ellisse by Lawrence Kasparowitz

ellisse*

(* Italian for ellipse)

a study of the elliptical form and the Baroque architecture of

Giovanni Lorenzo Bernini Francesco Borromini Guarino Guarini compiled and designed by Lawrence Kasparowitz

front cover image: Geometrical analysis of San Carlo alle Quattro Fontane, Rome (Francesco Borromini, architect)

First edition August 2017

Table of Contents

Introduction Baroque architecture 1 in Italy

Giovanni Lorenzo Bernini and 4 Piazza San Pietro, Rome Francesco Borromini and 10 San Carlo alle Quattro Fontane, Rome Guarino Guarini and 15 Real Chiesa de San Lorenzo, Turin Oval vs. Ellipse 21 Medieval Carpenters 23 Constructing an Ellipse 26 Bibliography 40 Colophon

Introduction During my academic studies at California Polytechnic State University at San Luis Obispo, I had the opportunity to spend my fourth year abroad in Florence, Italy. I fell in love with the city and the Renaissance which it represents. Forty ve years later I nd myself thinking about the Baroque period and Italian architects which represent that period in Italy. This study focuses on three architects - Bernini, Borromini and Guarini. I selected one work for each of them that represents their contribution. I am also fascinated with the application of geometry in architecture. This study is focused on the ellipse which to me represents the Baroque theme in architectural history. At rst I used the term ‘oval’, which I quickly learned was not the correct term for the geometric shape I was referring to. There is a discussion included which clari es why ‘ellipse” is the correct term and the correct shape. How did the master builders and designers in the Baroque period layout an ellipse? How does anyone draw an ellipse? The answer to those questions led me to include articles from a master carpenter (who obviously has a math background) as well as a professor of mathematics who gives the methods of constructing an ellipse. But how would the architects and builders of the seventeenth century build the ellipitical forms? Ross King in his book Brunelleschi’s Dome gives a glimpse of how Filipo Brunellschi would do it: “The master builders of the Gothic cathedrals regulated the curvature of such structures by rst plotting them in full-scale, like a giant set of blueprints, on to special tracing loors. These loors were covered in plaster of Paris onto which life-size geometrical designs of, say, a vault’s ribs would be drawn. Once these drawings were complete, carpenters used them to devise the wooden templates from which the stone for the ribs was shaped by the masons working at the quarry. The gypsum loor was a terward wiped clear and the next set of drawings incised into its surface. If facilities for tracing loors did not exist, an area of ground would be cleared and the designs sketched in the soil. “

Baroque Architecture in Italy The sacred architecture of the Baroque period had its beginnings in the Italian paradigm of the basilica with crossed dome and nave. One of the rst Roman structures to break with the Mannerist conventions exempli ed in the Gesù, was the church of Santa Susanna, designed by Carlo Maderno. The dynamic rhythm of columns and pilasters, central massing, and the protrusion and condensed central decoration add complexity to the structure. There is an incipient playfulness with the rules of classic design, still maintaining rigor. They had domed roofs. The same emphasis on plasticity, continuity and dramatic e fects is evident in the work of Pietro da Cortona, illustrated by San Luca e Santa Martina (1635) and Santa Maria della Pace (1656). The latter building, with concave wings devised to simulate a theatrical set, presses forward to ll a tiny piazza in front of it. Other Roman ensembles of the period are likewise su fused with theatricality, dominating the surrounding cityscape as a sort of theatrical environment. Probably the best known example of such an approach is trapezoidal Saint Peter's Square, which has been praised as a masterstroke of Baroque theatre. The square is shaped by two colonnades, designed by Gian Lorenzo Bernini on an unprecedented colossal scale to suit the space and provide emotions of awe. Bernini's own favourite design was the polychromatic oval church of Sant'Andrea al Quirinale (1658), which, with its lo ty altar and soaring dome, provides a concentrated sampling of the new architecture. His idea of the Baroque townhouse is typi ed by the Palazzo Barberini(1629) and Palazzo Chigi-Odescalchi (1664), both in Rome. Bernini's chief rival in the papal capital was Francesco Borromini, whose designs deviate from the regular compositions of the ancient world and Renaissance even more dramatically.

