Perspective properly refers to any of various graphic techniques for depicting volumes and spatial relationships on a flat surface, such as size and atmospheric perspective. The term perspective, however, most often brings to mind the drawing system of linear or artificial perspective.
Linear perspective is the art and science of describing three-dimensional volumes and spatial relationships on a two-dimensional surface by means of lines that converge as they recede into the depth of a drawing. While multiview and paraline drawings present mechanical views of an objective reality, linear perspective offers sensory views to the mind’s eye of an optical reality. It depicts how objects and space might appear to the eye of a spectator looking in a specific direction from a particular vantage point in space. While our eyes can rove about the surface of a plan or isometric guided by whim or reason, we are invited to read a linear perspective from a fixed position in space.
Linear perspective is valid only for monocular vision. A perspective drawing assumes the spectator sees through a single eye. We almost never view anything in this way. Even with the head in a fixed position, we see through both eyes, which are constantly in motion, roving over and around objects and through ever-changing environments. Through this constant scanning, we build up experiential data that the mind manipulates and processes to form our perception and understanding of the visual world. Thus, linear perspective can only approximate the complex way the eyes actually function.
Still, linear perspective provides us with a method for correctly placing three-dimensional objects in pictorial space and illustrating the degree to which their forms appear to diminish in size as they recede into the depth of a drawing. The uniqueness of a linear perspective lies in its ability to provide us with an experiential view of space. This distinct advantage, however, also gives rise to the difficulty often connected with perspective drawing. The challenge in mastering linear perspective is resolving the conflict between our knowledge of the thing itself—how we conceive its objective reality—and the appearance of something—how we perceive its optical reality—as seen through a single eye of the spectator.
Perspective projection represents a three-dimensional object by projecting all its points to a picture plane by straight lines converging at a fixed point in space representing a single eye of the spectator. This convergence of sightlines differentiates perspective projection from the other two major projection systems, orthographic projection and oblique projection, in which the projectors remain parallel to each other.
The direct projection method of perspective construction requires the use of at least two orthographic views: a plan view and a side elevation view. The side elevation is an orthographic projection that is perpendicular to the picture plane, but rotated 90° to be coplanar with the picture plane. The object, picture plane, and station point are shown in both views.
The perspective of any point is where a sightline from the station point to the point in question intersects the picture plane. To find the perspective projection of a point:
For a point behind the picture plane, draw a sightline from the point in question toward the station point until it meets the picture plane. If the point lies in the picture plane, simply drop the plan position vertically until it meets a horizontal line from the point in the elevation view. If the point is in front of the picture plane, draw a sightline from the station point, through the point, and extend it until it meets the picture plane.
To find the perspective projection of a line, establish the perspective projections of its endpoints and connect the points. If we can establish the perspective projections of points and lines in this way, we can also find the perspective projections of planes and volumes.
In theory, it is not necessary to use vanishing points in the direct projection method. However, the establishment and use of vanishing points greatly simplify the drawing of a linear perspective and ensure greater accuracy in determining the direction of receding lines.
The converging nature of sightlines in linear perspective produces certain pictorial effects. Being familiar with these pictorial effects will help us understand how lines, planes, and volumes should appear in linear perspective and how to correctly place objects in the space of a perspective drawing.
Convergence in linear perspective refers to the apparent movement of parallel lines toward a common vanishing point as they recede. As two parallel lines recede into the distance, the space between them will appear to diminish. If the lines are extended to infinity, they will appear to meet at a point. This point is the vanishing point for that particular pair of lines and all other lines parallel to them.
The first rule of convergence is that each set of parallel lines has its own vanishing point. A set of parallel lines consists only of those lines that are parallel to one another. If we look at a cube, for example, we can see that its edges comprise three principal sets of parallel lines, one set of vertical lines parallel to the x-axis, and two sets of horizontal lines, perpendicular to each other and parallel to the y- and z-axes.
In order to draw a perspective, we must know how many sets of parallel lines exist in what we see or envision and where each set will appear to converge. The following guidelines for the convergence of parallel lines is based solely on the relationship between the spectator’s central axis of vision and the subject.
We can categorize any line in linear perspective according to its relationship to the picture plane.
If oblique to the picture plane, a set of parallel lines will appear to converge toward a common vanishing point as it recedes.
