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Picturing Space: Projection and Perspective

Space is the breath of art.
— Frank Lloyd Wright

There are two kinds of space in advanced forms of mimetic painting: the flat, two-dimensional space of the canvas on which the painter places lines and colors, and the illusion of three-dimensional space, called pictorial space, which is suggested by those shapes, colors and lines.

Architectural Fantasy with Susanna and the Elders, Dirk van Delen, c. 1619-1671, Oil on canvas, 105 x 137 cm., Museum Boijmans Van Beuningen, RotterdamArchitectural Fantasy with Susanna and the Elders
Dirk van Delen
c. 1619–1671
Oil on canvas, 105 x 137 cm.
Museum Boijmans Van Beuningen, Rotterdam

In the earliest times, artists were concerned above all with the representation of objects, singularly or in groups. Untrained artists, as well as painters of ancient Egypt, Crete, India, Islam and pre-renaissance Europe, depicted objects and surroundings independently of one another. To do this they were viewed from a direction that presented their most characteristic features, rendering them as they are known to be, rather than as they are seen by the eye. Only when painters became increasing concerned with the representation of naturalistic appearances of the broader world did they begin to consciously ponder not only objects and their individual or symbolic qualities, but the spatial relationships among them.

For a complete list of pre-1900 perspective manuals (with subsequent republishings) consult the Russell Light's excellent PERSPECTIVE RESOURCES, from which the list below was derived.

Click on the links below to access PDF files of the treastises.

Cone of Vision (COV): The area of vision that emanates from our eyes, about 60 degrees wide, before distortion begins to affect what we see. Outside of the 60-degree angle, objects begin to blur. In linear perspective, the Cone of Vision is indicated with a 60 degree angle beginning at the station point it is 30 degrees to the left and right of the line of sight.

Distance Points & Distance Lines:Bruce MacEvoy, "Two Point Perspective," 2015. https://www.handprint.com/HP/WCL/perspect3.html. The two vanishing points on the horizon at which diagonal 45 degrees lines in the horizontal plane meet, are known as distance points. They are the same distance from the central vanishing point as the viewer is from the picture plane. If within a picture, a horizontal square parallel to the picture plane can be identified, extending the diagonals to the horizon will give the distance points. The distance of the viewer to the picture plane is then known, and it becomes possible, by working backwards, to create a plan of the space within the picture.

It is debatable whether the correct viewing distance was of any importance to the early users of perspective. In reality, however, there are paintings that show an approach that could not be considered to be purely Albertian. Many paintings show a floor grid with a recession that appears to be governed solely by the 45 degrees diagonals of the grid squares being drawn towards a point at eye level, often placed at the edge of the painting. This approach is often referred to as the 'distance point' method and these points are known as 'distance points' simply because the distance between them and the central vanishing point is the same as the distance between the viewer and the picture plane. It follows that if the vanishing point for the orthogonals is placed centrally, and the edge of the painting is used as a distance point, then the "correct" viewing distance is half the width of the painting. It also follows that the angle of view is 90 degrees. It has been generally assumed that these points have been placed at the edge of the paintings for completely practical reasons.

We do not know the precise moment at which the two lateral points received their theoretical explanation as the "point of distance." We do not know if Brunelleschi that their distance from the central vanishing point represented, according to the scale of the picture, the distance between the vantage point of an ideal spectator and the plane of the image.

Field of Vision (FOV). The area wider than the Cone of Vision, coming out from the viewer at 90˚, in which distortion begins.

Converging Lines: In perspective drawing, parallel lines that come together towards a single vanishing point.

Diminishing Forms or Diminutation: Refers to the apparent size of objects and how they become smaller when the distance between the object moves further away from the viewer/artist, a key tenant of linear perspective.

Foreshortening: Refers to the fact that although things may be the same size in reality, they appear to be smaller when farther away, and larger when close up. Foreshortening is often used in relation to a single object, or part of an object, rather than to a scene or group of objects.

An excellent example of this type of foreshortening in painting is The Lamentation over the Dead Christ (c.1470–1480, Pinacoteca di Brera, Milan), a work by Andrea Mantegna.