Acclaimed by later generations as a revolutionary in architecture, Borromini condemned the anthropomorphic approach of the 16th century, choosing to base his designs on complicated geometric gures (modules). Borromini's architectural space seems to expand and contract when needed, showing some a nity with the late style of Michelangelo. His iconic masterpiece is the diminutive church of San Carlo alle Quattro Fontane, distinguished by a corrugated oval plan and complex convex-concave rhythms. A later work, Sant'Ivo alla Sapienza, displays the same antipathy to the lat surface and playful inventiveness, epitomized by a corkscrew lantern dome. Following the death of Bernini in 1680, Carlo Fontana emerged as the most in luential architect working in Rome. His early style is exempli ed by the slightly concave façade of San Marcello al Corso). Fontana's academic approach, though lacking in the dazzling inventiveness of his Roman predecessors, exerted substantial in luence on Baroque architecture both through his proli c writings and through a number of architects whom he trained and who would disseminate the Baroque idioms throughout 18th-century Europe. The 18th century saw the capital of Europe's architectural world transferred from Rome to Paris. The Italian Rococo, which lourished in Rome from the 1720s onward, was profoundly in luenced by the ideas of Borromini. The most talented architects active in Rome — Francesco de Sanctis (Spanish Steps, 1723) and Filippo Raguzzini (Piazza Sant'Ignazio, 1727) — had little in luence outside their native country, as did numerous practitioners of the Sicilian Baroque, including Giovanni Battista Vaccarini, Andrea Palma, and Giuseppe Venanzio Marvuglia. Besides their complex ground-plans, the resultant curving walls were, therefore, the other outstanding characteristic of Baroque buildings. Not only did they accord with the conception of a building as a single entity, but they also introduced another constant of the Baroque, the idea of movement, into

architecture, by its very nature the most static of all the arts. And indeed, once discovered, the undulating motif was not con ned to walls. The idea of giving movement to an architectural element in the form of more or less regular curves and counter-curves became a dominant motif of all Baroque art. Interiors were made to curve, from the Church of S. Andrea al Quirinale by Giovanni Lorenzo Bernini, one of the main creators and exponents of Roman Baroque, to that of S. Carlo alle Quattro Fontane or S. Ivo alla Sapienza by Borromini, his closest rival. So too were facades, as in almost all Borromini's work, in Bernini's plans for the Palais du Louvre in Paris, and typically in the work of Italian, Austrian, and German architects. Even columns were designed to undulate. Those of Bernini's great baldacchino in the centre of St Peter's in Rome were only the rst of a host of spiral columns to be placed in Baroque churches. The Italian architect Guarino Guarini actually evolved, and put to use in some of his buildings, an 'Undulating order', in the form of a complete system of bases, columns, and entablatures distinguished by continuous curves. Even excepting such extremes, during the Baroque period the taste for curves was nonetheless marked, and found further expression in the frequent use of devices including volutes, scrolls, and above all, 'ears' - architectural and ornamental elements in the form of a ribbon curling round at the ends, which were used to form a harmonious join between two points at di ferent levels. This device was adopted primarily as a feature of church facades, where they were used so regularly as to be now perhaps the readiest way of identifying a Baroque exterior. In spite of their bizarre shape their function was not purely decorative, but principally a strengthening, functional one.

Giovanni Lorenzo Bernini and Piazza San Pietro, Rome

Giovanni Lorenzo Bernini was one of the most in luential and successful Baroque artists of all time. Bernini worked for several Popes before and a ter Alexander, and was beloved by most of them. His most signi cant title was “Reverenda Fabbrica”, the Architect to St. Peter’s, which he was given in 1629 by Pope Urban VIII. Bernini also had a number of projects all over Rome that he completed under the commissions of Popes, or other wealthy patrons. Particularly during Alexander’s reign, even if Bernini was not in charge of a project, he would o ten still have a say in how the project was to be completed. Bernini was busy working on his many projects up to his death in 1680. As with many of his papal patrons, Bernini had a very strong relationship with Alexander. Bernini and Alexander had fairly similar tastes in art which facilitated their friendship and their working relationship. Alexander would go to Bernini’s house, foundry or studio nearly every day to chat and discuss plans and progress for all of their projects1. Alexander’s agreement with and fondness for Bernini’s philosophy that “princes must build grandly or not at all” also contributed to their working relationship. It is clear that Alexander was familiar with Bernini’s work and had him in mind for the architect to design Piazza San Pietro before becoming Pope, because Alexander commissioned sketches for the colonnades from Bernini on the day of his election in 1655. Carlo Rainaldi, another architect, had been making plans for the piazza during Innocent X’s reign, and Alexander did give some consideration to these plans, but Bernini won the nal commission for

