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Liceo Scientifico G.Ferraris Taranto Maths course The ellipse Teacher Rosanna Biffi UTILIZZARE SPAZIO PER INSERIRE FOTO/IMMAGINE DI RIFERIMENTO LEZIONE

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An ellipse is a conic section that is formed by slicing a right circular cone with a plane, not passing through the vertex, forming an angle with the base plane of the cone. This effect can be seen in the following images. Conic section

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The ellipse belongs to a family of curves including circles, parabolas, and hyperbolas. All of these geometric figures may be obtained by the intersection a double cone with a plane, hence the name conic section. Conic section

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PF 1 +PF 2 =const The ellipse is the geometric locus of points P which moves so that, the sum of the distances from P to two fixed points, called foci, is a constant. Definition of ellipse P F2F2 0 F1F1

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The equation of the ellipse can be found by using the distance formula, to calculate the distance between a general point on the ellipse (x, y) to the 2 foci, for example: let PF 1 +PF 2 =2a where a is a positive constant. The ellipse equation given F 1 (-c;0) and F 2 (c;0) c > 0,

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From this relation, after eliminating radicals, and simplifying, we obtain the equation of the ellipse relative to the centre and the axes: (a 2 -c 2 )x 2 +a 2 y 2 =a 2 (a 2 -c 2 ) This is the ellipse equation in canonical form. where a > b. placed b 2 =a 2 -c 2 since a > c, the ellipse is described by this equation:

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How to draw an ellipse The gardeners method

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The ellipse parameters The intersection points of this curve with the x-axis are A 1 (-a,0) and A 2 (a,0), as with the y-axis are B 1 (0,-b) and B 2 (0,b). The foci in this case are found on x axis and we obtain c by In this picture a=4, b=2, c=3.46 The vertexes of the ellipse are defined as the intersections of the ellipse and the line passing through foci.

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The ellipse parameters The positive numbers a and b represent the measures of semi-axes. The distance between the vertexes is called major axis or focal axis and its length is 2a. The segment passing the centre and perpendicular to the major axis is the minor axis and its length is 2b.

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The distance between the foci is called focal length and its value is 2c. If the 2 foci are vertically aligned, then a < b: the minor axis will be found on the x-axis and the major axis on the y-axis, as shown in this picture. The midpoint of the segment connecting the foci is the centre of the ellipse.

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a = 4 is the semi-minor axis. b = 5 is the semi-major axis. c is given by: that is the distance from the centre to each focus. In this case the foci are found on the y axis. Example If a=4 and b=5, the major axis is vertical, then the equation becomes:

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We define eccentricity of the ellipse, the ratio of the focal lenght to the measure of the major axis. This ratio is denoted by e, that is e = 2c/2a, e = c/a. This number e is always between zero and one (0

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Two special cases exist: If e=0, the focal length is null, that is the 2 foci coincide and our ellipse is a circle. If e=1, that is c=a, the focal length coincides with the major axis and, as a consequence the ellipse flattens until it becomes a segment: it is a degenerate ellipse. The more the number e approaches 1, the more the ellipse flattens. Limit cases

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Keplers 1 st law: the law of ellipses All planets orbit the Sun in elliptical orbits with the Sun as one common focus. Kepler Brahe

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The Coliseum, originally the Flavian Amphitheatre

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St Peters Square

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Vault of St. Andrew Bernini St. Carlo alle Quattro Fontane Borromini

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Course Teacher Rosanna Biffi Linguistic Support Flaviana Ciocia Performed by Teacher: Rosanna Biffi Students: Biondolillo Alessia, Masella Angela, Nanni Alfredo (Grade 5 D - Secondary High School)

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Acknowledgement Marco Dal Bosco Headmaster Technical Support eni Director Rosanna Biffi Copyright 2012 © eni S.p.A.

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