Ground Line (G): A line drawn to establish the surface on which an object or objects rests; it is used to determine accurate vertical measurements in perspective drawings. The base or lower boundry of a picture plane. The term may also be applied to a similar construction line used anywhere in the picture to measure off points or to determine the scale of a figure.

The ground line is always parallel to the horizon line. In perspective drawings that show top and side views, the side view of an object is placed on the ground line. It is usually the plane supporting the object depicted or the one on which the viewer stands.

Horizon, Apparent Horizon, Visible Horizon, Skyline: The line at which the sky and Earth appear to meet. For observers near sea level the difference between the geometrical horizon (which assumes a perfectly flat, infinite ground plane) and the true horizon (which assumes a spherical Earth surface) is imperceptible to the naked eye (for someone on a 1000-meter hill looking out to sea the true horizon will be about a degree below a horizontal line).

Horizon Line (HL): The actual horizon, where earth and sky appear to meet, excluding obstructions like hills or mountains. In perspective drawing, the horizon is at the viewer's eye-level. Artists tend to use the term "eye level," rather than "horizon" because in many pictures, the horizon is hidden by walls, buildings, trees, hills etc. In perspective drawing, the curvature of the Earth is disregarded and the horizon is considered the theoretical line to which points on any horizontal plane converge (when projected onto the picture plane) as their distance from the observer increases.

Lines above the horizon line always converge down to it; lines below alwats converge upward to it.

Line of Sight: An imaginary line traveling from the eye of the viewer to infinity. In all paintings with perspective substructures, the line of sight is parallel to the ground. Lines which travel parallel to the line of sight are called orhtogonals, which in a perceptive drawing converge at the vanishing point.

One-point Perspective: A drawing has one-point perspective when it contains only one vanishing point on the horizon line. This type of perspective is typically used for images of roads, railway tracks, hallways, or buildings viewed so that the front is directly facing the viewer. Any objects that are made up of lines either directly parallel with the viewer's line of sight or directly perpendicular (the railroad slats) can be represented with one-point perspective. These parallel lines converge at the vanishing point.

One-point perspective exists when the picture plane is parallel to two axes of a rectilinear (or Cartesian) scene—a scene which is composed entirely of linear elements that intersect only at right angles. If one axis is parallel with the picture plane, then all elements are either parallel to the picture plane (either horizontally or vertically) or perpendicular to it. All elements that are parallel to the picture plane are drawn as parallel lines. All elements that are perpendicular to the picture.

Orthogonal: Orthogonal is a term derived from mathematics. It means "at right angles" and is related to orthogonal projection, a method of drawing three-dimensional objects. Orthogonal lines are imaginary lines which are parallel to the ground plane and the line of sight of the viewer. The are usually formed by the straight edges of objects. Orthogonal move back from the picture plane. Orthogonal lines always appear to intersect at a vanishing point on the horizon line, or eye level. Although we do not generally note the convergence of orthogonal lines in real life, sometimes they become apparent when standing in the middle of a road, train tracks or on a long straight urban street.

Parallel: Said of any two lines or surfaces that are always the same distance from each other.

Perpendicular: At a right, or 90 degree angle to a given line or plane. An absolutely vertical line and an absolutely horizontal line are perpendicular to each other.

Picture Plane (PP): In painting, photography, graphical perspective and descriptive geometry, a picture plane is an imaginary plane located between the "eye point" (or oculus) and the object being viewed and is usually coextensive to the material surface of the work. It is ordinarily a vertical plane perpendicular to the sight line to the object of interest. In painting, the surface of the artist's paper or canvas. The image that is created on the picture plane gives the impression that the subject is behind this surface.

Plane: In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. In colloquial language, any flat surface, such as a wall, floor, ceiling, or level field.

Prospettiva: from Latin perspicere, to "see distinctly."

Projection: From Latin proicere, "to throw ahead." A projection is a straight line drawn through different points of an object from some given point to an intersection with the plane of projection.

Receding: Moving away from the viewer. The opposite is Advancing.

