Research Paper FUNDAMENTALS OF DESCRIPTIVE … Int. J. Struct. & Civil Engg. Res. 2014 Andrea Donelli, 2014 FUNDAMENTALS OF DESCRIPTIVE GEOMETRY: APPLICATIONS FOR ARCHITECTURE AND

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Int. J. Struct. & Civil Engg. Res. 2014 Andrea Donelli, 2014

FUNDAMENTALS OF DESCRIPTIVE

GEOMETRY: APPLICATIONS FOR

ARCHITECTURE AND ENGINEERING

Andrea Donelli¹*

1 Department of Civil, Environmental, and Mechanical Engineering, University of Trento, Italy, via Mesiano, 77 – 38100 Trento, Italy.

*Corresponding author:Andrea Donelli � [email protected]

ISSN 2319 – 6009 www.ijscer.com

Vol. 3, No. 4, November 2014

© 2014 IJSCER. All Rights Reserved

Int. J. Struct. & Civil Engg. Res. 2014

Research Paper

INTRODUCTION

Architecture, engineering, and the environmentgenerate and form relationships amongstthemselves; geometric properties areessential for satisfying these conditions. Theprincipal and necessary reason forgeometrically representing drawings is definedby the biunivocal correspondence attributableto the projection between entities;consequently, it is also possible to verify theirmeasurements and dimensions. In addition,

This paper aims to reflect on the issues that concern the fundamentals of descriptive geometry,in particular the roles it plays in research, analysis, and design in the fields of architecture andengineering. Its foundations, like its history, are legitimate and up-to-date for the representationof drawings, thus rendering specific the contribution made by the rules of the elements ofprojective geometry. The fundamentals of projective geometry are, in fact, made explicit by theoperations defined by the properties of projection and section for the components of therelationships they describe by modelling, even in three-dimensions, the figures that cangeometrically satisfy such combinations. The research undertaken with projective geometry iscurrent/timely, because it refers to the need to finalize the project drawing for architecture,engineering, and the environment, as well as for its defined and codified method of representationthat establishes the feasibility of the project.

Keywords: Descriptive geometry, Fundamentals, Architecture, Engineering

values that are intrinsic are also representedgraphically, as they are specific and traceablefor and in the graphic representation. Suchrepresentations attest to the method andprocedure that belong to the logic of projectivegeometry. These alphanumeric values areexpressed by geometric entities; the mostwidely used and recognized, and immediatelyquantifiable, are specified by the differentscales of numerical ratio. Conversely, it is therelationship values that are determined by

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images understood as projections appliedwith the rule of orthogonal projection.

Perceptive geometric values obtained withthe application of perspective whose imageis fixed on the surface of the projection planeare also part of graphic representation. Thisplane is modelled and determined by arelationship that is set up and ensured by thegeometric operation, for example withoptimization for binocular vision. Anotherdescriptive system that may be used forgraphic research has the goal of representingthe relationships formed due to thepermanencies deduced from the environmentaldimension of the territory or urban area. Thesebelong to the historical design of theenvironment, or ground, as well as to its builthistory. This contribution is describedgraphically through concordances ofgeometric tracings necessary for recognizingwhat the geometric generators form,harmoniously coordinated in a drawing thatdefines systems from which one can derivelines, relationships, positions, measurements,and ratios.

Considering the codification of the doubleorthogonal projection (Monge) as anestablished and indisputable fact for drawing,descriptive geometry in its elements ofprojective geometry initiated a process ofordering through codification and formalizationapplied over a lengthy period, extending fromthe middle of the eighteenth through the entirenineteenth century. In recent times, drawing hasbecome a pragmatic technique for describingbuilt works. It has been disciplined andstandardized (ISO–UNI) and is still used in arigorous manner, particularly in the sphere ofmechanical drawing. The understanding of

descriptive geometry, as verified andoptimized, derives from centuries of knowledgeof drawing for building, from Villard deHonnecourt (XIII sec.) to Eugène EmmanuelViollet-le-Duc (1814-1880), as demonstratedby the eloquent case of stereotomy whoseprincipal goal was the cutting of wood andstones according to geometric principles. Inthe same way, the topic of statics, or graphicrational mechanics, is of significantimportance and without doubt deserves an in-depth chapter of its own.

