Our perception of space is dominated by perspective, in the sense of a reduction of the projected size of objects with distance. One of the key jobs of the visual brain is to decode this size diminution as distance in the third dimension, or egocentric distance. If the eye were a pinhole camera, the projection of the world onto the back plane would be in perfect linear perspective (and in perfect focus). The succession of images projects on the curved retina within the eye in what
Leonardo da Vinci termed
natural perspective, a series of distorted projections that need to be integrated over time in a representation in the brain as the eye moves around the scene. How the brain decodes the information in natural perspective into an accurate appreciation of the spatial layout has yet to be resolved.
Incorporating lens optics into the projection system introduces the potential for curvature in the projected image. Such internal curvature may consequently be a property of human perception at the extremes of the field, but this curvature would apply equally to the original scene and to its projection from the picture plane to the eye, so does not affect the external projection rules of geometric perspective. The key simplification in perspective construction is that the pictorial image is governed by linear projection through the point where the pupil is located, regardless of any optical distortions beyond that point.
Historically, space representation through perspective has engendered great conceptual effort. Perspective scene painting was a springboard of mathematical geometry even at the time of Plato and remained influential through the Hellenistic era, but did not re-emerge as an artistic technique until the 1300s. However, full mastery of analytic perspective took another six centuries to evolve. Accurate one-point perspective dominated the 1400s, being first used by Masolino da Panicale and his pupil Masaccio. Two-point perspective was initially described by Viator in 1505, although the two-point construction remained unknown throughout the Renaissance until 1650, becoming widely used in the 1700 and 1800s. Three-point and multi-point construction diagrams for mathematical treatises were attempted unsuccessfully by Piero della Francesca and Leonardo da Vinci in the late 1400s, but none appeared in art works until an isolated example by Tiepolo in 1744. The three-point construction seems to have been first introduced into 20th-century art by Georgia O'Keeffe in her New York Series in the mid-1920s. Far from springing into force during the early Renaissance, therefore, a full understanding of linear perspective was not achieved for 600 years. Interestingly, most of the conceptual advances in perspective construction were made by artists rather than geometers.
Linear perspective is the geometry of projection of the lines in a scene through a picture plane to a point in space (or centre of projection) corresponding to the pupil of the viewing eye (Fig. 1). The picture plane would be the canvas on which the painter wishes to depict the scene. For correct perspective, the picture will generate the same arrangement of light rays at the eye as did the scene behind it. When viewed from this point in space, therefore, the picture will form exactly the same image on the retina as did the original scene. The different forms of perspective construction concern the rules that apply to specific structures, but all are subcases of the same optical transform.
Principles of perspective
1. In geometric perspective, all straight lines in space project to straight lines (or points, if end on) in the picture plane. Rotating the line within the plane of projection will not introduce any curvature, just a change in its extent within the line of projection. In the limit when the line is viewed head on, the projected line will contract to a point in the picture plane.
Humans actually view scenes with two eyes, but the straight-line projections for each eye are both straight lines (Fig. 2
a left and right). The average, or binocularly fused, projection is the combination of two straight lines and is therefore also a straight line (see Fig. 2
b). No curvature is introduced by the geometry of binocular combination.
2. The projections of all lines that are parallel in space either remain parallel in the picture plane or intersect at a single vanishing point (see Fig. 1). Although the lines run to infinity in space, their projection to a picture plane has a vanishing point at a specific location. Each different set of parallel lines intersects at a different vanishing point. Thus, the first job in perspective projection is to identify all the lines in the scene that are parallel to each other, then make sure that they are drawn so as to project to a common vanishing point.
In the particular case of
central perspective, all the lines on the scene are either parallel with the line of sight or at right angles to it, parallel with the picture plane (Fig. 3). There is thus only one vanishing point, which is directly in front of the viewer's eye for any viewing position (a requirement widely violated by Renaissance painters). When the eye is at this centre of projection, the perspective geometry in the picture plane is independent of the direction in which the eye is looking.
Central perspective is illustrated by the earliest exponent of accurate perspective, Masolino da Panicale, in his
Herod's Feast (Fig. 4). The primary set of parallels is those horizontal and receding from a viewer toward the central vanishing point. The other sets of parallels consist of any lines at right angles to the first set, at any angle within the picture plane. These parallels, such as the verticals of the sides of buildings, will all remain parallel within the picture plane.