Station Point (SP or S): The position of the artist's eye relative to the object he or she is drawing. Sometiems referred to as "eyepoint," "point of veiw," or "viewpoint."

Transversal: Transversal lines are lines that are parallel to the picture plane and to one another. They are always at right angles to the orthogonal lines.

Two-point Perspective: A drawing has two-point perspective when it contains two vanishing points on the horizon line. In an illustration, these vanishing points can be placed arbitrarily along the horizon. Two-point perspective can be used to draw the same objects as one-point perspective, rotated: looking at the corner of a house, or at two forked roads shrinking into the distance, for example. One point represents one set of parallel lines, the other point represents the other. Seen from the corner, one wall of a house would recede towards one vanishing point while the other wall recedes towards the opposite vanishing point.

Two-point perspective exists when the painting plate is parallel to a Cartesian scene in one axis (usually the z-axis) but not to the other two axes. If the scene being viewed consists solely of a cylinder sitting on a horizontal plane, no difference exists in the image of the cylinder between a one-point and two-point perspective.

Two-point perspective has one set of lines parallel to the picture plane and two sets oblique to it. Parallel lines oblique to the picture plane converge to a vanishing point, which means that this set-up will require two vanishing points.plane converge at a single point (a vanishing point) on the horizon.

Vanishing Point (VP): Imaginary points on the horizon line in one- and two-point perspective. A point at which orthogonal lines receding into space appear to converge.

The vanishing point acts on the visual field as a point of attraction, somewhat like an open drain of a water basin which draws all the water to it.

Brook Taylor, Linear Perspective: Or, a New Method of Representing Justly All Manner of Objects as They Appear to the Eye in All Situations (1715) is said to have been the first to use the phrase "vanishing point."

The Jesuit friar Andrea Pozzo, the author of Perspectiva Pictorum et Architectorum (1693–1700) and the monumental ceiling of Sant'Ignazio in Rome, was the first commentator to systematize use of the "vanishing distance"point (punctum distantiæ) in order to resolve a broad spectrum of perspective problems. He even anticipated the geometrical drawing technique, from descriptive geometry proper, by introducing the simultaneous use of plan and elevation to originate a detailed solution to architectural ornamentation of the classical orders.

While the medium of the painter is to some degree complimentary to the appearance of natural objects—both present different degrees of color, brightness and texturepaint shares nothing with space, since space is not a substance but the lack of it. If single objects are accurately represented by the artist, they do produce the effect of the space that immediately surrounds them, in other words, the space that objects displace (e.g., the space inside a drinking glass or the space between two fingers of a hand), but the great arena of space which extends beyond them is in no way expressed. Only linear perspective can achieve this objective precisely and in a mathematically predictable manner.

Knowing the laws of linear perspective allows the artist not only to represent the shapes and positions of real objects correctly foreshortened in space, but to create such images from mathematical measurements. To do this, the shapes of objects are distorted, but according to consistent geometrical laws. Before linear perspective was codified by Leon Battista Alberti in 1435 (De Pictura), artists used various types of projection methods, sometimes referred to as "perspectives."

Methods of Expressing Spatial Dimensions on a Two-Dimensional Surface

Oblique Projection
Oblique projection, sometimes referred to as "parallel projection" or "parallel perspective," is a simple type of graphical projection used to produce two-dimensional images of three-dimensional objects. The objects are not in true perspective, because they do not correspond to any view of an object that can be obtained in practice. Nonetheless, oblique projection yields convincing and useful images that allow the viewer to perceive the semblance of spatial depth.

In oblique projection the horizontal edges of the opposing sides of an object remain parallel to each other. They produce no vanishing points, and no horizon line, so no part of the object gets smaller with distance as when projected with perspective. Oblique projection is "less faithful to appearance, but more faithful to fact; it shows things nearly as they are known to the mind. Since oblique projection removes the vanishing point, all things are represented as being the same distance away and the eye of the spectator is everywhere at once."Jan B. Deręgowsk and Denis M. Parker, "Convergent and Divergent Perspective," Perception, vol. 21, issue 4, 1992, 441–447. It would appear that the Chinese have never had any scientific interest in perspective or its rules and typically adopted the parallel projection when representing buildings or geometrically regular objects (fig. 1). Such objects are viewed obliquely and their orthogonal sides recede along two main angles.

