Beginning then, with the demonstation of the new property
of conics, let us first consider ellipse
where A straight line normal to ellipse
In order to prove this property of the ellipse, which is common
to all three conics, as further demonstrated, let us get at first,
the slope of the straight line Deriving the equation (1) it yields:
Hence, the slope of the straight line normal to the tangent
straight line Thus, it results that
is the slope being sought.
(but the reader should note that and beware). Again referring to Figure 2, making the calculation,
yields:
and since By substituting the previous expressions of
This represents the Cartesian equation of an ellipse for all
k Î Â, and by depending
on the parameter
- Identifying the semiaxis of E(M), we have,
Substituting those values into equation (6) gives rise respectively to the circumferences: ^{2} + y^{2} = (a + b)^{2}
x ^{2} + y^{2} = (a - b)^{2}- In the same way, making each one of the semiaxes equal to zero yields:
The equality With It can be easily proven that | a - b
Every generated point and generated conic remains defined
in their totality by a numerical parameter In the process of generating conics of the family E(M), we can find semi-straight lines or straight lines (depending on whether the generator conic is an ellipse, a hyperbola or a parabola). Thus, a straight line can be dealt with mathematically and geometrically as a conic belonging to the family. In the case of the generator ellipse, two circumferences are obtained that can be handled as conics within the family. As a corollary, the following can be stated: - A new concept appears: that of the "generator ellipse" or in general "generator conic" belonging to the corresponding family of generated conics
- It turns out that it is a novelty to handle the straight lines as ellipses with a null semi-axis .
This guideline for arranging the ellipses will be called the
'Arrangement Law' and shall correspond to a "generating
line" upon which all transformed points coming from a single
point of the generator ellipse shall rest. This means that each
Each point belonging to the generator conic creates lines
of the same family according to the 'Arrangement Law'. So, if
this 'Arrangement Law' is a linear function of or proportional
to the generating factor It is especially important to mention here in reference to matters of construction that those points belonging to the generating lines can be defined at once, by simply linking the homologous points between a straight line segment and a circumference. Thus a whole set of ellipses are obtained without the need to resort to traditional means for their generation. The simplicity of this mathematical treatment deserves to be noted as a remarkable property, in comparison to the fact that usually the only simple treatment of similar ruled surfaces in science and technology is a graphic one. It is outstanding that, thanks to the 'Arrangement Law', an infinity of surfaces,whether be linear or not, can be settled up. Substituting
which is the surface equation, whose cross-section curves are the ellipses of the family E(M), the z-coordinate being null in the generator ellipse plane, where all director curves or cross-section curves are located in planes parallel and perpendicular to the vertical axis Z. Although the study of this surface with this equation was
first undertaken with the aim of representing or characterizing
the corresponding mathematical model ( In Figure 5,
as it is demonstrated in APPENDIX III. Establishing the relation between the equation: and the fundamental trigonometric form: sin it is evident that:
and identifying the equations (8) and (9) it can be proved
that the paramenter z can be easily
removed to lead to equation (7),
- Semimajor axis
*A*and semiminor axis*B*of the elliptic base; - Height
*H*of the straight ridge; - Length
*L*of the straight ridge (parallel to*A*).
The general equation for all generated ellipses by the generator
ellipse with semiaxis
In the particular case for the base:
For the straight ridge, the corresponding ellipse would have
semiaxis
taking the center of the generator ellipse as the origin of
the coordinates, and considering it located between the straight
ridge and the base at a distance
From (11) and (13),
From (12) and (15),
From (13) and (16),
From (14) and (15),
With values
I
between the two straight lines (ellipses with a null semiaxis):
The two circumferences meet the condition that their semiaxis A and B are equal (A = B) for values
of k such that k = -a/b and k=
a/b.
All of what has been calculated above is shown in In order to obtain the straight lines (generating lines according to the concept
of ruled surfaces), it is sufficient to take several points on
the generator ellipse and apply them the values
Immisa
cross and the Greek cross. The Greek cross is a cross with
two arms of equal lenght that cross each other at right angles,
the point of intersection of the arms being at their centerpoints.
The Latin Immisa cross has arms of unequal lengths that
cross each other at right angles, the point of intersection being,
as in the Greek cross, at their centerpoints. This is different
from the more common Latin cross, in which the arms are unequeal
but which do not intersect at their middle point.The fact that each cross is composed of two arms perpendicular between each other, disregarding the difference in their lengths, made us think about the intersection of two surfaces with straight ridges that were rotated by 90°. On the other hand, since the base to be linked to the straight ridge consists of a circumference (in this particular case, of an ellipse), the only possibility we have, judging by all that has been dealt with in the preceding sections, is the employment of the family of ellipses E(M) expressed by equation (6), which we repeat here:
Endowing the former expression with a linear 'Arrangement
Law' of the form
where As was mentioned before, a surface that links the straight
ridge to the circular base will have to be obtained, and it can
easily be seen that such a surface comes about as the result
of extracting it from the general one (see the encircled detail
of The positioning of these figures is such that 'elemental surface
1' will correspond to the surface having both its major axis
of the generator ellipse and its straight straight ridge parallel
to the From what has been discussed up to this point, the project will consist of the interpenetration between the two surfaces rotated 90° with respect to each other, or in other words, all the cross-section or director ellipses of the two elemental surfaces, cutting each other perpendicularly. The following data is known: - Radius
*R*of the base circle; - Height
*H*between the ridge plane and the base one; - Lengths
*L*1 and*L*2 of the straight ridges of the elemental surfaces 1 and 2 respectively.