the piazza. Two years of discussion and planning went into the piazza. Bernini had to create a piazza that was pleasing to his patron, Alexander, but also satisfying to the Congregazione della Fabbrica di San Pietro. As with any major construction project, lots of thought was given to designing Piazza San Pietro which was subject to a number of requirements and considerations. One of the most fundamental purposes of the piazza is to be a place where mass can be said by the Pope. This implies that the area of the piazza should be large to hold as many people as possible. Since the former piazza was much smaller than Bernini’s creation, Bernini had to balance the need for grandiosity and a large open space with the number of buildings that would need to be demolished. Bernini chose to enlarge the piazza and remove the buildings to achieve a sense of awe, to hold large numbers of people, and also so that the piazza could serve as a parking lot for carriages. Bernini had many other requirements handed down from the Vatican that he had to take into consideration while drawing up plans for the piazza. The piazza had to be oriented in such a way that the Pope’s traditional location above the central entrance to St. Peter’s Basilica be visible to as many people in the piazza as possible. Also, nothing should obstruct the view of the piazza from the Pope’s private apartments in the Vatican. It was also important that the original entrance to the Vatican be retained, where Bernini later added the Scala Regia leading up to the Papal apartments. Much consideration was given to the basic shape of the piazza, while everything else was being kept in mind. A ter a great deal of discussion the Congregazione, Bernini and Alexander agreed on an elliptical shape. A trapezoidal component is attached to the ellipse which encloses the Piazza Retta directly in front of the basilica. This portion of the piazza was designed to compensate for an architectural mistake made by the previous Architect to St. Peter’s, Carlo Maderno. When designing and constructing the façade of the basilica, Maderno planned in space for two large towers, one on each side of the dome. It turned out that these towers were much too ambitious, and they were never successfully built (either by Maderno or later by Bernini). The lack of towers le t the basilica looking much too wide for its low overall height.

Bernini cleverly used the shape of the piazza and the colonnades that de ne that shape to create an optical illusion that helps the façade appear to have more traditional proportions. Since the straight portions of the colonnade lare out as they approach the basilica, the entire building appears closer to an observer in the main part of the piazza than it really is. Overall, Bernini did an incredible job utilizing previously existing elements in conjunction with considerations regarding the functionality of the piazza to create a very well-de ned shape to his piazza. Bernini created one of the most impressive examples of Baroque art of all time, and he managed to do it with numerous restrictions and constraints. He had the support of the Pope, but had to contend with the Congregazione of Cardinals to achieve his artistic goals. The beautiful curving oval colonnades that evoke emotions of re lection and awe achieve the theatrical result that is the goal of so many Baroque pieces. This particular piece was so e fective and so beautiful that it has never been signi cantly criticized, even during periods when it was popular to criticize Baroque art. Piazza San Pietro leaves no doubt in anyone’s mind that Bernini was a master of his art. Piazza San Pietro: the Rhyme and Reason Behind Great Art Jenne Driggers Honors in Rome - Summer 2006

Fig. 1 Axial layout of the Basilica and the Piazza.

