Conical

Euclidean purely geometrical definition

The conical train a family of curved plane resulting from the intersection of a plan with a circular cone.

According to the relative positions of the plan and cone, one obtains various types of conical:

conical the clean , when the plan is not perpendicular to the axis of the cone, and does not pass by its top. One distinguishes three kinds of conical clean according to the angle of inclination from the plan with the axis of the cone:

if this angle is higher than the aperture of the cone, the intersection is a ellipse;
if the angle of inclination is lower than the aperture, it is a hyperbole;
and if the two angles are equal, it is a Parabole.

conical the partially degenerated :

the intersection is a Cercle when the plan is perpendicular to the axis of the cone;
the intersection is a hyperbole équilatère when the angle of inclination of the plan is lower of 45° than the aperture of the cone;

and conical the completely degenerated , when the plan contains the top of the cone:

the intersection is a couple of right secant, if the angle of inclination of the plan with the axis of the cone is lower than the aperture of the cone;
the intersection is reduced on a line if these angles are equal.
finally it is reduced to a point if the angle of inclination is higher than the aperture.

Purely projective definition

It is a question of defining the conical ones without distances, without angles, just with the straight-edge, the pencil and a handle of axioms, in the purest tradition of Blaise Pascal and Girard Désargues: to see Treated projective of conical the

Monofocale definition

The definition monofocale of conical is still called definition by hearth and direct of these conical.

Definition

In a plan (p) , one considers a line (d) and a point $F$ not located on (d) . That is to say $e$ a strictly positive reality.

One calls conical right director (d) , of hearth $F$ and eccentricity $e$ the whole of the points $M$ of the plan (p) checking:

\ D (M, F) = E \ D (M, (d)) \ qquad E \ in \ mathbb {R} ^*_+

where

d (M, F)

measurement the distance of the point M at the point F

and

d (M, (d))

measurement the distance of the point M with the right-hand side (d)

It will be noted that the ellipses are closed and limited curves, that the parabolas are open and infinite, and that the hyperboles have two symmetrical branches compared to the point of intersection of their common asymptotes.

Setting in equation

That is to say O the orthogonal projection of the point F on the line (d) . In the plan (p) one defines the orthogonal Repère then ( O , (OFF) , (d) ).

That is to say p the distance from O to F ( p is called the parameter ). In the reference mark previously defined F has as coordinates ( p , 0).

For a point M of coordinates $(X, there)$ one can express the preceding distances using the two following formulas:

\ qquad D (M, F) = \ sqrt {(x-p) ^2 + (y-0) ^2}

\ qquad D (M, (d)) = \ sqrt {(x-0) ^2}

what implies while raising squared and while using and:

\ qquad (x-p) ^2 + y^2 = e^ {2} x^2

maybe after simplification:

\ qquad x^2 (1-e^2) + y^2 - 2xp + p^2 = 0

According to the values of E one obtains several types of curves:

If $0 a ellipse$
If $e=1$ a Parabole
If $e>1$ a hyperbole

The degenerated conical are obtained by modifying the preceding conditions

If F is on D , one obtains:

If $e<1$ the point O (which is also the point F );
If $e=1$ line perpendicular to (d) passing by F ;
If $e>1$ two secant lines;

If $e=0$ , the point O (which is also the point F ).

There thus does not exist definition of circle per hearth and director. On the other hand, if EP = R and if E tends towards 0 (what ad infinitum increases the distance between the hearth and the director), the ellipse approaches a circle of center F and R

Bifocal definition

The ellipse can be defined as the place of the points whose sum of the distances to two fixed points called hearths of the ellipse is constant and equal to a fixed value. This definition remains valid in the case of the circle, in which the hearths are confused.

The hyperbole can be defined as the place of the points whose absolute value of the difference of the distances to two fixed points called hearths of the hyperbole is constant and equal to a fixed value.

The parabola does not have a bifocal definition.

Analytical definition

Case closely connected

In analytical geometry refines , the conical ones are the algebraic plane curves of the second order, i.e. the plane curves whose Cartesian coordinates X and of the points are there solution of a polynomial equation of the second degree, form:

has x^2 + B X there + C y^2 + D X + E there + F = 0 \,

with one at least of the three coefficients has , B or C not no one so that the equation is indeed of the second degree (condition (1)).

According to the reference mark used, the expression of the equation will be more or less simple, but will always remain of the second degree. It is interesting to seek the reference mark in which the expression of the equation, known as reduced equation , will be simplest.

For that, we can notice initially that it is always possible to return the coefficient B no one by a rotation of the reference mark.

We look at then the coefficients has and C :

If the coefficient C is him also null, has is then inevitably not no one (condition (1)), and a translation along the axis of the X thus makes it possible to cancel the coefficient D .

* If E is null, by posing p = - F / has , the equation is reduced to:

x^2 = p \,

According to the sign of p , we obtain 0 with 2 parallel straight lines .