Shogi, Go and Ban-Sugorokufig. 1 Shogi, Go and Ban-Sugoroku
c. 1780
Torii Kiyonaga

Isometric Projection
Isometric projection is similar to oblique projection in that it allows to represent three-dimensional objects in two dimensions. Like oblique projection, the horizontal edges of an object in isometric projections remain parallel to each other, but the X and Z axes are inclined to the horizontal plane at the angle of 30 degrees. Because all edges of an isometric object are inclined at the same angle, they are equally foreshortened, which allows the measurement of each side of an object using the same scale making it particularity useful for technical and engineering drawing.

Looking Over Vermeer's Shoulder

Enhanced by the author's dual expertise as both a seasoned painter and a renowned authority on Vermeer, Looking Over Vermeer's Shoulder offers an in-depth exploration of the artistic techniques and practices that elevated Vermeer to legendary status in the art world. The book meticulously delves into every aspect of 17th-century painting, from the initial canvas preparation to the details of underdrawing, underpainting, finishing touches, and glazing, as well as nuances in palette, brushwork, pigments, and compositional strategy. All of these facets are articulated in an accessible and lucid manner.

Furthermore, the book examines Vermeer's unique approach to various artistic elements and studio practices. These include his innovative use of the camera obscura, the intricacies of his studio setup, and his representation of his favorite motifs subjects, such as wall maps, floor tiles, and "pictures within pictures."

By observing closely the studio practices of Vermeer and his preeminent contemporaries, the reader will acquire a concrete understanding of 17th-century painting methods and materials and gain a fresh view of Vermeer's 35 masterworks, which reveal a seamless unity of craft and poetry.

While the book is not structured as a step-by-step instructional guide, it serves as an invaluable resource for realist painters seeking to enhance their own craft. The technical insights offered are highly adaptable, offering a wealth of knowledge that can be applied to a broad range of figurative painting styles.

look-inside-icon

LOOKING OVER VERMEER'S SHOULDER
author: Jonathan Janson
date: 2020 (second edition)
pages: 294
illustrations: 200-plus illustrations and diagrams
formats: PDF
$29.95



CONTENTS

  1. Vermeer's Training, Technical Background & Ambitions
  2. An Overview of Vermeer’s Technical & Stylistic Evolution
  3. Fame, Originality & Subject Matte
  4. Reality or Illusion: Did Vermeer’s Interiors ever Exist?
  5. Color
  6. Composition
  7. Mimesi & Illusionism
  8. Perspective
  9. Camera Obscura Vision
  10. Light & Modeling
  11. Studio
  12. Four Essential Motifs in Vermeer’s Oeuvre
  1. Drapery
  2. Painting Flesh
  3. Canvas
  4. Grounding
  5. “Inventing,” or Underdrawing
  6. “Dead-Coloring,” or Underpainting
  7. “Working-up,” or Finishing
  8. Glazing
  9. Mediums, Binders & Varnishes
  10. Paint Application & Consistency
  11. Pigments, Paints & Palettes
  12. Brushes & Brushwork

Perspective

The Trinity, Andrej Rublëv, c. 1420-1430, Tempera on panel, 142 x 114 cm., Galleria Tret'jakov, Moscowfig. 2 The Birth of Saint John the Baptist: Predella Panel
Giovanni di Paolo
1454
Egg tempera on woo, 30.5 x 36 cm.
National Gallery, London

Divergent Perspective
Divergent (or "inverted," "reverse," or "Byzantine") perspective, was used by artists of the Russian Orthodox tradition, as well as by Byzantine, Chinese and Japanese artists, and many pre-Renaissance cultures (fig. 2). In divergent perspective, receding parallel lines appear to move away, or diverge, from one another. This has the visual effect that objects farther away from the viewing plane are drawn as larger, and closer objects are drawn as smaller, in contrast to the more linear perspective for which closer objects appear larger.