The word geometry derives from the Greekγεωμετρια, which is measurement of theground. The system of measurement can alsobe recognized in environmental, territorial, orurban relationships; analogously, one could saya “topography”, or correlation between thedesign of the ground with architecture andwaterways. These correspondences becomemeasureable, as has been said, through thesearch for geometric generators that bringback to a unity of relationships and cognitionsthe content of the drawing of the ground, findingtherein the set of relationships, codes, andcharacters of the historical built ground of theplaces in measurements.

DISCUSSION

The drawing, intended as descriptivegeometry, is the condicio sine qua non for allgraphic operations carried out in accordancewith the rule of logic. On this topic, Aldo Rossimade an intelligent consideration in what wasone of his final most important acts forarchitectural culture, and certainly among themost evolved for Italian architecture. Relativeto the “Terza Mostra Internazionale diArchitettura”, held in Venice in 1985, within the

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remarkable catalogue that accompanied theexhibition, Rossi inserted in his introductoryessay the term “drawing” as a prefix and suffix,attributing it to all of the other disciplines relatedto architecture and engineering. This culturedand attentive observation demonstrated howdrawing was without any doubt not only usefuland necessary, but in particular also leads tothe clarification of theoretical and constructivefacts. As if to say that the geometric partitionsof a built work transcend the architecturalspace and their constructive-structuraldimensioning. Considering geometric value asa means and defining the thicknesses withregular geometric figures derived from thespace of the compositional unit, we thenidentify exactly the points that in turn generatethe geometric arrangement of further parts ofthe building. The beginning of this graphic/geometric search for the architecture is crucial,as it is reflected in what one can generate inthe logical layout/tracing from the combinatorialsystem with the drawing of the ground. It shouldbe noted, therefore, that there exists a onenessbetween geometric and constructive thoughtthat gives rise to that which in the past was theintrinsic value of doing, as it was for measuring,composing, and drawing, or rather theintelligibility of things and facts.

Henceforth in this paper we will considerideally the terms “architecture” and“engineering” as a single expression, since theancients did not set forth differences orseparations between them, because they didnot exist. In the same way, the term drawingmay be considered synonymous with researchand project, or plan. In the past, in fact, it wasconventional wisdom to unite and give strengthto research and its implementation in the

concept of the indissoluble unity of things, inthe same way that projective geometric thoughtassumes the same value and meaning.

“To represent scientifically signifiesacquiring and transmitting the knowledge ofthe form of a real or imagined entity and therules that underlie this” (Catalano M G). Withthis assertion, appearing in an essay titled, “Atheorem for the unification of the methods ofthe science of representation,” by G MCatalano in the journal Disegnare n. 8/1994,the author expresses very well, and in brief,the concept of projective geometry. Drawingis therefore a science and it is thus because itconsists of rules that projective geometry hasdefined in considering the concept of unitypassed down from the experience of analyticalgeometry. The evolution that goes fromEuclidean thought to that of projection, with thecodification of double orthogonal projection,for the relationship of biunivocality of theentities projected onto two planes with thesubsequent overturning of a plane coincidentwith the other, was first expressed by GaspardMonge (1746-1818) in his treatise,“Géométrie descriptive”. Geometric thought asit relates to descriptive geometry is notdivisible or distinguishable a priori accordingto necessity or circumstances. It consists,rather, of a single attitude, that considers thewhole experience codified as a unitaryexperience that can be reproduced andrepresented (Figures 1 and 2). In this way, weclearly obtain a useful, ordered sequence, withthe single properties belonging to rationalgeometric thought (Figure 3). In fact, these arestudied initially as rules and are distinguishedin terms, considering obviously that theycombine as unitary projective operations in