Fig. 2 Site plan showing the basilica and the ellipse

Fig. 3 Aerial view (Google Earth)

Francesco Borromini and San Carlo alle Qua ro Fontane, Rome An iconic example of Italian Baroque architecture, San Carlo alle Quattro Fontane was designed and built between 1638 and 1641 by Francesco Borromini (1599–1667). The church was the Italian architect’s rst independent work a ter assisting on projects for his uncle, Carlo Maderno, and his eventual architectural rival, Gianlorenzo Bernini. It also became one of Borromini’s last works, as he returned to complete the façade nearly 30 years later, circa 1665–1667. Commissioned by the Spanish Trinitarian Order in Rome at the behest of Cardinal Francesco Barberini, San Carlo alle Quattro Fontane is located on the southwest corner of the intersection of Strada Pia and Strada Felice. A fountain marks each of the four corners of this intersection; thus the reference to the “quattro fontane” in the church’s name is a reference to its location. The church is also known by the name “San Carlino” (little Saint Charles) because of the awkwardly small plot on which it was constructed. The church is dedicated to the Trinity and to San Carlo Borromeo, the Counter-Reformation Archbishop of Milan. The plan for San Carlo alle Quattro Fontane demonstrates Baroque aesthetic sensibility because of its innovative spatial geometry. The oblong or “pinched oval” plan deviates from the Classical geometry that is characteristic of previous High Renaissance architectural plans. The main altar is situated in direct sight line of the main entrance to the church. Two side altars form a squeezed cross-plan, while the columns and entablatures of the wall decoration o fer undulating and rhythmic lines

throughout the interior space. Instead of a rounded dome, Borromini stretched the church’s dome to heighten the tension of the space. The interior decoration of the dome contributed to this tension by exaggerating the implied perspective of the co fers as they move toward the central lantern at its apex. Hidden windows, a characteristic feature of Baroque architectural and sculptural complexes, are tted into the base of the dome to illuminate its interior. Similarly, the exterior of San Carlo alle Quattro Fontane deviates from the Classical architectural vocabulary that was revived by High Renaissance architects in the period before it was built. Instead, Borromini chose distinctly curvilinear forms that contribute to the dramatic visual vocabulary of the Catholic Church during the Baroque period. Notably, Borromini’s façade for San Carlo alle Quattro Fontane is distinguished by its alternating concave and convex bays, which ebb and low in sculptural rhythm and suggest a greater continuity between exterior and interior space than High Renaissance lat façades. The undulating line of the cornice divides the lower and upper stories of the façade, whereas Corinthian columns thrust the design upwards. Deep niches feature Antonio Raggi and Sillano Sillani’s sculptures of San Carlo Borromeo as well as sculptures of the founders of the Trinitarian Order, Saint John of Matha and Saint Felix of Valois, both of whom were accorded special cult status by the pope in the 1660s. The façade has deeply cut recesses that emphasize light and shadow and give an overall sculptural feeling to the building. It is crowned with a large cartouche echoing the oval form. A second façade faces the corner and incorporates a pre existing fountain. Importantly, with San Carlo alle Quattro Fontane, Borromini did not copy a Baroque style so much as create one. Both the plan and the exterior of the church were imitated in northern Italy, northern Europe, and European colonies. In this way San Carlo alle Quattro Fontane set a precedent for Baroque architecture that helped to de ne the style as intense and theatrical, characterized by dramatic explorations of form. The College Board, 2012

Fig. 4 The ellipse within the sanctuary

Fig. 5 The building plan with garden

Fig. 6 View looking up at dome

Guarino Guarini and Real Chiesa de San Lorenzo, Turin Guarino Guarini was born in 1624 and died in 1683. As a youth, Guarini received his teachings in Modena where he was a member of the Theatine order. There, he learned philosophy, theology, astronomy, and mathematics. His study of mathematics led him to a career in architecture. As an architect, Guarini is well known for four buildings: S.Vincenzo in Modena, Sicily, Ste. Anne-la-Royale in Paris, France, S.Lorenzo and Santissima Sindone in Turin, Italy. He is also known for his two architectural treatises entitled Architettura Civile and Disegni d'architettura civile et ecclesiasticae as well as other literary works that concentrate on his mathematical knowledge. Moreover, there is a close alliance between his treatises and his architecture. In these works, Guarini discusses the four aforementioned structures and includes plates of drawings and plans for churches that were built and some not built. In addition, Guarini discusses Desargues projective geometry. "...it was this new geometry that supplied the scienti c basis for Guarini's daring structures, particularly of domes." Furthermore, he mentions the work of his contemporaries as well as periods of architectural history (Gothic) that he favors the most. According to R. Wittkower, "...each of Guarini's buildings (like each of his books) was an architectural 'summa'...". Guarini is known for two architectural devices: an openwork dome and the telescopic creation of vertical space. In the church St. Anne-la-Royale (begun in 1663 and not completed until 1720) one can see these typical aspects of Guarini's edi ces.

