DEFINITION OF THE CONIC ELLIPSE
Taking into account the ellipse illustrated in the Figure I.1, the following points are defined:
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:
SURFACES AND CURVES ON A SURFACE [Granero
Hence, the equations:
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.
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 .
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.
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
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.
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:
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.
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.
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:
which represent the parametric equations of a surface (two equations with a sole parameter whose isolation will give raise to the Cartesian form).
Since the vectorial equation of the surface is defined by means of the position vector of a given point P(x,y,z) (Figure III.1), we have:
will be its vectorial equation. Hence we will be able to say:
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 Â3.
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 Â3.
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.