Linear Perspective

The paper explores the systematic relationship of Duchamp's Readymades to the Large Glass. It relates the moment of the declaration of the Readymades to that temporal cut that - in the projection logic of the Large Glass (and the Green and the White Box) - is established as temporal analogon to the cut through the visual pyramid in the 'normal' perspective of the renaissance. The Readymades emerge thus as the incarnation or exemplification of that three-dimensional projection of the four-dimensional Mariée that Duchamp calls the ›pendu femelle‹.

A conciliação entre uma dimensão percebida e sensível com uma outra tangível e corpórea é condição inerente à prática cenográfica e museográfica de José Capela. Assente no domínio cognitivo e instrumental da arquitetura, a sua prática move-se na conformação de ilusões espaciais que questionam a perceção visual do espaço a par da sua receção háptica e intelectual.

By focusing on the architectural qualities inherent to quadratura painting, this essay embraces the acknowledgment of its perspective strategies, intended space/form, and achieved synthesis of both represented and built space. To accomplish such an inquiry, the imaginary architectures outlined by Luís Gonçalves Sena (1754) at Santarém’s Jesuit College were taken analyzed. In order to verify the compositional and spatial qualities underlying the outlined image, two methods are explored: (1) the perspective restitution, dismantling the quadratura through a reverse application of perspective procedures and removing parameters strictly related to the ars pingendi; (2) the reconstitution of the architectural idea by searching for its compositional matrix and confronting the illusory space with the wider practice of the scientia aedificandi. These methodologies allow a complete analysis of the quadratura’s architectural contents as well as the Baroque space perceived as a whole.

To students and practitioners of anamorphic art, the name of Jean-François Niceron is more than preeminent; it has become iconic. La Perspective Curieuse was first published in 1638. An augmented version was then translated into Latin by Mersenne in 1646. A newly amended and augmented version was retranslated into French by Roberval in 1652. This book is an English translation of the 1652 text, with reference to the 1638 and 1646 versions.
Considering the continued high reputation of the book, the mathematics have been checked for correctness and consistency, and the authors have provided a full commentary, pointing out the most difficult turns of the 17th-century French, the respective contributions of Niceron, Mersenne and Roberval, and explaining Niceron's greatest insights and weaknesses.

To students and practitioners of anamorphic art, the name of Jean-François Niceron is more than preeminent; it has become iconic. La Perspective Curieuse was first published in 1638. An augmented version was then translated into Latin by Mersenne in 1646. A newly amended and augmented version was retranslated into French by Roberval in 1652. This book is an English translation of the 1652 text, with reference to the 1638 and 1646 versions.
Considering the continued high reputation of the book, the mathematics have been checked for correctness and consistency, and the authors have provided a full commentary, pointing out the most difficult turns of the 17th-century French, the respective contributions of Niceron, Mersenne and Roberval, and explaining Niceron's greatest insights and weaknesses.

Many scholars believe that linear perspective existed in classical antiquity, but a fresh examination of two key texts in Vitruvius shows that 1.2.2 is about modularity and symmetria, while 7.Pr.11 describes shading (skiagraphia). Moreover, these new interpretations are firmly based on the classical understanding of optics and the history of painting (e.g., Pliny the Elder). A third text (Philostratus, Imagines 1.4.2) suggests that the design of Roman wall painting depends on concentric circles. Philostratus' system is then used to successfully make facsimiles of five walls, representing Styles II, III, and IV of Roman wall painting. Hence, linear perspective and its relatives, such as Panofsky's vanishing vertical axis, should not be imposed retrospectively where they never existed.