He used a lattice feature to clearly de ne the broken shapes of the upper regions in order to reveal the space beyond. In addition, this church uses pointed arches which reveals aspects of gothic architecture. San Lorenzo(begun in 1634 by the Theatines and Guarini started working on the church in 1666) used a typical Baroque inner dome, characterized by illusion. Also, he made use of the Islamic arch which, because of its geometry, held much mathematical appeal. Finally in 1668 Guarini began constructing Santissima Sindone from a pre-existing structure in order to house the holy shroud. The attempt to build a proper house for this relic began in 1430. So, when Guarini began, a circular plan, an established elevation, walls reaching to the second tier and a connection to the piano nobile of the Royal Palace were already in place. Throughout this chapel, Guarini made reference to the number three. This is seen in the number of pendentives, arches, and equilateral triangles. The use of three pendentives instead of four was an "unprecedented procedure". It is also in this chapel, where Guarini's architectural structures culminate in his diaphanous dome. As an architect of the seventeenth century, Guarini was in luenced by other architects of the same period, known to art historians as the Baroque period of art and architecture. Various features about art and architecture were explored during this time. One important feature is optical illusionism. Although this was explored by many, the typical expression of optical illusion was through paint; even in architecture, paint would be applied to the interior of a dome in some type of picture and this would create an illusion that the dome was receding up into space. Guarini, as a prominent and innovative architect as well as brilliant mathematician, explored optical illusion in a di ferent manner. In his Santissima Sindone, Guarini created a diaphanous dome; a geometrical optical illusion in the dome through the use of the actual structure. Furthermore, while the true Baroque style (as seen in Borromini's work) is a homogenous structure where each piece is easily read, Guarini o ten chooses an anti-homogenous structure that recalls aspects of the Mannerist tradition.

As Meek states: Guarini "...pushed architectural creativity and inventiveness beyond the familiar Baroque world of manipulation and trompe - l'oeil illusionism". Guarini was an outstanding architect of his time. Not only were his writings in luential, but his complicated structures surpassed anything that had been attempted before him. He was a brilliant mathematician as well as philosopher, teacher, writer and architect. Guarini's strong mathematical background is evident for he states in one of his treatises: 'Thaumaturga Mathematicorum miraculorum insigni, vereque Regali architectura coruscat' - 'The magic of wondrous mathematicians shines brightly in the marvelous and truly regal architecture'

Fig. 7 Floor plan showing loor patterns

Fig. 8 Cross sections through dome

Fig. 9 View looking upward at dome

Oval or Ellipse?

An oval is a curve resembling a squashed circle but, unlike the ellipse, without a precise mathematical de nition. The word oval derived from the Latin word "ovus" for egg. Unlike ellipses, ovals sometimes have only a single axis of re lection symmetry (instead of two). Ellipse[ih-lips] noun, Geometry.1753, from French ellipse (17c.), from Latin ellipsis "ellipse," also, "a falling short, de cit," from Greek ekleipsis (see ellipsis). So called because the conic section of the cutting plane makes a smaller angle with the base than does the side of the cone, hence, a "falling short." The Greek word was rst applied by Apollonius of Perga (3c. B.C.E.). to the curve which previously had been called the section of the acute-angled cone, but the word earlier had been technically applied to a rectangle one of whose sides coincides with a part of a given line 1. a closed plane curve generated by a point moving in such a way that the sums of its distances from two xed points is a constant : a plane section of a right circular cone that is a closed curve. 2. Italian: “sexy circle” (Kasparowitz).