In orthographic and oblique projection, the projectors remain parallel to each other. Therefore, the projected size of an element remains the same regardless of its distance from the picture plane. In linear perspective, however, the converging projectors or sightlines alter the apparent size of a line or plane according to its distance from the picture plane.
The farther away an object is from the picture plane, the narrower the angle between the sightlines to the object and the closer together are the intersections of the sightlines with the picture plane. The converging sightlines therefore reduce the size of distant objects, making them appear smaller than identical objects closer to the picture plane.
Note also that, as the object continues to recede, the more the sightlines to the object will approach the horizon line. For example, looking down on a tiled floor pattern, we can see more of the tiles’ surfaces in the foreground. As the same-sized tiles recede, they appear smaller and flatter as they rise and approach the horizon.
Foreshortening refers to the apparent change in form an object undergoes as it rotates away from the picture plane. It is usually seen as a contraction in size or length in the direction of depth so as to create an illusion of distance or extension in space.
Any facet of an object that is not parallel to the picture plane will appear compressed in size or length when projected. In perspective projection, as well as in orthographic and oblique projection, the amount of contraction depends on the angle between the facet of the object and the picture plane. The more a line or plane is rotated away from the picture plane, the less we will see of its length or depth.
In linear perspective, the apparent contraction in depth also depends on the angle between the sightlines to the object and the picture plane. The farther away an object is from the center of vision, the greater the angle between the sightlines to the object and the farther apart are the intersections of the sightlines with the picture plane. In other words, as the object moves laterally, parallel to the picture plane, its apparent size will increase. Note that this is the opposite of what occurs as the object recedes away from the spectator. At some point, the size of the object will be exaggerated and its form distorted. We use the cone of vision to limit our view in linear perspective and control this distortion.
While convergence, diminution of size, and foreshortening affect the apparent form of lines and planes, they also influence the compression of spatial relationships in a perspective drawing.
The spectator’s point of view determines the pictorial effect of a perspective drawing. As this viewpoint changes—as the spectator moves up or down, to the left or right, forward or back—the extent and emphasis of what the spectator sees also change. In order to achieve the desired view in perspective, we should understand how to adjust the following variables.
The height of the station point determines whether an object is seen from above, below, or within its own height. Assuming a level central axis of vision, as the station point—the eye of the spectator—moves up or down, the horizon line moves up or down with it. Any horizontal plane at the level of the spectator’s eye appears as a line. We see the tops of horizontal surfaces that lie below the spectator’s eye level and the undersides of horizontal planes that are above.
The distance of the station point to the object influences the rate of foreshortening that occurs in the perspective drawing. As the spectator moves farther away from the object, the vanishing points move farther apart, horizontal lines flatten out, and perspective depth is compressed. As the spectator moves forward, the vanishing points move closer together, horizontal angles become more acute, and perspective depth is exaggerated. In theory, a perspective drawing presents a true picture of an object only when the eye of the viewer is located at the assumed station point of the perspective.
The orientation of the central axis of vision relative to the object determines which faces of the object are visible and the degree to which they are foreshortened in perspective. The more oblique a plane is to the picture plane, the more it is foreshortened in perspective; the more frontal the plane is, the less it is foreshortened. When a plane becomes parallel to the picture plane, its true shape is revealed.
The location of the picture plane only affects the size of the perspective image. The closer the picture plane is to the station point, the smaller the perspective image; the farther away the picture plane is, the larger the image. Assuming all other variables remain constant, the perspective images are identical in all respects except size.
In constructing a perspective by hand, we must have experience in setting up the station point and the angle of view to predict and achieve a reasonable outcome. A distinct advantage in using 3D CAD and modeling programs is that once the necessary data is entered for a three-dimensional construction, the software allows us to manipulate the perspective variables and fairly quickly produce a number of perspective views for evaluation. 3D CAD and modeling programs, while following the mathematical principles of perspective, can easily create distorted perspective views. Judgment of what a perspective image conveys, whether produced by hand or with the aid of the computer, remains the responsibility of its author.
Illustrated are examples of computer-generated perspectives, showing how the various perspective variables affect the resulting images. The differences in the perspective views may be subtle, but they do affect our perception of the scale of the spaces and our judgment of the spatial relationships the images convey.
Only lines and planes coincident with the picture plane can be drawn to the same scale. The converging sightlines in linear perspective reduce the size of distant objects, making them appear smaller than identical objects closer to the picture plane. The converging sightlines also increase the apparent size of objects in front of the picture plane. Because of the combined effect of convergence and diminishing size, it is more difficult to make and draw measurements in linear perspective than in other drawing systems. But there are techniques we can use to determine the relative heights, widths, and depth of objects in the pictorial space of a perspective drawing.
