In the traditional account, Brunelleschi either discovered or verified linear perspective by literally painting views of buildings onto a mirror or windowpane. This "draw on glass" approach was probably not the method Brunelleschi actually used, but it fits well with the idea that a perspective painting is a mirror or window view of the world, and it soon became the standard way to convince drawing students that perspective really works, even when the scope of the problem is no bigger than a table top.

one more convert to perspective

from Charles Hayter, Introduction to Perspective (1813)

Want to try it? Punch a small hole in an index card, staple or tape the card to one end of a wood yardstick, then tape the other end of the yardstick to the back of a chair, so that the hole in the card is at your eye level as you straddle the chair. Put the chair in front of a large window, sit, and look through the hole in the card with one eye. Use an erasable marker pen or grease pencil to draw on the glass the buildings or objects in view, then hold a white paper behind the glass to see your finished design. As you draw, you may notice that this procedure feels confining. That's because it is  illustrating how far the assumptions of perspective differ from our fluid, constantly changing visual experience.

a renaissance perspective machine

from Albrecht Dürer, Underweysung der Messung mit Zirckel und Richtscheyt ... [Instruction How to Measure with Compass and Straight Edge ...] (1525)

This peephole or "peep show" tracing method does not make clear that linear perspective is really created by straight lines passing through a vertical plane. For that, the perspective "machine" (diagram, above) by the German artist Albrecht Dürer (1471-1528) is a better example. By looping a weighted string through an eye ring in the wall, one artist could hold the opposite end along the contours of a lute; a second artist could measure the height and horizontal position of the string where it passed through a rectangular frame, then transfer this point to a paper or canvas (shown as a hinged panel). By repeating the measurements dozens or hundreds of times, a "connect the dots" perspective view of the three dimensional lute could be constructed on a two dimensional surface. Through similar empirical methods, Renaissance artists worked out the geometric fundamentals of perspective technique.

Dürer's machine demonstrates that linear perspective explicitly depends on a single point of view in space. A perspective drawing of a building or landscape reveals the location, vantage and orientation of the viewer as precisely as it shows the physical form of the objects in view. The eye ring (diagram, above) represents the eye of the viewer or center of projection; the stretched string represents the straight beams of light or visual rays that converge on the eye from all objects in the field of view; the hinged surface is the artist's canvas or image plane. Everything comes down to that point of view defined by the wall ring or hole in index card: the point of view, not objects in space, is the fundamental perspective theme.

Linear perspective is fundamentally a geometrical method, not a mathematical one  which means you draw, you don't calculate. All you need are the standard construction tools from high school geometry: a straight edge (ruler), pencil and a compass (or a long strip of cardboard and some push pins). Unfortunately, these tools are awkward to use when the perspective view is extreme or the drawing is very large, and they make quantitative discussion of perspective issues difficult. So I explain numerical calculations, reduced drawings and other tricks that can help you through those problems. But calculations are not necessary to apply LP methods.

There is a sprawling and stale literature on linear perspective, but two points deserve mention. As James Elkins and Martin Kemp point out, the historical uses of perspective construction have always been opportunistic and evolving. Many seemingly "obedient" perspective paintings reveal, on closer examination, "fuzzy" or multiple vanishing points  some objects in the image are "in perspective" and other objects are not. Most artists chose to make "corrections" to apparently inconsistent perspective distortions, often in pursuit of a more pleasing finished composition: linear perspective often creates as many design problems as design solutions. In that limited sense, LP is not any better or worse for making "realistic" images than the various types of parallel or paraline perspectives used by architects and engineers  elevation, section and plan, or military, isometric and dimetric projections. In all these, similar distortions appear and are easily ignored. The moral is that perspective must be used gently: it is a very cool drawing tool and a compelling representational convention, not the ultimate code of drawing correctness.

The best approach is to rely on LP to lay out basic proportions and shapes, to clarify difficult drawing problems, and to guide your intuitions about forms and textures in space. Freehand perspective and trust in your visual judgment should not be pushed aside, but nourished and exercised by the insights linear perspective can provide. As Vasari wrote of Michelangelo, "he held his compasses, that is to say his judgment, in his eyes and not his hands."

I've divided these diverse materials into separate pages, and occasionally use nonstandard terminology in the interest of clarity. I hope to provide a practical and provocative reference to the topic.