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graphic renderings, in particular in theapplication of homology (Figures 4-8). Theproperties, therefore, are distinguished anddetermined in: definitions, concepts, orprimitive entities, which are the point, the line,and the plane; theorems, corollaries, inversetheorems, postulates or axioms and, finally,sets, which by their definition are part oftopological geometry. A pragmatic assumptionrelated to the concept of unity forrepresentation in descriptive geometry can beconsidered as a unified synthesis of graphicaloperations. This can also be demonstrated bymeans of a simple example. From a singlebasic shape drawn in plan and represented indouble orthogonal projection with relativeelevations, it is possible to integrate and obtainan axonometric projection and a perspectiveview, in the same way that it is always possibleto elaborate from the same initial basic shapein projection a homological relationship,including the consequent overturning of theplane. In conclusion, in a single definedrepresentation exist all of the operations ofdescriptive geometry (Figure 9). Traceable tothese through the projections, in fact, are therelationships of the entities and thebiunivocalities, which also describe the truesize and forms, the intrinsic passages, and theaccuracy of the graphic relationships. Througha further operation of projection that transformsthe geometric figures, altering theirappearance, which in any case between theoriginal form and its projections remains,permitting one to pass from one figure to thenext by means of a finite number ofcombinatorial operations of projection andsectioning; with these imagines, thus,homography is defined.

Figure 1: Terms of membership, rule: apoint belongs to a straight line when theprojections of the point belonging to the

projections of thehomonymson thestraight line

Figure 3: Graphic diagram of the basicstructure of the projection system by

finite distance and infinity

Figure 2: Orthogonal projections: thesystem of rotation of a plane of projection

of the plane containing the segment.(segment A -B in true greatness)

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Figure 4, 5, 6: Graphic diagrams of theconstruction of homological relationship.

The homology is the relation ofcorrespondence between the points of

two generic figures

Figure 9: With a single representationdefined there all the operations of

descriptive geometry

Figure 8: Drawing a simple holidayhome, showing the application ofhomology with overturningplan

Figure 7: Scheme derived from thebook : Migliari R., ”Geometriade

scrittiva”, Cittàstudiedizioni

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As we have mentioned, the most importantgeometric entities involved in geometricoperations are the values deduced fromEuclidean experience, that is the point, the line,and the plane, to which further relate thedirections of the lines and the position of theplanes (Figures 10-13). This spatial dimensionallows us to move from a finite andcircumscribed condition to a larger dimensiontending toward infinity by virtue of theoperations of direction of the straight lines andof the position of the planes, guaranteeing therelationships of belonging. Girard Desargues(1591-1661) set forth these concepts, in termsof the direction of the straight line and theposition of the plane, so that geometric thoughtwas amplified beyond what was Euclideanspace. The illustrative example isdemonstrated by the biunivocal, or one-to-one,relationship of a dotted line r and its projectionr’, just as we obtain the image I’r from thedirection of a line r. With these entities we havethe demonstration that these are also themeeting points forming a part of theperspective. Keeping in mind that orthogonalprojection originates from a point at infinity,unlike perspective whose centre is a properpoint (Figure 14).

Piero della Francesca (1417-1492) longbefore Desargues’ speculations and thecodification of double orthogonal projection byMonge, thought in a utilitarian way about theconcept of drawing in orthogonal projection.The use of perspective through the preparatoryuse of associated orthogonal projectionsaccording to the written contribution of AndreaCasale in Disegnare n. 12 /1996, shows usPiero aware of a concept that is controlled bythe projective system and above all by the

Figure 10: Representation of a straightrule: two straight lines parallel to each

other have the homonymsparallelprojections

Figure 11: Representation of the plan:conditions for belonging

Figure 12: Representation of the plane

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Andrea Palladio (1508-1580), unmindful ofMongian double projection for obviousreasons of chronology, may have been awareof speculations about perspective from culturalsources, first of all from Piero dellaFrancesca’s treatise “De prospectivapingendi” and then from studies by AlbrechtDürer (1471 – 1528). In his treatise, “I Quattrolibri dell’architettura”, Palladio drew plans andelevations using double orthogonal projection(Figure 16). Obviously, these representationsare incorrect according to Monge’s rules, sincePalladio was not specifically aware of therelationship between the projection planes,much less how to arrive at the second imageusing the line running perpendicular to theground line on the frontal plane. Consequently,his drawing is drawing of utilitas which,however, still guaranteed relationships ofgeometric origins, since the decision that laywith him did not involve, as it does in today’smodern culture, having to juxtapose theconceptualism between subject and object. Infact, what occurred with Palladio is a simpleconsideration, even in the absence ofperspective codifications that became explicitand more than ever crucial to clarifying and