In linear perspective, any line in the picture plane displays its true direction and true length at the scale of the picture plane. We can therefore use any such line as a measuring line.
Any line that can be used to measure true lengths in a projection drawing. While a measuring line may have any orientation in the picture plane, it typically is vertical or horizontal and used to measure true heights or widths. The groundline is one example of a horizontal measuring line.
Once we establish a height or width, we can transfer the measurement horizontally or vertically, as long as we make the shift parallel to the picture plane. Since parallel lines by definition remain equidistant but appear to converge as they recede in perspective, we can also use a pair of parallel lines to transfer a vertical or horizontal measurement into the depth of a perspective. We can transfer measurements made in this manner vertically or horizontally as long as the shift occurs in a plane parallel to the picture plane.
The measurement of perspective depth is more difficult and requires a certain degree of judgment based on direct observation and experience. The various methods of perspective construction establish depth in different ways. Once we establish an initial depth judgment, however, we can make succeeding depth judgments in proportion to the first.
For example, each time we halve the distance from the ground plane to the horizon line, we double the perspective depth. If we know how far away a point on the ground plane is from the spectator, we can subdivide on a proportional basis the height of the horizon line above the ground plane and establish the position of points further back in the depth of a perspective drawing.
There are two methods for subdividing depth measurements in linear perspective.
In any projection system, we can subdivide any rectangle into four equal parts by drawing two diagonals. For example, if we draw two diagonals across a rectangular plane in perspective, they will intersect at the exact center of the plane. Lines drawn through this midpoint, parallel to the edges of the plane, will subdivide the rectangle and its receding sides into equal parts. We can repeat this procedure to subdivide a rectangle into any even number of parts.
To subdivide a rectangle into an odd number of equal parts or its sides into a series of unequal segments, its forward edge must be parallel to the picture plane so that it can be used as a measuring line. On this forward edge, mark off the same proportional subdivisions to be made in the depth of the perspective. From each of the marked points, draw parallel lines that converge at the same point as the sides of the plane. Then draw a single diagonal. At each point where this diagonal crosses the series of receding lines, draw lines parallel to the forward edge. These mark off the desired spaces, which diminish as they recede in perspective. If the rectangle is a square, then the subdivisions are equal; if it is not, then the segments are proportional but not equal.
Since any line parallel to the picture plane can be subdivided proportionately to scale, we can use the line as a measuring line to subdivide any intersecting line into equal or unequal parts. First, define a triangle by connecting the ends of the measuring line and the adjacent line. Then, mark off the desired subdivisions on the measuring line to scale. From each of these points, draw lines parallel to the closing line of the triangle, which converge at the same vanishing point. These lines subdivide the adjacent line into the same proportional segments.
If the forward edge of a rectangular plane is parallel to the picture plane, we can extend and duplicate its depth in perspective. First establish the midpoint of the rear edge opposite the forward edge of the rectangle. Then extend a diagonal from a forward corner through this midpoint to meet an extended side of the rectangle. From this point draw a line parallel to the forward edge. The distance from the first to the second edge is identical to the distance from the second to the third edge, but the equal spaces are diminished in perspective. This procedure can be repeated as often as necessary to produce the desired number of equal spaces in the depth of a perspective drawing.
The perspective view shows four rectangular planes in space. Assume the forward edge of each plane is parallel to the picture plane. Make three copies of the perspective view. On the first copy, subdivide the depth of each of the planes into four equal parts.
On the second copy, subdivide the depth of each of the planes into five equal parts.
On the third copy, double the depth of each of the planes.
Assume the forward face of each cube is parallel to the picture plane. First halve the depth of each cube. Then double the original depth of each cube.
Once we are familiar with how lines parallel to the three principle axes of an object converge in linear perspective, we can use this rectilinear geometry as the basis for drawing perspective views of inclined lines and circles.
Inclined lines parallel to the picture plane retain their orientation but diminish in size according to their distance from the spectator. If perpendicular or oblique to the picture plane, however, an inclined set of parallel lines will appear to converge at a vanishing point above or below the horizon line.