Cinco chaves essenciais no ensino da Geometria JORNADAS DIDÁCTICAS 2018, Faculdade de Belas Artes da Universidade do Porto Resumo O presente texto, que resulta da comunicação por nós proferida nas Jornadas Didácticas 2018 que ocorreram na Faculdade de Belas Artes da Universidade do Porto, aborda em síntese uma parte da minha experiência como docente das unidades curriculares de Geometria e de Métodos de Representação Rigorosa na Faculdade de Belas Artes de Lisboa, que envolvem os vários sistemas de representação e vários níveis de ensino. Neste texto, que incide sobretudo na minha experiência didáctica e pedagógica, enunciam-se cinco chaves essenciais que consideramos imprescindíveis para o ensino da Geometria e da transmissão dos conhecimentos e dos métodos que fazem parte daquela ciência. Podendo haver outras chaves possíveis, neste texto enunciamos cinco que consideramos fundamentais. Abstract This text, which is based on our communication in the Jornadas Didácticas 2018 that took place in the Faculty of Fine Arts of the University of Porto, addresses the issue of my experience as a teacher of Geometry and Methods of Rigorous Representation curricular units in the Faculty of Fine Arts of Lisbon University, which involve the various systems of representation and various education levels. In this text, which focuses in our particular didactic and pedagogical experience, we enunciate five essential keys that we consider essential for the Geometry education and for the transmission of knowledge and methods that are part of that science. While there may be other possible keys, in this text we enunciate five that we consider fundamental. Introdução Para Ensinar Geometria Descritiva com eficácia nos seus vários sistemas de representação, com projecções centrais ou cilíndricas, paralelas, para além da obrigatoriedade dos conhecimentos do Professor, é factor fundamental a forma como se ensina e como se transmite o conhecimento. Neste sentido, factores lúdicos e variáveis concorrem para a transmissão do conhecimento. Na nossa experiência com os alunos na Faculdade de Belas Artes da Universidade de Lisboa, verifico que a percepção e a assimilação dos conhecimentos dos conteúdos das unidades curriculares que lecciono são mais receptivos e assimilados quando o autor utiliza cinco meios, a que designo de chaves essenciais. O simples recurso ao moderno quadro branco não é por si suficiente para a transmissão do conhecimento,

Linear perspective, which is the discovery of the Renaissance, is momentous in terms of the appearance of a new " way of seeing " that is central to the " subject " and emergence of a " subjective " " way of representing " the world. It is first mediated into the field of architecture by Flippo Brunelleschi and also practiced throughout his architectural productions. Thus, linear perspective as a mode of architectural representation is considered to be both an " instrumental " and " conceptual " tool for the perception and production of architectural space. Referring back to the naves of the Renaissance cathedrals designed by Brunelleschi, this paper will put remark on the status of perspective projection which renders possible the experience of the spatial effects before they are constructed, in better words " the production of space as an image ". Accordingly, it will promote the spatial concept of " perspectiv-ated space " which addresses a more perspectival effect spatially in the space constructed and visually in the space represented. This concept will reassert spatial issues that reinforce the perspectival effect. This paper will trigger a reinterpretation for the concept of perspectiv-ated spaces –that has its roots in the Renaissance-and a possible re-reading on the spaces of Modern Architecture regarding METU Faculty of Architecture Building as an exceptional construction for the reconsideration of the concept. It is claimed that the necessary elements of linear perspective; the central vanishing point and the structural grid, " literally " exists in the faculty building which will be represented visually with the images of architecture.

Discovery of the linear perspective of the Renaissance is momentous in terms of the appearance of a new “way of seeing” that is central to the “subject” and emergence of a “subjective” “way of representing” the world. With the assumption that “perspective” is a cultural constituent of the “Renaissance paradigm” that generates the representational notions of art and architecture, perspective is considered as a “convention” and a “mode of representation” that has been influential until the “paradigm shifts” in the nineteenth and twentieth century. With the influx of the modernist avant‐gardes, “subjective” linear perspective is replaced with the “instruments” of the new paradigm, and “parallel projection” becomes the mere tool for the representation and production of art and
architecture introducing a new “objective” structure. The aim of this study is to reawaken and reconsider perspective through the “conventions” and the “aberrations” it reproduces. In this context, the course in the curriculum of METU Department of Architecture entitled Arch524 Architecture and Different Modes of Representation is covered as a “pretext” for this study. The theoretical framework and the structure of the course has a great impact on this study, since the course reconsider Renaissance perspective and its architectural production in a new context; within the modern architectural space of METU Faculty of Architecture Building. Following the traces of the selective works from the final exhibitions of the course, creating “illusionistic spaces,” and recalling the subject “participation,” it is questioned how perspective is “exhibited” and how the architectural space is reproduced.