References. There are many books on linear perspective, and they are wildly uneven in quality and practicality. For beginners, Perspective Made Easy by Ernest Norling (Norton, 1967), first published in 1939, provides a very lucid, entertaining and highly practical general introduction. John Raynes's A Complete Guide to Perspective (Collins & Brown, 2005) introduces perspective problems through an attractive selection of photographs and diagrams, but is rather informal about construction solutions, especially for shadows and reflections. The discussion in Perspective for Artists by Rex Vicat Cole (Norton, 1976) is a more extensive and practical presentation, though somewhat out of date. The most comprehensive practical instructions, tailored to the tools and methods of architects and draftsmen, can be found in Perspective Drawing: A Step-by-Step Handbook by Michael E. Helms (Prentice Hall, 1997), available as a facsimile reprint. Another useful though occasionally inaccurate guide is Perspective: From Basic to Creative by Robert W. Gill (Thames & Hudson, 2006). The elegant "circle of view" approach adopted here is based on Linear Perspective: Its History, Directions for Construction, and Aspects in the Environment and in the Fine Arts by Willy Bärtschi (Van Nostrand Reinhold, 1981), regrettably now out of print but probably too formal for most artists.

The following sources expand on the historical and artistic context for perspective techniques. The Science of Art by Martin Kemp (Yale University Press, 1990) is a careful study of the historical development and use of linear perspective. The Poetics of Perspective by James Elkins (Cornell University Press, 1994) is a stimulating if academic study of the development and artistic impact of perspective techniques from the early Renaissance to early Baroque. E.H. Gombrich's classic Art and Illusion (Princeton University Press, 1960) is the most articulate defense of linear perspective as an accurate description of the visual world. On that theme, Optics, Painting and Photography by M.H. Pirenne (Cambridge University Press, 1970) experimentally (and with clear photographic examples) explores and confirms the accuracy and limitations of perspective as a description of natural human vision. The most famous dissenting view is Erwin Panofsky's Perspective as Symbolic Form (1924; Zone Books, 1997), treacherous for the uninformed but an amusement for adepts. There are also many web pages devoted to perspective, most of them useless; start with the online translation of Alberti's Della Pittura, which is available in paperback from Penguin Books. There are many books describing perspective for use in CAD programs, which are less useful for a painter.

PAGE INDEX

Page 1. Perspective in the World

The Texture of Space. Perspective originates in the common appearance of the real world, where texture is as important as geometrical buildings in our experience of distance.

Four Perspective Facts. Everything in perspective arises from four perspective facts  the straight line path of light, a fixed viewpoint, the visual cone centered on the direction of view, and the image cross section through the visual cone.

Creating the Perspective View. The four perspective facts provide the basis for a framework to represent the three dimensional world on a two dimensional surface.

The Perspective Setup. We discover the structure of the perspective view by "looking at" a metric grid; this perspective structure becomes the 90° circle of view framework that allows us to solve any perspective problem.

Basic Rules of Perspective. Nearly all perspective constructions can be explained in terms of 15 perspective rules (and a perspective glossary).

Image Plane, Viewpoint & Direction of View. The orientation of the image plane to the viewer's head is governed by the display convention, viewing convention and projection assumption; changing the direction of view changes the perspective in the image; the height of the viewpoint is always shown by the virtual horizon line.

Perspective Distortions. Viewing a perspective image from any location other than the center of projection creates distortions in the image objects: these are "cured" by reducing the circle of view, or can be manipulated for expressive effect.

Page 2. Central Perspective

The Visual Ray Method. Artists and mathematicians of the 18th century discovered the "double fold" that allows perspective problems to be analyzed in terms of visual rays from the viewpoint to the plan view of an object.

One Point Perspective. In 1PP or central perspective, parallel lines define horizontal and vertical dimensions and a single vanishing point (the principal point) defines recession in depth.

Orthogonals & Central Recession. Orthogonals define the viewer's central recession, which is unaffected by the orientation of objects in view; they project a unit dimension in depth and, with the diagonal vanishing points, can define perspective distance from the viewer.

Slanting & Sloping Planes. Simply by rotating the circle of view or raising the horizon line, central perpective can represent slanting or sloping surfaces.

Perspective Gradients. Perspective controls the appearance of textures or patterns on all natural and artificial surface: object distance and ground height on the viewing plane are inversely proportional.

Distance & Size. Triangular proportions are the geometric core of perspective; using them we can define the exact size of objects we want to draw, provided we know the viewing distance to painting and the distance or actual size of the objects.

Scaling the Drawing. Discussion of central recession and format dimensions, with step by step instructions for scaling the circle of view, object distance, image size and image location using the anchor point and measure bar.