We can draw any inclined line in perspective by first finding the perspective projections of its end points and then connecting them. The easiest way to do this is to visualize the inclined line as being the hypotenuse of a right triangle. If we can draw the sides of the triangle in proper perspective, we can connect the end points to establish the inclined line.
If we must draw a number of inclined parallel lines, as in the case of a sloping roof, a ramp, or a stairway, it is useful to know where the inclined set appears to converge in perspective. An inclined set of parallel lines is not horizontal and therefore will not converge on the horizon line. If the set rises upward as it recedes, its vanishing point will be above the horizon line; if it falls as it recedes, it will be appear to converge below the horizon line.
To determine the vanishing point for any inclined set of parallel lines:
A line along which all sets of parallel lines within a plane will appear to converge in linear perspective. The horizon line, for example, is the vanishing trace for all horizontal planes.
The steeper the inclined set of parallel lines, the farther up or down on the vanishing trace will be its vanishing point. If an inclined set of parallel lines rises upward and another set in the same vertical plane falls at the same but opposite angle to the horizontal, the distance of their respective vanishing points above and below the horizon line are equal.
The circle is the essential basis for drawing cylindrical objects, arches, and other circular forms. The perspective view of a circle remains a circle when it is parallel to the picture plane. The perspective view of a circle is a straight line when the projectors radiating from the station point are parallel to the plane of the circle. This occurs most frequently when the plane of the circle is horizontal and at the height of the station point, or when the plane of the circle is vertical and aligned with the central axis of vision.
In all other cases, circles appear as elliptical shapes in perspective. To draw a circle in perspective, first draw a perspective view of a square that circumscribes the circle. Construct the diagonals of the square and indicate where the circle crosses the diagonals with additional lines parallel to the sides of the square or tangent to the circumference of the circle. The larger the circle, the more subdivisions are necessary to ensure smoothness of the elliptical shape.
In a plan view of the perspective setup, the sightlines from the station point to tangent points on the circumference of the circle define the widest part of the circle in perspective. This width, which is the major axis of the ellipse representing the circle in perspective, is not coincident with the actual diameter of the circle. Just as the forward half of a square in perspective is greater than the rear half, so is the nearer half of a perspective circle fuller than the far half.
We tend to see things as we believe them to be. So while a circle in perspective appears to be an ellipse, we tend to see it as a circle and thus exaggerate the length of its minor axis. The minor axis should appear to be perpendicular to the plane of the circle. Checking the relationship between the major and minor axes of elliptical shapes helps to ensure accuracy of the foreshortening of circles in perspective.
In the perspective view below, use the principles of perspective geometry to construct the following:
In any rectilinear object, such as a cube, each of the three principal sets of parallel lines has its own vanishing point. Based on these three major sets of lines, there are three types of linear perspective: 1-, 2-, and 3-point perspectives. What distinguishes each type is simply the spectator’s angle of view relative to the subject. The subject does not change, just our view of it and how the sets of parallel lines will appear to converge in linear perspective.
If we view a cube with our central axis of vision perpendicular to one of its faces, its vertical lines are parallel with the picture plane and remain vertical. The horizontal lines parallel with the picture plane and perpendicular to the central axis of vision also remain horizontal. The lines parallel with the central axis of vision, however, will appear to converge at the center of vision. This is the one point referred to in one-point perspective.
If we shift our view so that the same cube is viewed obliquely but keep our central axis of vision horizontal, then the vertical lines will remain vertical. The two sets of horizontal lines, however, are now oblique to the picture plane and will appear to converge, one set to the left and the other to the right. These are the two points referred to in two-point perspective.
If we lift one end of the cube off the ground plane, or if we tilt our central axis of vision to look down or up at the cube, then all three sets of parallel lines will be oblique to the picture plane and appear to converge at three different vanishing points. These are the three points referred to in three-point perspective.
Note that each type of perspective does not imply that there are only one, two or three vanishing points in a perspective. The actual number of vanishing points will depend on our point of view and how many sets of parallel lines there are in the subject being viewed. For example, if we look at a simple gable-roofed form, we can see that there are potentially five vanishing points since we have one set of vertical lines, two sets of horizontal lines, and two sets of inclined lines.
The one-point perspective system assumes that two of the three principal axes—one vertical and the other horizontal—are parallel to the picture plane. All lines parallel to these axes are also parallel to the picture plane and therefore retain their true orientation and do not appear to converge. For this reason, one-point perspective is also known as parallel perspective.
