APPENDIX I. DEFINITION OF THE CONIC ELLIPSE
The ellipse as a conic section can be considered as generated by the cutting of a cone by a plane that cuts all its generating lines, and it is therefore defined as the loci of points of the plane whose sum of distances between two points is constant.

Figure I.1

Taking into account the ellipse illustrated in the Figure I.1, the following points are defined:

• The fixed points F, F', are called focus of the ellipse;
• The distance = 2c is called focal length;
• The distances and are called radius vectors;
• The sum , is a constant value, as it can be demonstrated according to the definition for the ellipse itself;
• The difference will be called b² and will be positive, since a² > c²;
• The line segment distance = 2a will be called major axis of the ellipse;
• The line segment distance =2b, will be called minor axis of the ellipse.

Taking the line segment as the horizontal axis and the perpendicular to the point in the middle of the line segment as the vertical axis (figure I.1), and from the definition of the ellipse given above, it comes to:

By operating upon that equation and, taking into account the relation a²-c²=b², the following expression is obtained for the canonical equation of the ellipse:

(I.1)

APPENDIX II. SURFACES AND CURVES ON A SURFACE [Granero Rodriguez 1985]
In the affine three-space, the term curves applies to the entire collection of points C Ì Â³ of coordinates (x(t), y(t), z(t)), such that x(t), y(t), z(t) are continuous functions at a certain interval I.

Hence, the equations:

(II.1)

are called the parametric equations of the curve C (tridimensional line).

Considering the existing isomorphism between the free vectors of space and the points of E₃, it is often more suitable to define curve C by a free vector of components (x(t), y(t), z(t)).

In this case it is clear (Figure II.1) that by changing the parameter t, the vector end will draw the curve C.

Figure II.1

The equation is called vectorial equation of C.

When the curve is located in a plane is called a plane curve. Otherwise it is called a nonplane curve.

In the affine three-space the term surface applies to the entire collection of points S Ì Â³ of coordinates [x (l, m), y(l,m), z(l,m)], such that x(l,m), y(l,m), z(l,m) are continuous functions in a certain domain D Í Â² .

The equations

(II.2)

are called parametric equations of the surface S.

In the same way, based on the isomorphism between free vectors and points, we would be able to define the surface S by a free vector or position vector of components (x(l,m), y(l,m), z(l,m)). In these conditions (Figure II.2), by varying the parameters l and m, the position vector end will travel through the surface S.

Figure II.2

The equation will be called the vectorial equation of such a surface.

Often it is feasible to remove the two parameters l and m among the three equations (II.2), giving rise to an equation of the form F(x,y,z) = 0, or z = f(x,y), which is called the Cartesian equation of the surface. Reciprocally, if the surface is defined by z = f(x,y), making x = l, y = m yieds the following parametric equations

(II.3)

Likewise, let us notice that whatever relationship is established between both parameters, as for instance l = f(m), the position vector will depend solely on one of those parameters, and consequently, a curve upon the surface will be drawn. Again from what has preceded, the result will be that when one of the two parameters is fixed, or when these are in the relationship l = f(m), or , which is the same, we will have a curve on the surface.

Example:

Given the surface S defined in parametrics as:

by isolating l and m in terms of x and y between the two first equations, which are linear, by substituting the third we will have z = f(x,y). However in this case we can very easily obtain z = f(x,y) by just noticing that:

where .
Therefore, or represents the Cartesian equation of the surface.

It is very simple to display the preceding surface, since all its intersections through planes z = k parallel to the horizontal are circumferences the radii of which become uniformly shorter until becoming null at point (0,0,16).

Likewise, let us notice that

are less dificult parametric equations of this surface than the preceding ones.

APPENDIX III. RULED SURFACES [Granero Rodriguez 1985]
It is well known from analytic geometry that every straight line in space can be expressed in the form:

(III.1)

where m ,n,a,b Î Â.

It is clear that when the coefficients m,n,a,b are fixed, the system (III.1) will represent a unique straight line in space, or in other words, that to every quadruple set of (m,n,a,b) Î Â⁴ will correspond a unique straight line in such a space.

If we make this four coefficients be in terms of a parameter z Î Â, then for every value of z we will have a well-defined straight line such that by varying this parameter the result will be a loci of straight lines, that is, in a surface made up of straight lines.

A ruled surface is that which is generated by a straight line (generating line) which moves according to a given mathematical law.

Let (III.1) be a variable straight line whose coefficients m,n,a,b depend on a parameter z.

In these conditions we will have that the equations:

(III.2)

which represent the parametric equations of a surface (two equations with a sole parameter whose isolation will give raise to the Cartesian form).

Figure III.1

where

will be its vectorial equation. Hence we will be able to say:

(III.3)

are also parametric equations for that surface.

Let us note that last equation of the preceding system can be verified by every point of space and in consequence z = z will be all ³.

It is clear that the parametric equations (III.2) and (III.3) represent the same surface, since the system (III.3) can be considered as intersection of the surface defined as the system (III,2) with z=z, that is, with ³.

Furthermore, taking into account that APPENDIX II demonstrated the parametrical expression of the surfaces through three equations with two parameters l and m, in this paper we have choosen to achieve it with system (III,3), and for convenience t and z have been selected as parameters.

REFERENCES
Granero-Rodríguez, Francisco. 1985. Álgebra y geometría analítica. Madrid and Mexico City:McGraw-Hill.