* If E is nonnull, a translation along the axis of the cancels F there. By posing p = - has / E , we obtain the reduced Cartesian equation of a PARABOLA :

there = p \, x^2 \,

If the coefficient has is null, one obtains the symmetrical situation of the preceding one where X and sees their exchanged roles there. One thus obtains still:

* If D is null, 0 with 2 parallel straight lines ,

* If D is nonnull, a PARABOLA of reduced equation:

X = Q \, y^2 \,

If the coefficients has and C are all two the nonnull ones, a translation along the axis of the X cancels D , and a translation according to the cancels E there. The equation is thus reduced to:

has x^2 + C y^2 = - F \,

* If has and C is of the same sign:

- if F is him also same sign, it does not have there not corresponding curve ;

- if F is null, the curve is reduced to a point ;

- if F is of opposed sign, we can pose has ² = - F / has and B ² = - F / C ; we thus arrive at the reduced Cartesian equation of a ELLIPSE :

(X/a)^2 + (there/b)^2 = 1 \,

* If has and C are opposite signs:

- if F is null, the curve is reduced to 2 secant lines (= which cross);

- if F is sign of has , we can pose has ² = F / has and B ² = - F / C ; we thus arrive at the reduced Cartesian equation of a HYPERBOLE :

(X/a)^2 - (there/b)^2 = -1 \,

- if F is sign of C , we can pose has ² = - F / has and B ² = F / C ; we thus arrive at the other reduced Cartesian equation of a HYPERBOLE :

(X/a)^2 - (there/b)^2 = 1 \,

Projective case

In analytical geometry projective , the conical ones is still the algebraic plane curves of the second order, i.e. the plane curves whose points have projective coordinates X , Y and Z which checks a homogeneous polynomial equation of the second degree of the form (see homogeneous Coordonnées):

has X^2 + B X Y + C Y^2 + D X Z + E there Z + F Z^2 = 0 \,

One thus works in the projective Plan where a generic point has as homogeneous coordinates , and two homogeneous coordinates proportional ( $X:\lambda Y:\lambda Z$ and , for some $\ lambda$ ) indicate the same point of the plan. Our projective plan contains several specimens of the plan closely connected; in particular the admitting whole of the points of the homogeneous coordinates of the form .

One can note whereas for Z = 1, one finds the equation of the case closely connected. In fact, one a:

X = X/Z \,

and

there = Y/Z \,

A first question which one installation is then: while limiting itself to the image of conical in the plan closely connected above definite, which type of conical closely connected does one find? (and even, one finds conical closely connected well?). With this intention, one ad infinitum looks at initially their behavior (presence of asymptotes or parabolic branches,…). To make tighten X and towards the infinite come down to make there tighten Z towards 0. For Z = 0, the preceding equation is reduced to:

has X^2 + B X Y + C Y^2 = 0 \,

This equation is called equations with the asymptotic directions , because the report/ratio Y / X gives the slope then ad infinitum curve, i.e. its asymptotic direction.

If C = 0:

* if B = 0, the equation has a solution X = 0 of multiplicity double, which corresponds to a slope ad infinitum infinite, therefore to a double vertical asymptotic direction; the curve is thus either a parabola of vertical axis, or 0 on 2 parallel vertical lines;

* if B is nonnull, we obtain two simple asymptotic directions, one vertical, the other not; the curve is thus either a hyperbole , or 2 convergent lines;

If C is nonnull, by posing T = Y / X , the equation becomes:

has + B T + C t^2 = 0 \,

* if the discriminant of this equation is strictly positive, we obtain 2 distinct simple asymptotic directions, and the curve is either a hyperbole , or 2 convergent lines;

* if the discriminant of this equation is null, we obtain an asymptotic direction doubles, and the curve is either a parabola , or 0 on 2 parallel straight lines;

* if the discriminant of this equation is strictly negative, the curve does not have asymptotic direction, therefore not of infinite branches, and the curve, if it exists, is either a ellipse , or a point.

However, the true interest of the use of the projective geometry is elsewhere. The classification which was made in the case refines, and reinterpreted within the projective framework, bases itself on changes of coordinates closely connected; and which can be interpreted, the plan refines being seen like part of the projective plan, as the projective changes of coordinates which leave fixes the line ad infinitum (i.e. points of the projective plan of the form ). There exists obviously much of other projective changes of coordinates, and to be authorized to use them will make it possible to soften the classification of the conical ones largely. In fact, the classification of conical projective comes directly from that of the symmetrical bilinear form on the vector space of dimension 3 subjacent with our projective plan.

Barycentric case

In analytical geometry barycentric , the conical ones is always the algebraic plane curves of the second order, i.e. the plane curves whose points have barycentric coordinates λ , μ and ν which checks a homogeneous polynomial equation of the second degree of the form:

A_ {11} \ lambda^2 + A_ {22} \ mu^2 + A_ {33} \ nu^2 + 2 A_ {12} \ lambda \ driven + 2 A_ {23} \ driven \ naked + 2 naked A_ {31} \ \ lambda = 0 \,

One can identify this equation with the preceding one, while posing:

\ lambda = X, \ \ driven = Y, \ \ naked = Z

One obtains then, with a multiplicative coefficient near:

A_ {11} = has, \ A_ {22} = C, A_ {33} = F, A_ {12} = B/2, A_ {23} = E/2, A_ {31} = D/2 \,

Bonds between the definitions

Monofocale definition and bifocal definition

The parabolas admit one and only one couple hearth/direct within the meaning of the monofocale definition, and the corresponding eccentricity is worth 1.

Ellipses and hyperboles admit exactly two couples hearth/direct within the meaning of the monofocale definition, and those correspond to the same value of the eccentricity. They are symmetrical compared to the center of the ellipse or at the point of intersection of the asymptotes of the hyperbole. These hearths are the points intervening in the bifocal definition.

Geometrical definition and bifocal definition

The hearths and the directors of conical can be geometrically given within the framework of the definition of conical like intersection of a cone and a plan not passing by the center of this one.

There exists, according to the orientation of the plan compared to the axis of the cone, one (case of the parabolas) or two (case of the ellipses and the hyperboles) tangent spheres at the same time in the plan and the cone; they are spheres centered on the axis, located in the same half-cone (case of the ellipses) or in opposite half-cones (case of the hyperboles).

Each one of these spheres defines one of the hearths of conical (it is the tangential point of the sphere and the plan) as well as the line associated director (it is the intersection of the plan of conical and the plan containing the circle of tangency of the sphere and the cone).