The use of divergent perspective was regarded as arbitrary or even as erroneous in spite of the fact that entire schools of art exist in which this kind of perspective was regularly used. Ian P. Howard, Robert S. Allison, "Drawing with Divergent Perspective, Ancient and Modern," Perception, 2011, vol. 40, 1017–1033. Curiously, when asked to draw a simple cube, about 50% of a group of university students unfamiliar with perspective used divergent perspective.Dominique Raynaud, Studies on Binocular Vision: Optics, Vision and Perspective from the Thirteenth Centuries, New York: Springer, 2016, 15.

Architectural Fantasy with Susanna and the Elders, Dirk van Delen, c. 1619-1671, Oil on canvas, 105 x 137 cm., Museum Boijmans Van Beuningen, Rotterdam
fig. 3
Death of the Virgin
Duccio di Buoninsegna
Maestà (recto)
Panel, 41.5 x 54, Museo dell'Opera del Duomo, Siena

Convergent Perspective
Convergence is a type of distortion caused by the medium of human sight, although it is largely overridden by the perceptual mechanisms, such as distance constancy. In fact, convergence is rarely noticed except in highly specific conditions, which rarely occurred until man began to modify nature and create constructions which produce many long parallel lines.

Giotto (1267–1337) may have been the first post-classical artist to intuit that lines which run parallel to the line of sight appear to eventually intersect at a single point, whereas physically, they do not. Both Duccio (c. 1255/1260–c. 1318/1319) (fig. 3) and Giotto employed convergence, but the projected lines of different objects in the same picture (ceiling rafters, floor tiles or furniture) converging at different locations, not at a single vanishing point as is the case in linear perspective.

The Avenue at Middelharnis, Meindert Hobbema, 1689, Oil on canvas, 104 x 141 cm., National Gallery, Londonfig. 4 The Avenue at Middelharnis
Meindert Hobbema
1689
Oil on canvas, 104 x 141 cm.
National Gallery, London
Museum Boijmans Van Beuningen, Rotterdam

Linear Perspective
Linear perspective is often referred to as "geometric perspective" or more simply, "perspective." Perspective is a measurable relationship between object, pictorial plane and observer. The two most characteristic features of perspective are that objects are smaller as their distance from the observer increases; and that they are subject to foreshortening, meaning that an object's dimensions along the line of sight are shorter than its dimensions across the line of sight.

Linear perspective creates a picture on a flat surface in such a manner that the various objects represented in it appear to have the same sizes, shapes and positions, relatively to one other, that the actual objects as located in actual space would have if viewed from a single immobile point of view in a single direction. With the aid of perspective geometry the relative sizes of different objects can be assessed at a distance for the first time.James Burke, The Day the Universe Changed, London: Little, Brown & Company, 1995, 76. Distant objects could be reproduced with fidelity, or created to exact specifications in any position in space and then manipulated mathematically.

Because perspective reproduces the way our eyes see objects, it creates distortions that oblique projection does not. Although a perspective projection of objects is considered more realistic than an oblique projection, oblique projections are useful in technical applications, since the parallelism of an object's lines and faces is preserved. Perspective itself rejects the property of parallelism for the principle of a gradual reduction in size, reproducing as closely as possible the conditions of natural vision."Science Art and Technology, What is Perspective?" (website no long available).

From a painter's point of view, linear perspective may be thought of as a practical means of establishing a measurable relationship between the shapes of objects as definitely located in space when observed from a single point and direction of view (fig. 5). It is essentially the same as tracing some scene on a window while keeping one eye closed and the other eye absolutely stationery.

Architectural Fantasy with Susanna and the Elders, Dirk van Delen, c. 1619-1671, Oil on canvas, 105 x 137 cm., Museum Boijmans Van Beuningen, Rotterdamfig. 5 Architectural View
attributed to Francesco di Giorgio
c. 1490–1500
Oil on poplar wood, 131 x 233 cm.
Staatliche Museen Berlin, Gemäldegalerie, Berlin

Unlike aerial perspective, which utilizes gradations of tone, color and detail to evoke the sensation of distance, linear perspective makes use of visible or implied lines associated with the objects represented. Although particularly useful for the portrayal of man-made geometric objects, such as buildings and furniture, linear perspective may be applied to organic objects as well, particularly when they are the same height (fig. 4 & 6). Through linear perspective the viewer becomes aware of a tactile and spatial sensations through the sense of sight, which sees objects in perspective.