"Critique after Modern Monetary Theory" is the opening chapter of the book, DECLARATIONS OF DEPENDENCE: MONEY, AESTHETICS & THE POLITICS OF CARE. 

https://www.amazon.com/Declarations-Dependence-Aesthetics-Politics-Provocations/dp/1496201922

Declarations of Dependence rethinks the historical relationship between money and aesthetics in an effort to make critical theory newly answerable to politics. Scott Ferguson regrounds critical theory in the alternative conception of money articulated by the contemporary heterodox school of political economy known as Modern Monetary Theory. Applying the insights of this theory, Ferguson contends that money, rather than representing a private, finite, and alienating technology, is instead a public and fundamentally unlimited medium that harbors still-unrealized powers for inclusion, cultivation, and care.

Ferguson calls Modern Monetary Theory’s capacious ontology of money the “unheard-of center” of modern life. Here he installs this unheard-of center at the heart of critique to inaugurate a new critical theory that aims to actualize money’s curative potential in a sensuous here-and-now. Declarations of Dependence reimagines the relation between money and aesthetics in a manner that points beyond neoliberal privation and violence and, by doing so, lends critical theory fresh relevance and force.

According to the early Renaissance philosopher Nicholas of Cusa, we are by divine providence destined to develop our own view on the world that we inhabit. Only the free actualisation of a plurality of concordant perspectives can enable us to find our path to the point of unity in which oppositions coincide in a meaningful way. However, we are neither masterly spectators of a unified space, as the Florentine tradition of modern art suggested, nor are we able to anticipate the coincidental point of unity, as linear perspective paintings suggest. Rather, our path to the point where oneness and concordance emerge requires, what Jean-Luc Marion called, a movement of ‘anamorphosis’: Instead of relying on the default-settings of the ‘imaging technics’ we have to change our point of view and feel our way to the turning point of its emergence. Like in Holbein’s painting The Ambassadors, we must shift our gaze in order to find the point where the ‘invisible’ gives itself to our view.
However, the world that we inhabit is not reducible to the theatrical space of Holbein’s age, in which artificial strategies ensured that the relevant ‘effects’ are securely brought about. After all, the creation is an image of God: it is not designed to uncover the memento mori of a skull, but the fullness of life. If anywhere, this life shouts outside in the alleys and squares – its ‘givenness’ is always already inscribed in the inhabited spaces of a temporalized ‘politics of glory’ that reveals the dignity and nobility of places, responsibilities, rules and professions.
This has far reaching implication not only for our understanding of the scientific modelling of ‘complex spaces’ but also for our understanding of the forces that govern the emergence of representative democratic spaces. Cusa’s account of complex spaces undermines not only the absolutist world-picture of Descartes and Newton, but also the totalising basic assumptions of modern liberalism. This essay develops a phenomenological reconstruction of Cusa’s late Gothic concept of complex spaces in conversation with the related, 20th century discussion on the concept of iconicity in Pawel Florensky and Jean-Luc Marion.

In the year of 1708, the College of Santo Antão integrated the wedding festivities of D. João V with Maria Anna of Austria (daughter of the Holy Roman Emperor Leopold I), staging a play under the auspices of the Jesuit Father Inácio Vieira (1678-1739) who considered it the "most complete tragedy made in Portugal during our times”.
If, in relation to the wedding there is nothing to report about the chosen location, the theatrical play was severely criticized at the time by the press invoking the desecration of religious space and obstruction to its use over a long period.

“[…] performed in the church (place for its decency, indecent for similar play) preventing, with its structure, the use of the main chapel and church for a long time, jeopardizing the performance of ministries, masses or divine offices during the days of this act” .

Although, as we shall see, the fact can be pragmatically explained due to lack of alternatives, the truth is that such kind of performance was no novelty in that space, but rather a common and legitimate use founded over the college's tradition in theatrical practice, an application of the lectures on perspective that Inácio Vieira, himself, contemporaneously ministered at the College’s Aula da Sphera.
Indeed the church, and specifically its main chapel, would be the place by excellence for the theatrum Sacrum but also, when necessary, would serve the performance of classical tragedies testing out, in either case, teachers and students’ ability in defining spatial perspectives, as well as all the scenery machinery associated with its implementation and transformation.
As for the royal wedding seems to be no doubt of Inácio Vieira’s direct involvement, based upon his written production on perspective, but also on optics, catoptrics and dioptrics, compliant to the framework of the Jesuit culture, highly skilled, on these topics. As such we propose to develop in this paper an exploration on the characterization of scenographic space probably designed by this Jesuit priest for the event.