The third principal axis is horizontal, perpendicular to the picture plane, and parallel with the central axis of vision. All lines parallel to this axis converge on the horizon line at the center of vision. This is the particular vanishing point referred to in one-point perspective. The convergence of a major set of parallel lines at this central vanishing point is the dominant visual characteristic of one-point perspective.
A one-point perspective may not effectively explain the three-dimensional form of a rectilinear object if the receding lines and planes that impart depth are not visible in the perspective view. In depicting spatial volumes, however, the one-point perspective system is particularly effective since the display of three bounding faces provides a clear sense of enclosure. For this reason, designers often use one-point perspectives to present experiential views of street scenes, formal gardens, courtyards, colonnades, and interior rooms and spaces. We can also use the presence of the central vanishing point to focus the viewer’s attention and emphasize axial and symmetrical arrangements in space.
The diagonal point method for constructing a one-point perspective enables us to obtain accurate depth measurements directly within the perspective view without making projections from a plan view. It requires only an elevation or section view and is therefore especially useful in constructing section perspectives.
The method uses the geometry of a 45° right triangle and the principles of convergence to make depth measurements in perspective. We know that the perpendicular sides of a 45° right triangle are equal in length. Therefore, if we can draw one side of a 45° right triangle to scale, the hypotenuse will mark off an equal length on the perpendicular side.
The technique involves establishing one side of the 45° right triangle in or parallel to the picture plane so that we can use it as a measuring line. Along this side, we measure a length equal to the desired perspective depth. From one endpoint of this length, we draw the perpendicular side that converges at the center of vision. From the other endpoint, we draw the hypotenuse that converges at the vanishing point for lines making a 45° angle with the picture plane. This diagonal marks off a perspective depth along the perpendicular side equal to the scaled length of the parallel side.
We begin with an elevation or section view perpendicular to the spectator’s central axis of vision and coincident with the picture plane. The scale of the elevation or section view establishes the size of the perspective drawing.
Refer to the discussion of perspective variables to review how varying the distance from the station point to the subject, raising or lowering the horizon line, and locating the picture plane all affect the pictorial nature of a perspective view.
To utilize the diagonal point method, we must locate the vanishing point for a set of parallel lines that make a 45° angle with the picture plane. The vanishing point for any set of parallel lines is that point where a sightline from the station point, drawn parallel to the set, intersects the picture plane. Therefore, if we draw a 45° line from the station point in a plan view of the perspective setup, it will intersect the picture plane at the vanishing point for all 45° diagonals. We call this vanishing point a diagonal point or distance point.
There is one diagonal point for horizontal lines receding to the left at a 45° angle to the picture plane, and another for horizontal lines receding to the right at a 45° angle to the picture plane. Both diagonal points lie on the horizon line, equidistant from the center of vision. From the geometry of the 45° right triangle, we also know that the distance from each diagonal point to the center of vision is equal to the distance from the spectator’s station point to the picture plane.
If we understand this geometric relationship, we need not set up a plan view of the perspective setup directly above the perspective view. We can simply locate either or both diagonal points directly in the perspective view, on the horizon line, at a distance from the center of vision equal to the distance of the spectator from the picture plane. For a 60° cone of vision, the distance from the center of vision to either diagonal point should be equal to or greater than the width of the elevation or section view.
For example, if the spectator is standing 20 feet away from the picture plane, the diagonal point on the horizon line will be 20 feet away to the left or right of the center of vision. This distance, measured at the same scale as the picture plane, establishes the vanishing point for all 45° lines receding to the left or right.
If we move the diagonal points toward the center of vision, this is equivalent to the spectator moving closer to the picture plane and seeing more of the receding faces of the space. If we shift the diagonal points farther away from the center of vision, the spectator also moves farther away from the picture plane and the receding faces of the space become more foreshortened.
The basic steps in using a diagonal point to make depth measurements are:
The diagonal points for both sets of 45° lines in the ground, floor, ceiling, and any other horizontal plane are located on the horizon line. The diagonal points for both sets of 45° lines in a sidewall or any other vertical plane perpendicular to the picture plane are located on a vertical vanishing trace drawn through the center of vision. All four diagonal points are equidistant from the center of vision and lie on the circumference of a circle whose center is the center of vision. While only one diagonal point is required to measure perspective depths, knowing there are three others gives us flexibility in the actual construction of the perspective view.
