Liceo Scientifico “G.Ferraris”Taranto

Maths course

The ellipse

Teacher

Rosanna Biffi

UTILIZZARE SPAZIO PER INSERIRE FOTO/IMMAGINE DI RIFERIMENTO LEZIONE

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.

Conic section

PF1+PF2=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

F20F1

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 PF1+PF2=2a

where “a” is a positive constant.

The ellipse equation

given F1(-c;0) and F2(c;0) c > 0 ,

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)

This is the ellipse equation in canonical form.

where a > b.

placed b2=a2-c2 since a > c,

the ellipse is described by this

equation:

How to draw an ellipse

The gardener’s method

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 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.

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.

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:

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.

Ellipse eccentricity

This picture shows two ellipses

with different eccentricities,

e1= 0.8 , e2= 0.94

e1<e2.

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

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

Course TeacherRosanna Biffi

Linguistic Support Flaviana Ciocia

Performed byTeacher: Rosanna Biffi

Students: Biondolillo Alessia, Masella Angela, Nanni Alfredo

(Grade 5 D - Secondary High School)

Acknowledgement Marco Dal Bosco 

Headmaster

Technical Support

eni

Director Rosanna Biffi

 

Copyright 2012 © eni S.p.A.