برز عمل ألبرتي التنظيري في مرحلة خلت من التنظير الرياضي والهندسي. فألبرتي منظر  في الرياضيات ويمارس الرسم وبذلك مثل النموذج مثالي في النهضة للرسام الطموح والعالم المنظر الذي ساهم في تطوير الفنون بفضل كتبه التي تشرح المبادئ الأساسية للهندسة والبصريات لتجعل من عمل الفنان يتأسس على العقلانية وترقيه من الحرفية اليدوية.
نلقى لدى ألبرتي توجيه للرسامين بشكل ممنهج إذ يفصل في شرح مبادئ الهندسة التي تتقاطع مع عمل الفنان. يبدأ ألبرتي بتحديد النقطة ثم الخطوطه والأحجام لينتقل إلى شرح كيفية صياغتها فوق مساحة اللوحة المسطحة. إذ قدم ألبرتي التعليمات الهندسية التي يحتاجها الرسام لبناء المكان الهندسي الذي يصيغه الرسام وهي عبارة عن عمليات معقدة تحول المكان المسطح للوحة إلى فضاء منفتح ومنسجم وممتد بشكل يوهم بلانهائيته.
هكذا يولي ألبرتي أهمية خاصة لفعل الصياغة الهندسية للمكان المسطح للوحة، كما اهتم بشكل واضح بفعل الرؤية أي الإدراك من خلال تعاطيه للبصريات التي تشرح وتفصل عملية الإدراك البصري وكيفياتها. يخلص ألبرتي إلى أن عملية الرؤية أي الإدراك تحدث بفضل وجود الأشعة، أشعة الضوء التي تحيط بالأشياء. وبذلك خدمت نظريته هذه التي اقتبسها من البصريات عمل الرسامين وقدمت خدمة كبيرة على المستوى المعرفي، فالقاعدة الأولى التي طرحها ألبرتي هي كيفية انتشار الضوء فوق الأشياء المرئية وتأثيرها على حجم تلك الأشياء، فكلما استعملوا أشعة أكثر كلما بدا حجم الأشياء أكبر.
ميز ألبرتي بين نوعين من الأشعة، وهي الأشعة الخارجية التي تمتد على مجمل المساحة المرئية بحيث تكون سياجا حول عموم تلك المساحة. لذلك ينطلق فعل الرؤية من انعكاس الضوء فوق عدسة العين. ويأخذ الضوء فوق مساحة اللوحة شكل الهرم الذي تنتظم فيه الأشعة هندسيا تكون قاعدته السطح المرئي، وهي الأشعة البصرية التي يسميها الأشعة الخارجية. أما رأس الهرم فيستقر داخل العين حيث يصبح نقطة التقاء زوايا الأحجام على شكل مثلثات. وما يحدد الرؤية بالنسبة لألبرتي، هو عنصر الضوء أو الشعاع الضوئي الذي ينبعث من الأشياء المرئية وبذلك يرتبط بوضعية الشعاع المركزي من جهة وبوضعية المشاهد من جهة أخرى.
يذهب دورر متبعا في ذلك فرضية دي لافرانسيسكا التي تعتبر عملية الإدراك تستلزم الشروط التالية. ويتم الإدراك البصري على مستويين تحدده عناصر أساسية هي العين المشاهدة والأشياء المرئية. ويحدد فعل الرؤية أي الحكم واللون وحجم الضوء والظل المسافة التي تفصل بين الأشياء المدركة والمشاهد.  ويستنتج من هذه النظرية أن عملية الإدراك تتم بإزالة المسافة التي تفصل بين الشيء المرئي والعين المشاهدة. يمكن شرح هذا المسلسل البصري حسب هذين الرسامية، بأن الإدراك البصري هو انعكاس للأشياء المرئية بواسطة الأشعة التي تبعثها نحو عدسة العين المبصرة. وكأن الأشياء المرئية تلج صورها في قلب العين المشاهدة بحيث تربط عالم الأشياء المرئية وهي ذات وجود مستقل تفرضه على العين التي تدرك الأشعة الخارجية والمركزية هو ما نسميه الحقل البصري الذي يواجه المشاهد ويجعله يستجيب له بإبصاره .
نستنتج مما سبق الكيفية التي تتم بها عملية الإدراك في مستواها الفيزيوجي، أما الإدراك في بعده الذهني والذي عمد ألبرتي على بيانه وتوضيحه للرسام فيتركز في كيفية صياغة المكان الفيزيائي للوحة بحيث يحيل يبني أشكال هندسية بواسطة النقط والخطوط والأحجام مواضيعه بشكل متناسق ومنسجم وممتد وهذه تعتمد على قواعد ثابتة لها من الدقة الرياضية ولكنها ليست بدقة العلماء. وهذه الشروط النظرية لبناء المكان الهندسي هي نفسها التي تحدد إدراك المشاهد للمشهد فوق اللوحة. لأن القواعد الرياضية التي يشتغل بها الرسام تحيل بدورها للشروط السيكولوجية التي تحدد الإدراك البصري حيث يقترن التعيين الاصطناعي ب"نقطة الرؤية" الذاتية للفضاء التصويري.
يفترض "جان بياجي" أن عملية بناء المكان في فضاء تصويري ترتبط بقدرة الإنسان على الإدراك العقلي، لكون الإنسان يمتلك هته القدرة  بشكل طبيعي، فما دام الكائن البشري ذا امتداد معين، أي امتداد حجمه أو جسده داخل الفضاء، فإن إمكانية الإدراك بهذا الشكل تكون فطرية لديه. ويبقى أن هذا النمط من الإدراك هو إدراك حسي وتلقائي ومباشر وعفوي. أما صياغة المكان هندسيا فيعتمد على قواعد هندسية لبناء فضاء اللوحة عقليا، أي بناء مكان ذي الخصائص الأقليدية وهي الانسجام والامتداد والوحدة. وهذا يتطلب قدرات عقلية تقتضي التمكن من وسائل وتقنيات هندسية، لأن الأمر هنا يتعلق بإعادة بناء هندسي لموضوعات الرسام فوق مساحة مسطحة تمتلك البعد الثنائي فقط ويراد صياغتها لتوهم بالعمق والبعد الثالث.