If a diagonal point is too far removed from the center of vision to be accessible, we can use a fractional diagonal point to establish depth measurements. This technique is based on the geometric principle that corresponding sides of similar triangles are proportional.
To establish a fractional diagonal point, we divide the true distance from the center of vision to either diagonal point by a factor of two or four. A half diagonal point will mark off two units of depth for every unit of width measured parallel to the picture plane; a quarter diagonal point will mark off four units of depth for every unit of width measured parallel to the picture plane.
Assume the spectator is standing 15 feet away from the front face of a 10-foot cubic volume and that this face is coincident with the picture plane. Locate the following points in linear perspective:
Assume the spectator is standing 15 feet away from the picture plane and looking toward a wall that is 16 feet wide, 12 feet high, and 30 feet away. Construct a one-point perspective of the space. Within this perspective view, construct:
A perspective grid is a perspective view of a three-dimensional coordinate system. The three-dimensional network of uniformly spaced points and lines enables us to correctly establish the form and dimensions of an interior or exterior space as well as to regulate the position and size of objects within the space. Several types, varying in scale and point of view, are commercially available. We can also use the following procedure to construct a one-point perspective grid:
Construct a one-point perspective grid on an 81⁄2" × 11" sheet of quality tracing paper or vellum. Assume a scale of 1⁄2" = 1'-0" for the picture plane and a horizon line 5 or 6 feet above the ground line. When completed, the perspective grid may be photocopied and enlarged or reduced to any desired scale. By laying tracing paper over the grid, we can utilize the perspective structure to more easily sketch freehand perspective views of both exterior and interior spaces.
The section perspective combines the scaled attributes of a section drawing and the pictorial depth of a perspective drawing. Therefore, it is able to illustrate both the constructional aspects of a design as well as the quality of the spaces formed by the structure. We begin a section perspective with a building section drawn at a convenient scale. Since the section cut is assumed to be coincident with the picture plane of the perspective, it serves as a ready reference for making vertical and horizontal measurements for the perspective drawing.
The schematic building section below is drawn at a scale of 1⁄4" = 1'-0". Given the horizon line, center of vision, and left diagonal point indicated, convert the section into a section perspective.
In order to transform a two-dimensional floor plan into a three-dimensional view, we can draw a plan perspective—a one-point perspective view of an interior room or exterior space as seen from above.
We assume the spectator’s central axis of vision is vertical and the picture plane is coincident with a horizontal plane passing through the tops of the walls of the space.
The two-point perspective system assumes that the spectator’s central axis of vision is horizontal and the picture plane is vertical. The principal vertical axis is parallel to the picture plane and all lines parallel to it remain vertical and parallel in the perspective drawing. The two principal horizontal axes, however, are oblique to the picture plane. All lines parallel to these axes therefore appear to converge to two vanishing points on the horizon line, one set to the left and the other to the right. These are the two points referred to in two-point perspective.
The pictorial effect of a two-point perspective varies with the spectator’s angle of view. The orientation of the two horizontal axes to the picture plane determines how much we will see of the two major sets of vertical planes and the degree to which they are foreshortened in perspective. The more oblique a plane is to the picture plane, the more it is foreshortened in perspective; the more frontal the plane is, the less it is foreshortened.
Two-point perspective is probably the most widely used of the three types of linear perspective. Unlike one-point perspectives, two-point perspectives tend to be neither symmetrical nor static. A two-point perspective is particularly effective in illustrating the three-dimensional form of objects in space ranging in scale from a chair to the mass of a building.
In depicting a spatial volume, such as the interior of a room or an exterior courtyard or street, a two-point perspective is most effective when the angle of view approaches that of a one-point perspective. Any perspective view that displays three bounding faces of a spatial volume provides a clear sense of enclosure. The spectator then becomes an integral part of the space rather than a mere observer looking in from the outside.
The common method of constructing a two-point perspective is also known as the office method. It requires the use of two orthographic projections: a plan view and an elevation view. The scale of the plan and elevation views establishes the scale of the picture plane in the perspective view.
While the plan, elevation, and perspective views are shown some distance apart for clarity, they can be arranged in a more compact manner to fit a smaller workspace. To do this, move the plan and elevation closer to or under the sheet used for the perspective view, being careful to maintain the proper horizontal and vertical relationships between the three views.
Refer to the discussion of perspective variables to review how varying the distance from the station point to the subject, raising or lowering the horizon line, and locating the picture plane all affect the pictorial nature of a perspective view.
