Conic sections

(notes by Roberto Bigoni)


1. Definitions.

Given in a plane Π a straight line d and a point F distant f from d (f > 0), a conic section (or, shortly, a conic) γ is the locus of the points P of Π such that, said H their orthogonal projection on d, the ratio PF/PH has constant value e (e>0).

Eqn1.gif

The straight line d is named the directrix of γ and the point F is named the focus of γ.

The ratio e is named the eccentricity of γ, f is called the focal distance and the product p=ef is called conic parameter or semilatus rectum.

figura1


2. Polar equation.

In order to analytically describe a conic section with eccentricity e and parameter p, we can take in Π a polar reference system with pole in F and polar semiaxis given by a ray from F, perpendicular to d and outward oriented with respect to d.

Within this system every point P is in bijective correspondence with a pair of real numbers (ρ;θ), respectively called radial coordinate and angular coordinate of P, such that

Eqn1.gif

where X is a point of the polar semiaxis and the angle θ is counterclockwise oriented, measured in radians and ≤ 2π.

In this reference system the (1.1) becomes

Eqn2.gif

Eqn3.gif

Eqn4.gif

Eqn5.gif

The (2.3) is the polar equation of a conic with parameter p and eccentricity e.

Since the cosine is an even function, that is it has the same value for opposite arguments, the (2.3) shows an important attribute of all the conics: they are symmetrical with respect to the polar axis.

Moreover, from (2.3) we can deduce that, when θ is a right angle, ρ coincides with p; so p can be geometrically interpreted as the distance of F from the points of γ with polar angles respectively Eqn6.gif.

 


3. Cartesian equation in the system xFy

In order to obtain the equation of a conic in cartesian coordinates, we can take a cartesian orthogonal reference system xFy with origin in the focus F, with abscissa axis coincident with the polar axis and with ordinate axis from F.

In this system we have

Eqn1.gif

From the first equation in (3.1) we have

Eqn2.gif

and, by squaring both sides of the equations in (3.1),

Eqn3.gif

The substitution of the (3.2) into the (3.3) gives

Eqn4.gif

It must be observed that the (3.4), is more general than (3.1), so the locus given by the latter is a subset of that given by the former one. The name of conic is commonly extended to the loci given by the (3.4).

 


4. Tangents to a conic in a point on it (doubling rule)

The (3.4) can be rewritten as

Eqn1.gif

It can be demonstrated that, if P(xP;yP) is a point on the conic, then the equation of the tangent to the conic in P has equation

Eqn2.gif

This equation can be authomatically deduced from that of the conic:

 


5. Circumference.

If the eccentricity e=0, the (2.3) simply becomes

Eqn1.gif

that is the conic is the locus of the points with constant distance from the pole F. The conic therefore is the circumference with center F and radius p.

The equation of a circumference in a cartesian plane comes from the (3.4) for e=0

Eqn2.gif

In a Cartesian frame system XΩY, with axes X and Y parallel to those of the former system and origin Ω(;), we have

Eqn003.gif

In this system the coordinates of the center are α e β and the equation (5.2) becomes

Eqn004.gif

In general, every equation like the (5.3) is the Cartesian equation of a circle with center C(α;β) and radius ρ.

If we write the variables as x and y, the equation (5.3) becomes

Eqn005.gif

If we expand the squares of binomials, we obtain

Eqn006.gif

Eqn007.gif

If we introduce the parameters

Eqn008.gif

the equation (5.5) becomes

Eqn009.gif

so the equations like (5.7) represents circumferences with center C(α;β)

Eqn010.gif

and radius ρ

Eqn011.gif

The circumference is real only if the radicand a2+b2-4c is positive.

Circumference through 3 points.

 


6. Parabola

If e=1, the conic is the locus of the points equidistant from F and d and, adopting the word first proposed by Apollonius of Perga, is named parabola.

In this case p coincides with f and from the (2.3) we have

Eqn1.gif

Since the denominator in the right hand side cannot be null, it follows that the points of a parabola can not have polar angles like 0 or 2π and that when the polar angle approaches these values the polar radius goes to the infinity.

The polar radius has a minimum when the polar angle is π; in this case the polar radius is half of f and the corresponding point V, called the vertex of the parabola, is on the polar axis at equal distances from F an K.

The equation of a parabola in the orthogonal cartesian system xFy follows immediately from the (3.4) with e=1:

Eqn2.gif

In order to obtain the equation of the same curve in a reference system with the same x-axis and with origin in V, we must apply to the (6.2) the translation

Eqn3.gif

which gives

Eqn4.gif

From the (6.3) the expression of x in terms of y is

Eqn5.gif

In general we shall say that every equation

Eqn6.gif

represents a parabola symmetrical with respect to the x-axis, with vertex at the origin and focus and directrix given by

Eqn7.gif

If, as we did with the circumference, we assume a reference system XΩY, with axes parallel to those of the previous system and with origin Ω(;), the relations between the coordinates of the two systems are given by

Eqn003.gif

In XΩY the coordinates of the vertex are α e β and the equation (6.5) becomes

Eqn010.gif

therefore, each equation with the form

Eqn011.gif

is the equation of a parabola with the axis of symmetry parallel to the abscissa axis and vertex V(α ; β) such that

Eqn012.gif

that is

Eqn013.gif

Analogously, every equation

Eqn8.gif

represents a parabola symmetrical with respect to the y-axis, with vertex at the origin and focus and directrix given by

Eqn9.gif

and every equation with form

Eqn014.gif

is the equation of a parabola with the axis of symmetry parallel to the y-axis and vertex V(α ; β) such that

Eqn015.gif

Parabola through 3 points.