Architectural Fantasy with Susanna and the Elders, Dirk van Delen, c. 1619-1671, Oil on canvas, 105 x 137 cm., Museum Boijmans Van Beuningen, Rotterdamfig. 6 from: La perspective pratique, necessaire a tous peintres, graveurs, sculpteurs, architectes, orphevres, brodeurs, tapissiers, […]
Jean Dubreuil
1642

Perspective—perspectiva in Latin means "see through"—enables not a view of the world of objects in space, but rather, a view of the world as it would appear seen through a theoretical window with an immobile eye, and drawn on the window pane.

Specifically, a perspective projection is an intersection of a plane, called the picture plane in painting, with a virtual pyramid or cone of sight (fig. 6 & 7). As Leonardo da Vinci wrote, perspective is "nothing else than seeing a place behind a plane of glass, quite transparent, on the surface of which the objects behind that glass are to be drawn. These can be traced in pyramids to the point in the eye, and these pyramids are intersected on the glass plane."

Assuming the a photographic camera has a simple single lens, and the viewing screen is flat, then the perspective geometry of the image will be exactly the same as that of a conventional perspective drawing. Paradoxically, perspective, which is indispensable for creating highly realist images, does so through by the distortion, often extreme, of tactile reality.

La Perspective Curieuse, Jean-François Nicéron, 166fig. 7 La Perspective Curieuse
Jean-François Nicéron
1663
Le due regole della prospettiua prattica, Iacomo Barozzi da Vignola, 1682Le due regole della prospettiua prattica
Iacomo Barozzi da Vignola
1682

There are basically three types of linear perspective: one-point, two-point and three-point perspective. Vermeer employed both one- and two-point perspective in his interior scenes, sometimes simultaneously in the same composition. He did not use three-point perspective.

from: La perspective pratique, necessaire a tous peintres, graveurs, sculpteurs, architectes, orphevres, brodeurs, tapissiers, […], Jean Dubreuil, 1642fig. 8 One-point perspective
from: La perspective pratique, necessaire a tous peintres, graveurs, sculpteurs, architectes, orphevres, brodeurs, tapissiers, […]
Jean Dubreuil
1642

One-point perspective is typically used to draw the exteriors or interiors of buildings whose facades or background walls are directly facing the viewer (fig. 8). One-point perspective is highly adapted for representing objects within such environments that are positioned parallel or perpendicular to the viewer's line of sight and whose principal structural edges are at right angles to each other. All lines that are parallel to the picture plane, called transversals, are drawn as parallel lines on the surface of the image. All lines that run parallel to the viewer's line of sight, referred to as orthogonals, converge at the so-called vanishing point. The vanishing point falls on the horizon line.

"In brief, one-point perspective begins with a horizon line, which defines the farthest distance of the background and a central vanishing point. To this vanishing point, orthogonals may be drawn from the bottom of the picture plane, which defines the foreground of the space. The orthogonals, vanishing point, and horizon line establish the space in which the artist may arrange figures, objects, or architecture such that they appear to exist in three dimensions. Once these basic elements have been set in place, the artist may add further elements to create a more complicated, yet more realistic, space. For example, to represent a square-tiled floor, the artist chooses another point on the horizon line, called the distance point, and draws a line through the orthogonals to a point at the bottom of the picture plane. The points at which this line bisects the orthogonals establish the points at which horizontal lines, called transversals, may be placed. These lines represent the perspectively correct regression of the square tiles into space. These elements of linear perspective link the science of three-dimensional geometry with the art of illusionistic representation."Bruce MacEvoy, "Two Point Perspective," 2015. https://www.handprint.com/HP/WCL/perspect3.html.