Fig. 10 Two examples of ovals

Both of these gures are ovals, but neither are ellipses. RULES: 1. All ellipses are ovals. 2. Not all ovals are ellipses. 3. A circle is a special case of an ellipse. (one focus, not two)

String Theory on Medieval Masons and Carpenters drawing an Ellipse While looking thru books written 500 plus years ago on medieval cathedrals, stemotomy or geometry in French, German and Spanish, it's easy for me to overlook or misinterpret a lot of information. However, as a carpenter I see things di ferently than architectural historians. An easy example of seeing things di ferently is in the book "Traict Five Orders of Architecture" written by Frenchman Pierre Dumb in 1645. It was a reprint of Andrea Palladio's (1508 – 1580) "The Four of Architecture", Venice, 1570. Dumb added several drawings of French mansard roofs to the reprint of Andrea Palladio's book. François Mansart (1598 - 1666) was a French architect whose buildings or drawings of his Mansard roofs were probably widespread by 1645. The Dutch published a counterfeit copy of Dumb's book a year later in 1646. One of the drawings Dumb added is a drawing of what we refer to as a Dutch Gambrel roof, even though the French carpenters still call it a French Mansard roof. So the Dutch Gambrel Roof is a counterfeit French Mansard roof in more ways than one. Misinterpreting Information One of the things that is confusing to me, a ter reading the history of the Greeks and Romans, is the classi cation of Greek and Roman architecture on mouldings. If the curves of mouldings, in Roman architecture, were most generally composed of parts of circles; while the Greeks architectural mouldings were almost always elliptical, or of some form of the conic sections, then why isn't there more information of the geometry of drawing out elliptical Greek mouldings? What we call the "garden method" of drawing out an ellipse with a string today in the United States might have been such common knowledge that there wasn't any need to include it in their books. Or

as architects so o ten do they'll only present what is considered worthy of knowing to their fellow architects. The use of a rope or string for drawing an ellipse was rst brought to light in the 6th century by Anthemius of Tralles , a Greek professor of Geometry in Constantinople, where they built 336 groin vaults underground in Constantinople for water storage. So what about all the Roman groin vaults 400 years earlier? Where the intersection of two barrel vaults intersect in an elliptical cross diagonal. Maybe, Anthemius of Tralles was the only one that documented the string theory of the ellipse. Or was it the only documentation that has survived from the Greek's? German Renaissance artist Albrecht Dürer's book “Underweysung der Messung”. A Manual of Measurement, (1525 ) makes acknowledgement of Euclid in his book. Dürer makes no mention of how to draw out an ellipse with a string when he shows how to draw the ellipse using ordinates. Again, maybe the string method wasn't worthy of showing to his fellow readers of Euclid. Also, Dürer favors the two dimensional geometry methods of Ptolemy over Euclid. Again, I think Ptolemy sundials were the basis of medieval builders use of ordinates for drawing an ellipse. Philibert De l'Orme, in 1515, documented a lot of the stereotomy (masonic projection) techniques. Maybe he didn't feel the need to include the string method in his book. The Mathurin Jousse book "The Art of Carpentry"(1627) doesn't show the string method either. A reprint of the Jousse book by Gabriel-Philippe de LaHire in 1702 doesn't show the string method either. LaHire's father was a French mathematician and astronomer who published an extensive book on geometry called " La Gnomonique ou Methodes Universelles". It's quite surprising that no diagrams of geometry were included in the reprint of Jousse book by LaHire. Practical art of carpentry tracing - the "trait de charpente". The French Scribe method for timber framing traces it's roots back to the 13th century. "The symbolic aspects of instruction in the French scribe method was

revealed in the context of the “Compagnonnage” (French guild system) only in a con dential manner during joining and induction ceremonies. These very particular circumstances are not open to outsiders, nor are they communicated to the outside world in any way.. Just about every medieval cathedral has an octagonal baptismal. The octagonal or hexagonal structure required the carpenters to draw out the elliptical hip ra ters of the vaults for the centering using stereotomy to develop the elliptical hip ra ter side cuts at the peak of the octagonal vault and for the hip ra ters of the octagonal roof itself. So maybe the ordinates used by the Compagnonnage to draw out ellipses was as much as a secret as the Freemasons “masonic projections technique”. Conclusion All of the modern day geometry books have a sidebar where they explain how to draw an ellipse using a string, because it's common knowledge. If the string method was common knowledge in the middle ages, why aren't there side bars in the books explaining the string method? If I'm going to build a cathedral, I'm going to hire carpenters and masons who know how to draw out an ellipse using ordinates and not a carpenter or mason who draws out the ellipse using a string. So the carpenters and masons who draws out the ellipse using a string were probably not documented by the architects of the cathedrals. Even you can draw an ellipse with a string, you need to know what part of the ellipse to use for the cross diagonals of the vault. Using ordinates, you wouldn't have to know anything about the type of curve the ordinates produce or what section of the curve to use for the cross diagonals of the vault. The scribe tradition in French timber framing, or the “trait de charpente” as the carpenters call it, makes it possible to design complex wooden buildings in three dimensions. The Masonic projection and French Scribe method both used stereotomic drawings, not strings. Sim Ayers, SMB Builders November 2010