A particular case of the lack of distortion in central projection is the case of circles, and circular arches (such as those at right in Fig. 4). When they are facing us, i.e. parallel to the picture plane, circles and parts of circles remain circular in the projection to the picture. The projection to the retina distorts them to ellipses, but they should be shown as circular in the picture.
3. The viewer's angle to any pair of vanishing points is the same as the angle between the generating parallels in space. In particular, the vanishing points for any 90-degree angle in space form a 90-degree angle at the viewer's eye (Fig. 5). However the angle is rotated in space (even to complete foreshortening), the vanishing points will none the less hold to a 90-degree angle at the viewer's eye. The physical distance between the vanishing points depends on the intended viewing distance, but a good rule of thumb is at least twice the width of the picture. Leonardo da Vinci recommended 10–20 times the height of the largest objects depicted.
4. All sets of parallel lines lying within a particular plane in space have vanishing points that fall on the horizon line defined by that plane. In Fig. 5, three sets of converging parallels are shown converging on the same horizon. The fourth set, the transversals, may be said to converge at the horizon at infinity to the left and right.
5. Although perspective distorts rectangles to asymmetrical trapezoids in general, the properties of circles are such that they always project to an ellipse of some orientation, with the centre of the resulting ellipse generally displaced forward from the centre of the projected circle (Fig. 6). Geometrically, ellipses may be drawn by attaching a string to the foci and drawing a perimeter by keeping the string fully stretched. However, there is no known geometric method for placing the foci for a required ellipse in the perspective construction.
6. Spheres in space also project to ellipses in the picture plane, although generally with much less distortion than for circles (Fig. 7). Spheres always project to circles when at the centre of projection. The elongation arises only because of marginal distortion, the stretching of the image as the picture plane itself recedes from the viewer at increasing angles of view.
In conclusion, perspective proves to be a rich paradigm for revealing the interplay of the mind with the space that it inhabits. Its problems posed in Greek antiquity have challenged the human intellect into the 21st century. Some issues, indeed, still resist a geometric solution. Moreover, virtually nothing is known about the representation of perspective in the brain itself. It will be fascinating to follow the investigation of the neural processing structure and dynamics of this profound modality of space representation by present and future brain-imaging techniques.
Fig. 1. Projection of parallel lines in space. Three parallel lines in the scene at left are projected through a rectangular picture plane to the point where the observer's eye is located. The light rays projecting to the eye are shown by dashed lines. The three parallel lines project as straight lines in the picture, but not as parallel because their orientation is not parallel to the picture plane. The point in the picture where the projected lines converge is termed the vanishing point (VP).
Fig. 2. Binocular perspective. a. The different left (l) and right (r) eye views of the three parallel lines projecting onto the picture plane. The two straight lines from each eye are combined in the brain by binocular fusion into the straight lines of fused image, b. whatever their geometric relationships.
Fig. 3. a. A painter at his canvas painting a scene of a rectangular grid with ziggurat. Dashed lines in the grid are parallel to the canvas (transversals), solid lines are perpendicular to it. b. Depiction of the scene as projecting to the plane of the canvas, with the perpendicular lines converging at the central vanishing point, while the transversals remain parallel to it.
Fig. 4. Early example of central perspective. Herod's Feast by Masolino (1435), where many receding horizontal lines project to a single central vanishing point. This is one-point perspective because the horizontal (black) and vertical (white) sets of parallel lines project as parallel in the picture.
Fig. 5. Viewing geometry for perspective constructions. The viewer's angle to the vanishing points is 90 degrees, matching the angle between the edges of the cube that generated the vanishing points.
Fig. 6. Perspective projection of a circle to an ellipse. a. The circle inscribed on the face of a cube. b. The simple ellipse, with its centre marked. c. The projection when the cube is on the visual axis forms an ellipse whose centre (and vertical axis) is displaced from that of the projected circle. d. For a circle above the visual axis, the major axis of the ellipse is rotated from the vertical, making its construction a challenging geometric problem with no known geometric solution.
Fig. 7. Projection of spheres. Spherical objects always form a circular cone projecting to the eye whatever the angle of view. Unless this cone is perpendicular to the picture plane, the sphere will project as an ellipse in the picture plane.
(Published 2004) — Chris Tyler
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