Transcript Verdana 30 pt

Liceo Scientifico “G.Ferraris”
Taranto
UTILIZZARE SPAZIO
PER INSERIRE
FOTO/IMMAGINE DI
RIFERIMENTO LEZIONE
Maths course
The ellipse
Teacher
Rosanna Biffi
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
0
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)
let PF1+PF2=2a
where “a” is a positive constant.
c>0,
From this relation, after eliminating radicals, and simplifying, we
obtain the equation of the ellipse relative to the centre and the
axes:
placed b2=a2-c2
(a2-c2)x2+a2y2=a2(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 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
If a=4 and b=5, the major axis is
vertical, then the equation becomes:
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.
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
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)
Acknowledgement
Marco Dal Bosco
Headmaster
Technical Support
eni
Director
Rosanna Biffi
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