Constructing Ellipses Foci and Axes Given the major and minor axes of an ellipse, you can always ﬁnd the foci. You need the foci for some construc on methods. Just draw radii of length a from the ends of the minor axis. Given the foci, however, you can't uniquely determine the axes. You need addi onal informa on such as the length of one axis. However, the major axis is always along the line through the foci and the minor axis always perpendicularly bisects the line between the foci.

Pin and String

Mostly useful as a heuris c tool to help students visualize ellipses, but it can be useful for construc ng large ellipses. Put a pin in each focus and e a string to each pin leaving slack with length 2a. Pull the string taut with a pencil point and slide the pencil to draw the ellipse. This makes use of the fact that an ellipse is the locus of points whose distances from two ďŹ xed points have a constant sum. h p://www.sahraid.com /has a spiďŹ&#x20AC;y tool that makes this method much handier to employ.

Trammel Method

27Â

In my view the best method. With the major and minor axes constructed (and extended) mark a piece of paper with points O, A and B so that OA = a and AB = b. Slide O along the minor axis and B along the major axis. Point A traces out the ellipse. With any reasonable care this method is quite accurate and it is very fast.

28Â

For cases where the axes are similar in size, the method above may be inaccurate. It is also possible to draw an ellipse using an external trammel.

29Â

Envelope Method Given the focus and a circle whose diameter equals the major axis, draw a radius from the focus to the circle, then a line perpendicular to that. The perpendiculars sweep out the ellipse. This is fast and can be quite accurate. If you use an index card to draw the perpendiculars you can dispense with drawing the radii. Just let one edge of the card serve as a radius and use the other to draw the perpendicular. For a quick and dirty way to sketch an ellipse this rivals the trammel method. It roughs out the area enclosed by the ellipse a bit faster.

30Â

The Draftsman's Method This method is given in a lot of dra ing texts. For extreme accuracy it's probably the best method. It's convenient for use on a dra ing board with T-square and triangles. Construct the major and minor axes and draw circles with each axis as diameter. Also construct radii as shown. Angles aren't cri cal so radii can be closer in areas where greater accuracy is needed.

31Â

Draw ver cal lines from the intersec on of each radius with the outer circle and horizontal lines from the intersec on of each radius with the inner circle. The intersec on of each pair of corresponding lines is a point on the ellipse. It's easy to see this is simply an implementa on of the parametric equa on of an ellipse: x = a cos t, y = b sin t.

32Â

When the desired number of points are drawn, construct the ellipse.

33Â

Parallelogram Method Some mes you know an ellipse can be enclosed within a parallelogram, for example, a foreshortened view of a circle or a spherical object sheared out of shape. Construct the parallelogram and divide it into quarters. Divide all the lines into equal numbers of segments and number them as shown. From the midpoints of opposite sides, draw straight lines through the ck marks as shown.

34Â

Con nue the construc on for all quadrants of the parallelogram.

35Â

Connect all the 1-1, 2-2 intersec ons, etc., to construct the ellipse. This construc on will work perfectly well if the parallelogram is a rectangle, so it will work to construct an ellipse if the major and minor axes are known. If the parallelogram is a square, the resul ng ellipse is a circle. The intersec ng lines are perpendicular, and the construc on is the famous one of construc ng a right angle inside a semicircle. The general construc on here simply works by deforming the construc on so an ellipse results.