The vanishing point for any set of parallel lines is the point where a sightline from the station point, drawn parallel to the set, intersects the picture plane.
Any line in the picture plane displays its true length at the scale of the picture plane. Therefore, we can use any such line as a measuring line. While a measuring line may have any orientation in the picture plane, it is usually vertical or horizontal and used to measure true heights and widths.
If we know where the base of a vertical line meets the ground plane in perspective, we can determine its perspective height in two additional ways:
A second method for determining the perspective height of a vertical line involves the height of the horizon line above the ground plane. If this height is known, we can use it as a vertical scale to measure vertical lines anywhere in the depth of a perspective.
Once we find the perspective length and location of major vertical lines, we can draw the planes and volumes the lines establish by following the principles of convergence. As a general rule, work from points to lines to planes to volumes, and first establish the perspective of the major forms of the subject before working out the secondary forms.
We can transfer heights and widths forward or back into the depth of a perspective drawing as long as we make the shift perpendicular to the picture plane, along imaginary planes whose parallel sides converge at the center of vision. We can also transfer depth measurements vertically, horizontally, or diagonally as long as we make the shift in a plane parallel to the picture plane.
For inclined lines and circles, see the principles outlined in the section on perspective geometry.
Construct a two-point perspective of the structure shown in the perspective setup.
How far back would you have to shift the picture plane in the plan view to double the perspective image?
Double the height of the horizon line and construct another two-point perspective of the structure.
Double the distance of the station point from the structure and construct another two-point perspective of the structure.
The perspective plan method allows an entire perspective drawing to be laid out from measurements made entirely in the picture plane of the perspective view. It does not require the direct use of an orthographic plan or elevation.
Follow the same procedure as outlined in the common method to construct a plan diagram of the perspective setup. We use this plan diagram to establish the position of the picture plane, station point, vanishing points for major sets of horizontal lines, and locations of vertical measuring lines.
We also use the plan diagram to locate measuring points. A measuring point is a vanishing point for a set of parallel lines used to transfer true dimensions along a measuring line in the picture plane to a line in perspective. The diagonal point in one-point perspective is one example of such a measuring point.
In two-point perspective, there are two measuring points for transferring dimensions along a horizontal measuring line in the picture plane to the perspective of a horizontal line in the subject. To determine the location of these measuring points in the plan diagram:
Note that chord SP-MPL is parallel to chord AB. MPL is therefore the vanishing point for AB and all other lines parallel to it. We use this set of parallel lines to transfer scale dimensions along the ground line in the picture plane to the perspective of baseline BC in the subject.
We may construct the perspective plan on the floor or some other horizontal plane of the subject. If this plane is too close to the horizon, however, the perspective plan can become too foreshortened to precisely determine where lines intersect. Being able to discern these intersections is necessary when transferring scaled dimensions along a measuring line in the picture plane to a line in perspective. Because of this, we usually construct the perspective plan at some distance above or below the horizon line in the perspective drawing.
Construct the perspective plan according to the following procedure:
If the scale measurements along the ground line extend beyond the limits of the perspective drawing, we can use a fractional measuring point. To establish a fractional measuring point, we divide the normal distance from the vanishing to the measuring point by a factor of two or four. A half measuring point requires halving the normal units of measurement along the ground line; a quarter measuring point requires using a quarter scale along the ground line.
Once we complete the perspective plan, we begin to construct the perspective view.
Use the perspective plan method to construct a two-point perspective of the structure at twice the scale shown in the perspective setup.
A perspective grid is a perspective view of a three-dimensional coordinate system. The three-dimensional network of uniformly spaced points and lines enables us to correctly establish the form and dimensions of an interior or exterior space as well as regulate the position and size of objects within the space.
Several types, varying in scale and point of view, are commercially available. We can also use the perspective plan method to construct a two-point perspective grid.
In the perspective view:
Over this perspective grid, we can lay tracing paper and freehand or draft a perspective view. It is important to see the perspective grid as a network of points and lines defining transparent planes in space rather than solid, opaque walls enclosing space. The grid of squares not only allows us to plot points in three-dimensional space but also regulates the perspective width, height, and depth of objects and guides the drawing of lines in proper perspective.