Figure 13: Applications: projectingplanes and straight lines of a plane

Figure 14: Graphic diagram ofperspective in painting vertical pointcorrelation method and verification

exercise by other processes

geometric value that he looks for in expressingexperiences between measurement andperception (Figure 15). If the measurement isobtained by verifying the representation inorthogonal projection, the perception evokingother “measurements” is obtained fromperspective representation .

Figure 15: Drawing orthographicprojection deduced by Pierodella

Francesca. See diagram presented inthe journal “Disegnare” n. 12/1996 - pp.

15 -23 by Andrea Casale

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recognizing the passage of integrationbetween Euclidean geometry and projectivegeometry. In Palladio’s day, the main concernwas to bring theoretical dignity and scientificstructure to knowledge. Palladio himselfinitially began working as a stonecutter, first inthe workshop of Cavazza and later in the moreprestigious bottega of “Pedemuro”, where helearned the basics of drawing. However, helater expressed his potential at the Accademiadel Trissino, where he completed classical,

mathematical, and theoretical studies.Measurement, logic, the rationality of building,and understanding the raison d’être ofarchitecture, as a practical activity, as scientificand theoretical knowledge, and as artisticexpression were the factors that motivatedPalladio to study the past, and this occurred,above all, through his survey drawings andredrawing of ancient architecture. Despite thedifficulties and dangers of the time, Palladiotravelled to Rome because of his interest inthe ancient world and for the city as it was andas it had been, which he initially studied andthen experienced first hand in order tounderstand its architecture, seizing the relativelogic of know-how and observing that which islasting, distinguishing it from the superfluous.In fact, this approach leads back to ameasurable order, in representation, or ratherit generates that which can be defined in themeasure of reason, and therefore thisstipulates consequential and reciprocalrelationships between drawing (project) andconstruction, as is, indeed, the raison d’êtreof a principle of indissoluble unity.

The continual clarification of studiesregarding descriptive geometry is attested toand improved upon by the graphic experienceobtained through the operations of projectionand section and, in particular, once again withthe codification of the double orthogonalprojection of Gaspard Monge. Monge’s goalwas three-dimensional representation, therealization of controlled models regardingfigures. He carried out these speculations bothanalytically and graphically, clarifying with greatattention the title of the study and thedisciplinary expression; in fact, he refers toanalytical descriptive geometry and graphical

Figure 16: House Villa of Andrea Palladio,drawing taken by facsimile relatingto the Treaty of the “Four Books ofArchitecture”edizioniHoepli, Milano

1980. p.64.

Note: How the concept of dual by projecting orthogonal codingis different from the Gaspard Monge. In fact, there areno distinct from the stakeout sign of hatching of theprojections, and as the first projection (building plan),considered as the horizontal section is placed on the(floor of the framework and not as a first screening onwhich it is geometral.

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descriptive geometry. On Monge’sfundamental contribution to analytical and,above all, graphical geometric thought, see inparticular the Neapolitan school and VincenzoFlauti (1782-1863). Flauti explained andadded, according to the study conducted byprofessor Vito Cardone, some definitions thatwere not always spelled out clearly in Monge’swritings. One particularly interesting exampleis his contribution to projection planes(Cardone V, 1998). Flauti had no otherparticular accolades, while very interesting isthe debate set in motion by the magazine XYdimensioni del disegno with regard to thefigure of Flauti, a Monge scholar. See on thissubject the contributions of Riccardo Migliariand Vito Cardone listed in the bibliography.