See also: Area of a parabolic segment

See also: An optical property of the focus

 


7. Ellipse

If 0 <e <1, the conic is named ellipse.

In this case, the denominator in the right hand side of the (2.3) is always positive, so every point of the ellipse has finite distance from the focus.

In particular:

These two points are the vertices of the ellipse; they, like F, lay on the polar axis.

In the reference system xFy their abscissas results from the (3.4) with y=0

Eqn3.gif

Eqn4.gif

The segment V''V' is called the major axis of the ellipse; its length is

Eqn5.gif

Let a be the half of this measure:

Eqn6.gif

Let c be the abscissa of the middle point of V''V':

Eqn7.gif

From the (7.3)

Eqn8.gif

and from the (7.4) and the (7.2)

Eqn9.gif

We can deduce an alternative cartesian equation of the ellipse if we put the origin of the cartesian reference system in the middle point of V''V'.

The equations of the translation are

Eqn10.gif

Using these equations, from the (3.4) we obtain

Eqn11.gif

Eqn12.gif

By setting

Eqn13.gif

we have

Eqn14.gif

and finally

Eqn15.gif

The (7.6) is the canonical equation of the ellipse and the reference system is its canonical system.

Since in this equation there are only the squares of the variables, the curve must be symmetrical with respect to both the axes and to the origin. For x=0 we have y = ± b. The parameter b is the measure of the minor semiaxis.

Furthermore, the symmetry of the curve implies that it has another focus beyond that used as pole in the (2.3), which in the canonical system has abscissa -c. So the ellipse has two foci with abscissas ±c.

The equation (7.6) implies a>b. Otherwise, if a<b, the equation represents an ellipse in which the focal axis coincides with the y-axis and the distances |c| of the foci from the origin at the center of the ellipse are given by

Eqn016.gif

The sum of the distances from these foci to the same point of the ellipse is a constant equal to the measure of the major axis, that is 2a.

If we assume a reference system with origin at Ω(;) and axes parallel to those of the canonical system, the equation of the ellipse becomes

Eqn017.gif

Ellipse through 4 points

See also: Construction of the ellipse, given the semiaxes a e b. The area of the ellipse. The area of the elliptic segment.

 


8. Hyperbola

A conic with eccentricity e >1 is named hyperbola.

Since ρ is positive by definition, also the denominator in the right hand side of the (2.3) must be positive, so the polar angles of the points of a hyperbola are such that

Eqn1.gif

that is

Eqn2.gif

The more θ approaches the extremes of this interval, the more ρ increases going to the infinity. Hyperbolas, like parabolas but unlike ellipses and circumferences, are open curves.

The point V' with polar angle θ=π is the nearest to the focus

Eqn3.gif

V' is the vertex of the hyperbola. But if we let y=0 in the (3.4), we obtain the (7.1). In order to make evident that, if e>1, the two abscissa are both negative, the (7.1) may be rewritten as

Eqn4.gif

Let V'' be the point with lesser abscissa (the secondary vertex). The segment V''V' is called the real axis of the hyperbola. Its length is

Eqn5.gif

Let a be the half of this distance. We can write

Eqn6.gif

As we have seen for the ellipse, it is possible to deduce a cartesian equation alternative to the (3.4) by translating the origin of the reference system from F to O, the middle point of the real axis. In the original system xxFy the abscissa of O is

Eqn7.gif

therefore

Eqn8.gif

With the (8.4) from the (8.2) we obtains

Eqn9.gif

The equations of the translation to the new reference system are

Eqn10.gif

that, applied to the (3.4), give

Eqn11.gif

Eqn12.gif

With

Eqn13.gif

we have

Eqn14.gif

and finally

Eqn15.gif

The (8.6) is the canonical equation of the hyperbola and the reference system is its canonical system.

Since in this equation there are only the squares of the variables, the curve must be symmetrical with respect to both the axes and to the origin. Because of this symmetry, the curve has another focus. The two foci in the canonical system have abscissas ±c.

The absolute value of the differences between the distances from these two foci to any point of the hyperbola is a constant, equal to the measure of the real axis, that is 2a.

If we rewrite the (8.6) as

Eqn16.gif

we can see that the curve doesn't have real points in the interval (-a<x<a), so it is formed by two separated branches.

From the same equation we can deduce that the more the absolute value of x increases, the more the equation approaches the following

Eqn17.gif

which represents a pair of straight lines passing through the origin and having slopes

Eqn18.gif

These lines are called the asymptotes of the hyperbola.

The equation (8.7) implies that the real axis of the hyperbola coincides with the axis of abscissas. Equations of form

Eqn019.gif

represent hyperbolas with focuses on the ordinate axis.

See also:

Construction of the hyperbola, given the semiaxes a e b.

The area of the hyperbolic segment.

 


last revision: November 2016