The equation from which we begin the further mathematical
development of the project is given by equation (22) in its Cartesian
form, where the linear relationship is given by The nex step consists of calculating the intersection between
each pair of perpendicular director ellipses that are located
in the same horizontal plane for every value of the parameter
The intent of the preceding considerations is to build three
point arrays, each one to store the coordinates ( The quest for the values of the semiaxis
The values The steps that follow consist of particularizing the aforementioned
values of
Immisa cross as that with arms that are unequal
and centered at a point located upon the vertical axis Z,
we will designate as L1 the length of the minor arm of
the cross parallel to the X axis and belonging to elemental surface
1. We will designate as L2 the length of the major arm
of the cross parallel to the Y axis and belonging to elemental
surface 2, in such a way that the values of the semiaxis and
the relative locations of every one of the elemental surfaces
will be determined according to the expressions (23).The expression for elemental surface 1, that is, that with
a minor arm that concurs with the direction of the
The expression for elemental surface 2, that is, that whose major arm concurs with the direction of the Y-axi, is deduced from equation (22) by a rotation of 90°, and is given by:
Additionally, we want to take into account the following relationships:
where the values
It is interesting to mention as well, as it is proved by making the calculations that: which justifies the fact that yields to B Eventually it can be verified that the sum of the semiaxis
We want to emphasize the fact that the generator ellipses
of the elemental surfaces do not lie in the same plane ( When the system made up of equations (24) and (25) is resolved, the result yields the loci of the space points common to the two surfaces from which the eventual cathedral is formed, loci that will be also be the outcome of the process of interpenetration. In consequence of what has been discussed above, the equations
that define the nonplanar curves that result from the process
of interpenetration for the case of Latin
The process of interpenetration consists, then, of first obtaining
the four possible points of intersection generated by each of
the ellipses belonging to elemental surfaces 1 and 2 situated
on the same plane defined by parameter For the next phase, the starting
point will be the previous calculation process that consisted
of selecting a determined number of direction curves that are
common to the two surfaces of the cathedral, calculating their
respective dimensions with reference to the plane containing
the generating curve and which will be determined by parameter
The next step is to determine all the points at which each
previously selected generating straight line cuts each and every
one of the direction curves also selected in accordance with
what was discussed in the paragraph above. Having obtained all
the points of intersection, a point matrix is created for each
of the From the point of view of construction, this cathedral project could be accomplished in two ways. One would involve the technique of reinforced concrete shells [Candela 1985; Faber 1970] for spans which are not too large, while the other would involve the use of metal structures for large spans. In either of these methods, one important feature is the absence of intermediate pillars, allowing completely open and diaphanous spaces. In those structures where the use of laminar reinforced concrete involves high building costs due in part to an excessive increase in thickness of the concrete layers, and where membrane stress calculations are no longer reliable for this type of structure, it is preferable to resort to metal structures involving the use of stronger, lighter and more resistant steel, making it possible to achieve spans of up to 300 metres without intermediate supports. Having obtained the point matrices, according to the construction
process using the technique of reinforced concrete laminar structures,
the phase of erecting the wooden formwork would involve raising
wooden piers of length After the formwork, the necessary reinforcing bars are positioned and the concrete poured. Once the concrete has set, the formwork is removed, revealing a curved structure in the form of a concrete shell of a certain thickness capable of providing adequate rigidity and strength to withstand and absorb the stresses generated. The final surface will create a totally open interior space with no internal columns to take away light and space, one of the merits of this type of construction, devised by Spanish architect Félix Candela [Candela 1985; Faber 1970]. Designed with care, these surfaces would allow the creation of openings to accommodate rose windows, and other large apertures to let light enter. Glazing these openings with suitably coloured glass and engraving with appropriate religious images, in conjunction with the design of the top part of the space as a transparent cross, would project an extraordinary halo of light onto the altar into the interior of the cathedral. The result of our cathedral project with the Latin
It can be easily verified, as in the case of the Latin In the discussion up to this point, we have worked out the
values of
For elemental surface 2, whose straight ridge lies parallel
to the
In the previous expressions it can be observed that
In The final result of the Greek cross project is illustrated
in
Click here to go to Appendix I, Appendix II and Appendix III.
Candela, Félix. 1985. Faber, Colin. 1970. Granero-Rodríguez, Francisco. 1985.
Martín-Zorraquino, J.V, Francisco Granero-Rodríguez,
José Luis Cano-Martín, J.J. Doria-Iriarte, J.J.
1999. "A Novel Version of the Cathedral Inspired in the
Already Built one in Rio de Janeiro". Pp. 357-364 in Martín-Zorraquino, J.V.. Francisco Granero-Rodríguez, J.J.Doria-Iriarte, J.J. 1997. "Otras Aportaciones y Resultados sobre la Nueva Familia de Figuras Geométricas Elementales con Cónicas". 9º International Congress of Graphic Engineering. Bilbao - S. Sebastián: UPV-EHU. Martín-Zorraquino, J.V., Francisco
Granero-Rodríguez, J.J. Doria-Iriarte. 1998. "New
Properties of Conic Sections and the Ruled Surfaces Deriving
from them. Selected Applications in Architecture and Engineering".
Pp. 431-438 in Martín-Zorraquino, J.V., Francisco Granero-Rodríguez, J.J. Doria-Iriarte. 1993."Superficies Tridimensionales de Secciones Cónicas". Patent number P9301662/1993.
Dr. Granero Rodriguez is industrial engineer and Lecturer in Applied Mathematics at the High Technical School of Engineering in Bilbao. He has written several research workings and several books on mathematics, five of them published by the publishing house McGraw-Hill. Dr. Cano Martín is industrial engineer. He works as an engineer for a large firm located in the Basque Country.
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