1 The need for a perspective technique able to be utilized for designing in the perspective drawing is the focus of this thesis project. 2 The proportional method is analyzed since it requires minimal set up time and speed and convenience are gained. The rules are few and simple to remember. 3 Many of the parameters involved in design process require the establishment of a few key points making the proportional method idea. 4 Through the establishment of a base and a horizon line, the basis of measurements of width and height are set up and possess a direct linear relationship. 5 Heights are found by setting a scale on 0 at the base line and the number 5 (of same scale) on the horizon line. The objects height is marked off. 6 Depth measurements are a little more complex, but the method is so directly derived from the " size varies inversely with distance " law that the procedure should not easily be forgotten. Depth is obtained through proportional analysis. 7 If the distance between the base line and the horizon line 8s 80 ft. (a given prerequisite) then half that distance is 40 ft. or halfway in the picture plane. 8 This simple method of locating points with a minimum number of steps allows the proportional great flexibility as a design tool in creating quick study concepts for three dimensional interpretation.

In a previous paper, orthogonal projection gives: 1 1/a = (1 / b xy – b yx) and 2 b/a = (1 / b yx – b xy) 3 y bar = b1 + b/a x bar ; b1 = (y bar – b/a x bar) ; and 4 y subi = x subi 1/a III Deviation 5 (y subi – y bar) =-b1 + b/a x bar + 1/a x subi 6 Y = B X ; Y = (y subi – y bar) ; X = (x subi – x bar) 7 B = (1/a – b/a) 8-b1 = B + b/a x bar – 1/a x subi 9-b1 = B + 1/a (b x bar – x subi) 10-b1 = B + 1/a [ (1 – b) x bar-(x subi – x bar) ] IV Derivation Hence, Y is the orthogonal projection of X 6. But B 7 is the difference of two slopes 3 and 4. But 1/a 1 and b/a 2 are the difference of two slopes b yx and b xy. Hence the projection of the ordinary regression to the orthogonal regression is the same from the orthogonal progression of X to Y. It results in a difference of slopes. V Linear Perspective Wen the above angles 1/a 1 and b/a 2 are used in linear perspective, their slope differences can result in contruction or foreshrtening. Hence, projection or orthogonal angles crea slope differences and these result in construction of lines. Hence, a conversion of axes projective to linear perspective is possible. VI Time travel.