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

Conic section 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 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.

Definition of ellipse 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. P F1 F2 PF1+PF2=const

The ellipse equation 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: given F1(-c;0) and F2(c;0) c > 0 , let PF1+PF2=2a where “a” is a positive constant.

This is the ellipse equation in canonical form. From this relation, after eliminating radicals, and simplifying, we obtain the equation of the ellipse relative to the centre and the axes: (a2-c2)x2+a2y2=a2(a2-c2) placed b2=a2-c2 since a > c, the ellipse is described by this equation: where a > b. This is the ellipse equation in canonical form.

How to draw an ellipse The gardener’s method

The foci in this case are found on x axis and we obtain “c” by The ellipse parameters The intersection points of this curve with the x-axis are A1(-a,0) and A2(a,0), as with the y-axis are B1(0,-b) and B2(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.

The ellipse parameters 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. The positive numbers “a” and “b” represent the measures of semi-axes.

The midpoint of the segment connecting the foci is the centre of the ellipse. 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.

Example 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. If a=4 and b=5, the major axis is vertical, then the equation becomes:

Ellipse eccentricity 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<e<1) and tells us how the ellipse is flattened. This picture shows two ellipses with different eccentricities, e1= 0.8 , e2= 0.94 e1<e2.

Limit cases 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.

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

The Coliseum, originally the Flavian Amphitheatre

St Peter’s Square

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

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