Display Geometry & Image Impact. The geometrical relationships in a perspective image strongly affect its visual impact and esthetic qualities.

Anamorphic Images. Images can be distorted, using projection methods similar to linear perspective, so that they appear flat or veridical when viewed obliquely or in a curved mirror.

Constructing a 1PP Cube. The step by step method for constructing a three dimensional rectangular solid using 1PP or central perspective projection methods, with emphasis on the artistic decisions that are part of the process.

Early Renaissance Methods. The earliest uses of central perspective were based on practical drawing methods, not abstract geometry.

Page 3. Two Point Perspective (2PP)

Two Point Perspective. The basic geometry of 2PP projections, in which there are two vanishing points for the sides of a rectangular solid seen at an angle.

Rotating the Vanishing Points. The correct method for locating 2PP vanishing points in relation to the 90° circle of view.

Locating the Measure Points. The isoceles triangle that defines the geometry of measure points; locating measure points from established vanishing points.

Constructing a 2PP Cube. The step by step method for constructing a three dimensional rectangular solid using 2PP projection methods.

Inclined Lines & Inclined Planes. Methods for finding the vanishing point of an inclined line and the vanishing line of an inclined plane.

Distance Point Projection. A tradition method for constructing 2PP drawings relies on the diagonal vanishing points to project a square and its diagonal into perspective space.

The Ground Line Framework. The alternative perspective setup commonly used by draftsmen and architects.

Who Has a 12 Foot Table? It's not uncommon for the vanishing points in 2PP to be 12 feet or more apart. There are three different remedies for this problem.

VP Spacing from an Object Drawing. There are inflexible geometrical relationships between the size of the object, its distance from the viewer, and the spacing of the vanishing points in 2PP drawing. These limit your freedom to guess at vanishing point relationships.

Where Is the Center of Projection? The circle of view approach allows you to find the center of projection implied by any 2PP image that contains defining diagonal elements.

Page 4. Three Point Perspective (3PP)

Three Point Perspective. The basic geometry of 3PP projections, in which there are three vanishing points for all sides of a rectangular solid that is both rotated and tilted to the image plane.

The Perspective Sketch Construction Method. The step by step method for constructing a 3PP projection of a cube by starting with a sketch of the desired perspective proportions.

Constructing a 3PP Cube (Sketch Method). The step by step method for constructing a three dimensional cube using 3PP projection methods, with two methods to define the measure points.

The Horizon Line Construction Method. The geometrical logic for rotating the 3PP vanishing points, which permits them to be located and the drawing scaled exactly.

Constructing a 3PP Drawing (Horizon Line Method). Step by step explanation of how a 3PP drawing is made, from rotating the vanishing points to scaling the image size.

Page 5. Advanced Perspective Techniques

Perspective of Complex Plane Figures. Projecting complex two dimensional figures into perspective space, including an ellipse, a pentagon, a city map, and three ways to project a circle.

Perspective of Complex Solid Forms. Complex solid forms can be projected into perspective space by using various strategies; examples include Uccello's chalice, a sphere, the Leaning Tower of Pisa, two types of cones, a rectangular solid, a series of arches, a staircase, an octahedron, a dodecahedron, and the human figure.

Buildings From Blueprints or Plans. Buildings and large objects can be projected into perspective space by deriving measure bars and vanishing points from plans, elevations and profile drawings.

Paraline Perspectives. Perspective systems based on nonconverging (parallel) lines, which can represent all three dimensions and all internal right angles at equal scale and without recession.

Curvilinear Perspectives. Recent perspective systems based on curved vanishing lines, designed to represent the subjective curvatures of natural vision.

Page 6. Shadows, Reflections & Atmosphere

Basic Rules of Perspective Shadows. The basic geometry and terminology for perspective shadow constructions.

Shadows from Solar Light. Different construction problems arise depending on whether the shadow creating edge and the shadow receiving surface are vertical, horizontal or inclined.

Shadows from Local Lights. Nearby light sources complicate shadow perspective because they are not point sources and produce enlarged or diffuse shadow shapes.

Perspective of Reflections. Reflections from mirrors or still water surfaces present simple construction problems; ripples in water produce more complex periodic patterns; convex or spherical surfaces create optical distortions.

Aerial Perspective. Landscape distance is suggested by the many subtle changes in detail, color and tone that occur when air filters light across large distances, or smoke or fog filters light across a romantic rendezvous.

Rainbows. Rainbows combine color, atmosphere, landscape and light in panoramic proportions of beauty that vary with weather, time of day, geographic location and season. They signal the end of our perspective studies.