Two-point perspective, from: La perspective pratique, necessaire a tous peintres, graveurs, sculpteurs, architectes, orphevres, brodeurs, tapissiers, […], Jean Dubreuil, 1642 fig. 9 Two-point perspective
from: La perspective pratique, necessaire a tous peintres, graveurs, sculpteurs, architectes, orphevres, brodeurs, tapissiers, […]
Jean Dubreuil
1642

Instead, a two-point perspective drawing contains two vanishing points. Two-point perspective is used to draw the same type of objects that can be represented with one-point perspective, but rotated, for example, looking obliquely at the corner of a house (fig. 9) or the inside of a room. For objects viewed at an angle, the parallel lines of left-hand side of an object recede towards a vanishing point to the left of the central vanishing point (of the one-point perspective system) while those of the right-hand side converge at a vanishing point located to the right. As in one-point perspective, vertical lines that are parallel to the picture plane remain vertical. Distance points increasingly move away from the object as the observer recedes from the object; and vice versa, the points converge as an observer approaches it.

Two-point perspective greatly expands the range of pictorial effects and emotional associations of a perspective image, creating a palpable contrast between the viewer and the object, highlighting the accidental or arbitrary viewpoint of the viewer, or the distinctive physical presence of the object. Richard Talbot, "Speculations on the Origins of Linear Perspective Including Analyses of Masaccio's Trinity and Piero's Flagellation," Nexus Network Journal, vol. 5, no. 1, 2003.

When both one-and two-point perspectives are combined in the same drawing, all lines converge at vanishing points located on the horizon line. Objects with curves can also be rendered with one linear perspective but the process is much more complicated.

Architectural Fantasy with Susanna and the Elders, Dirk van Delen, c. 1619-1671, Oil on canvas, 105 x 137 cm., Museum Boijmans Van Beuningen, Rotterdamfig. 10 from: La perspective pratique, necessaire a tous peintres, graveurs, sculpteurs, architectes, orphevres, brodeurs, tapissiers, […]
Jean Dubreuil
1642

The principles of one-point perspective were eventually used to develop the shape of shadows cast by solid objects (fig. 10).

Three-point perspective is often used for buildings seen from above or below. In addition to the two vanishing points from before, one for each wall, there is now one for how the vertical lines of the walls recede. For an object seen from above, this third vanishing point is below the ground. For an object seen from below, as when the viewer looks up at a tall building, the third vanishing point is high in space.

Three-point perspective exists when the perspective is a view of a Cartesian scene where the picture plane is not parallel to any of the scene's three axes. Each of the three vanishing points corresponds with one of the three axes of the scene. One, two and three-point perspectives appear to embody different forms of calculated perspective, and are generated by different methods. Mathematically, however, all three are identical; the difference is merely in the relative orientation of the rectilinear scene to the viewer.

The Role of Perspective Manuals in Practice

During the Renaissance and Baroque periods, the study of perspective was a crucial aspect of artistic training and practice. Perspective manuals played a significant role in teaching artists the principles of linear perspective, which was a fundamental technique for creating the illusion of depth and three-dimensionality on a two-dimensional surface. For many artists, the mastery of perspective was seen as a mark of their technical proficiency and understanding of the natural world. As artists strived to depict reality in increasingly accurate ways, the knowledge and application of perspective were key components of achieving that goal.

These manuals were not just theoretical texts; they were practical guides that provided artists with step-by-step instructions on how to construct accurate and convincing spatial representations. The most ambitious artists of the time were deeply interested in exploring the mathematical and optical principles that underpinned perspective, as it allowed them to achieve a higher level of realism and naturalism in their works.

Perspective manuals covered a range of topics, including one-point, two-point, and even three-point perspective, as well as techniques for depicting complex architectural structures, foreshortening and other visual effects that could enhance the illusion of depth. These manuals often included diagrams, illustrations, and geometric constructions to help artists understand how to apply these principles to their own artwork.

† FOOTNOTES †

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If you discover a or anything else that isn't working as it should be, I'd love to hear it! Please write me at: jonathanjanson@essentialvermeer.com