles passing through the focus and tangent to a line, i.e. the P directrix. Tangent at the vertex: set of pedal points M of F lines from F perpendicular to the tangents of M the parabola. d V Under the reflection in the parabola, a ray E emitted from the focus will be parallel to the axis. Geometrical Constructions 2 Conic Sections 18 Osculating Circle at the Vertex of Parabola s i x a K F d V The radius of the osculating circle at the vertex: r = dist(F,d) Geometrical Constructions 2 Conic Sections 19 Hyperbola t Definition: axis P 2a Let two points F1, F2 conjugate Hyperbola is the set of r1 foci and a distance 2a points, whose difference of r1 r2 be given. distances from F1 and F2 is equal to the given distance. r2 traverse F1 F2 axis |r1 - r2|= 2a. Dist(F1, F2 )=2c, a<c. The hyperbola is symmetrical with respect to the straight line F1F2 , to the perpendicular bisector of F1F2 and about their point of intersection O. The tangent at a point is the bisector of the angle of r1 and r2. Geometrical Constructions 2 Conic Sections 20 Asymptotes of Hyperbola y Equation of hyperbola: x2 y 2 − =1 a2 b2 c where b is determined by the Pythagorean b equation: 2 2 2 a x a+ b = c. F1 F2 The limit of the ratio y/x can be found from the equation: ⎛ y 2 ⎞ ⎛ b2 b2 ⎞ ⎜ ⎟ = ⎜ − ⎟ lim ⎜ 2 ⎟ lim ⎜ 2 2 ⎟ →x ⎝ ∞ x ⎠ →x ∞⎝ a x ⎠ y b Construction: lim = ± 1) Draw the tangents at the vertices.
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