A final consideration, as revealed by thecontent of the present paper, has to do withthe concept of measurement obtained from thedesign of the ground in relationship to builtform, or rather the search for geometricgenerators. The purpose of this research is todetermine the relationship of measurementsthat occur between the historical design andthe built work over time. In fact, it establishesprecise points that clearly define geometry withthe construction of built works, which impliesan order derived from the measurement of theground. Historically relevant to these ideas isthe reading of maps of plans and cities byOpicino de Canistris, in the science ofmeasurement, whose purpose is to juxtaposenumbers and figures referring to the form ofthe ground that contribute to the magnitude ofthe architectural form. A further task undertakenwith the search for geometric generators is tobring back these concepts to design as well:that is, to translate the dimensions that emerge

from the relationships into another set ofrelationships for the new building. To this endwe propose an example of research carriedout in the locality of Vigasio, in the provincesof Verona in Italy. The drawing that describesthe habitat of Vigasio reveals the search forconnections between the parts derived fromthe drawing of the ground in relation to thehistorical, rural building. The research inquestion was conducted between December2004 and January 2005, in part on the basisof work already carried out in 1984 withprofessor U Tubini at the IUAV, department ofarchitecture in Venice, who proposed theanalysis of the Piazza dei Signori in Veronaand later with the students in the fifth-yearArchitectural Composition course, during the2003-2004 academic year, in the degreecourse in Civil Engineering in Trento, on theagri and urbis form of the city of Trento. Thetheme of the paper concerning the graphicrepresentation is the geometric generatorsderiving from the ancient systems that belongto the surface reading of the nature of theground with their historical design. Thecircumferential arcs intercept a system ofparallel and oblique lines obtained andrendered graphically by the recognisability ofthe Roman system of centuriation, obtainingthe value attributed to the unit of the meter of710. Other, more ample circumferential arcsintersect, with their radii, the lines belongingto the drawing of the ground, which in turndefine the rural structures by tangency points.Additionally, starting from the point that definesthe origin of the radius, one can drawsubsequent sub-multiples of circumferentialarcs which in turn define other tangential pointswith the tracing of the ground corresponding

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to the built structures, in the same way that wedetermine the entry point of the so-called“Leona” trench, the essential waterway of theTartaro river, its main waterway. From this pointarise other tracings of lines that are geometricgenerators which, passing through otherknown points, in turn form, with other lines, anintersection coincidental with the dimension ofthe historical or rural buildings (Figure 17).These results describe graphically how evenan historical drawing of the ground interactswith the site, establishing a dimension thatserves as the measurement and geometricconstruction of the same. From theseconsiderations it is possible to determine thecontrol of the built form within the geometricrelationships that fix the ancient measurementsof the ground of their division and subdivisionwith the superimposition of the measurementsdeduced from the cadastres: measurementsthat taken together determine distances,alignments, and lineaments useful forcontextualising, even according toenvironmental criteria, the system of the designof the built work.

CONCLUSION

The ways of describing graphically are rooted

in the ground of our culture. As is evidenced in

literature as well, in Plato’s Timaeus, in

particular in Chapter XXI, the explanation of

the solid figures of geometric expression fixes

a principle of regulation of geometry itself with

nature, and consequently man. Analogously, in

his treatise, “Writings”, Le Corbusier argues

that civilization is founded on geometry and

that man lives, practically, by geometry alone.

He continues, affirming that “The specificity of

man is to establish what is at right angles with

respect to him; this causes him to classify,

order, and see clearly in front of himself”. Again

referring to civilization, he makes the point that

through geometry man has found the way to

measure space by using coordinates on three

perpendicular axes. The task that Le Corbusier

wisely suggests we take on derives from the

fact that: “geometry is reflected in that work made

by man that extends from the house to the site”.

In fact, broadening the study based on graphic

geometric-mathematical relationships, one can

observe that in addition to the established solid

figures such as the cube, the cylinder, the sphere,

and the round, pyramidal cone, generated by the

rotation of a right-angled triangle around a

cathetus, the contribution made by descriptive

geometry takes the form of measurements which,

in turn, generate logical relationships such that

we define an indivisible unit according to a

rationality that fixes universal graphic

geometrical and mathematical codes applicable

to search for an understanding of space for the

project in architecture and engineering.

Figure 17: Historic town ofVigasio(Verona) Italy:Search for generating

geometrical measure of historical designof the soil and geometric relationships

defined by the intersection of thesystems of the relationship between soil

and historical buildings

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