36Â

Five-Center Method Draw the major and minor axes of the ellipse and draw a rectangle around them. Construct XY and draw P-C1 perpendicular to XY. Let X-C2 = r and locate Q such that YQ = 2r. Draw a circle of radius r centered on C2. By symmetry, everything on the right side of the diagram is repeated exactly on the le . Using C1 as center and C1-Q as radius, draw an arc through Q. Call the intersec on of this arc with the circle X-C2 C4.

37Â

Construct line C1-C4 and extend it to T. Using C1 as center, draw arc YU. Construct line C4-C2 and extend it to L. Using C4 as a center, construct arc UV. The successive arcs YUVX are an approxima on to the ellipse.

38Â

The true ellipse is shown in red, the approxima on in purple. The approxima on is quite good for slightly or moderately eccentric ellipses but becomes obviously incorrect for very elongated ellipses.

Steven Dutch Department of Natural and Applied Sciences University of Wisconsin - Green Bay Created June 17, 2005 Last Update January 30, 2012

Bibliography General - Baroque Duvernoy, Sylvie. Baroque Oval Churches: Innovative Geometrical Patterns in Early Modern Sacred Architecture Hart, Vaughan; Hicks, Peter, eds. (1996), Sebastiano Serlio on Architecture Volume One: Books I-V of 'Tutte L'Opere D'Architettura et Prospetiva', New Haven & London: Yale University Press. Morrissey, Jake. The Genius in the Design: Bernini, Borromini and the Rivalry That Transformed Rome. New York: Harper Collins, 2005. Varriano, John. Italian Baroque and Rococo Architecture. New York: Oxford University Press, 1986. General - Ellipse/Oval Akhtaruzzaman, Md and Sha e, Amir A.. Geometrical Substantiation of Phi, the Golden Ratio and the Baroque of Nature, Architecture, Design and Engineering. Grunova,Zuzana and Holesova,Michaela. Ellipse and Oval in Baroque Sacral Architecture in Slovakia. Huerta, Santiago. Oval Domes: History, Geometry and Mechanics Mozo, Ana López. “Ovals for Any Given Proportion in Architecture: A Layout Possibly Known in the Sixteenth Century”, Nexus Network Journal, October 2011. Rosin, Paul. On Serlio's Construction of Ovals, Nexus Journal 2000.

________ and Sylvie Duvernoy, Paul L. Rosin - The Compass, the Ruler and the Computer, Nexus Journal 2006. Bernini Parulkar, Rutika and Gavande, Shreyas. The Making of St. Peter’s Basilica. February 25, 2013. St. Peter’s Square: Embracing All Mankind. Borromini Hill, Michael. “Practical and Symbolic Geometry in Borromini’s San Carlo alle Quattro Fontane”. Journal of Architectural Historians. December 2013. Hatch, John G., "The Science behind Francesco Borromini's Divine Geometry" (2002). Visual Arts Publications. Paper 4. McCrossan, John. Francesco Borromini’s Drawings of San Carlo alle Quattro Fontane, Rome, 1634 - 1667, Trinity College Dublin. Guarini Badillo, Noé. Ocularium Lucis: Light and Optical Theory in Gurino Guarino’s Church of San Lorenzo. University of Arizona, thesis 2012. Goetting, Carol Ann. Guarino Guarini: His Architecture and the Sublime. University of California, Riverside, 2012. Ranseck, Kelsey. Guarini’s Grand Goal: The Awe-Inspiring Optics and Exploration in the Dome of San Lorenzo. Roero, Sylvie. Guarino Guarini and Universal Mathematics, Nexus Network Journal · January 2009.

Colophon This book was produced on a Toshiba Chromebook 2. Google applications such as Docs, Drive, Draw, etc. were used to layout the pages, insert the images and format the text. There are two fonts that were used: Abril Fat face for for the project titles and page numbers Alegreya for the text Printing is by Createspace. Ful llment and shipping are through Amazon.

Images All images were obtained through internet sources.

Gratitude I have had the good fortune to have a group of remarkable professors of architecture. I must offer thanks to Don Koberg and Carleton Winslow of Cal Poly and Earl Moursund at the University of Oregon.

other titles in this series…

plan section model sketch isometric spatial composition geometry of architecture

notes & sketches