To draw an object within a space, begin by laying out its plan or footprint on the grid of the ground or floor plane. Then elevate each of the corners to their perspective heights using either a vertical grid or the known height of the horizon line above the ground plane. Complete the object by drawing its upper edges, using the principles of convergence and the grid lines to guide their direction. We can use the grid to plot inclined and curved lines as well.
Construct a two-point perspective grid on a sheet of quality tracing paper or vellum. Assume a scale of 3⁄8" = 1'-0" for the picture plane and a horizon line 5 or 6 feet above the ground line. When completed, the perspective grid may be photocopied and enlarged or reduced to any desired scale.
Once constructed, a perspective grid should be saved and reused to draw perspective views of interior and exterior spaces of similar size and scale. Each unit of measurement can represent a foot, four feet, a hundred yards, or even a mile. Rotating and reversing the grid can also vary the point of view. We can therefore use the same grid to draw an interior perspective of a room, an exterior perspective of a courtyard, as well as an aerial view of a city block or neighborhood.
These perspective drawings use the perspective grid developed over the last three pages. In each case, the spectator’s height has been selected to portray a specific point of view and the scale of the grid has been altered to correspond to the scale of the subject matter.
In both one-point and two-point perspectives, the spectator’s central axis of vision is horizontal and the picture plane vertical. The three-point perspective system assumes that either the object is tilted to the picture plane or the spectator’s central axis of vision is inclined upward or downward. In the latter case, because the picture plane is always perpendicular to the central axis of vision, the picture plane is also tilted. Since all three of the principal axes are oblique to the picture plane, all lines parallel to these three axes will appear to converge at three different vanishing points. These are the three points referred to in three-point perspective.
The convergence of parallel vertical lines is the most striking visual characteristic of three-point perspectives. While not widely used, the three-point perspective system can effectively depict what we see when we look up at a tall building or down into a courtyard from a second-story balcony.
We can use the three points of a triangle as the vanishing points for a cube seen in three-point perspective. One side of the triangle is horizontal and connects the left and right vanishing points for horizontal lines. The third vanishing point for vertical lines is located above or below, depending on our point of view.
Using an equilateral triangle assumes the faces of the cube are at equal angles to the picture plane. Extending the vanishing point for vertical lines further away from the horizon alters our point of view and the perspective effect.
We begin drawing a three-point perspective view of a cube by selecting a point A close to the center of the equilateral triangle. From this point, draw lines to the three vanishing points. Once we establish one edge of the cube, length AB, we can complete it by using diagonals. The vanishing points for these diagonals lie midway between the three major vanishing points.
If we rotate this page 180°, we can see a three-point perspective of the same cube, but in this case, we are looking up at it.
The casting of shades and shadows in linear perspective is similar to their construction in paraline drawings, except that the sloping lines representing the conventional or actual light rays appear to converge when oblique to the picture plane. Light sources behind us illuminate the surfaces we see and cast shadows away from us, while sources in front of us cast shadows toward us and emphasize surfaces that are backlit and in shade. Low light angles lengthen shadows, while high light sources shorten them.
To determine the vanishing point for the light rays, construct a triangular shadow plane for a vertical shade line in perspective, having a hypotenuse establishing the direction of the light rays and a base describing their bearing direction. Because the bearing directions of light rays are described by horizontal lines, their vanishing point must occur somewhere along the horizon line.
Extend the hypotenuse until it intersects a vertical trace through the vanishing point of the bearing direction of the light rays. All other parallel light rays converge at this point. This vanishing point represents the source of the light rays, and is above the horizon when the light source is in front of the observer, and below the horizon when behind the observer.
Since a vertical edge casts a shadow on the ground plane in the direction of the light ray, the shadow converges at the same vanishing point as the bearing direction of the light ray.
Since a horizontal edge is parallel with the ground plane and thus casts a shadow parallel with itself, the shadow cast by that edge converges at the same point as the casting edge itself.
When the light rays originate from either the observer’s right or left and are parallel to the picture plane, they remain parallel in perspective and are drawn at their true angular elevation above the ground plane. Their bearing directions remain parallel to each other and the horizon line, and are drawn as horizontal lines.
Given the shadow plane ABC, construct the shade and shadows for the structure in two-point perspective.
Reflections occur on the horizontal surfaces of bodies of water, the mirrored surfaces of glass, and the polished surfaces of floors. A reflecting surface presents an inverted or mirror copy of the object being reflected. Anything in front of or above a reflecting surface also appears in back of or below the reflecting surface in a direction perpendicular to the surface. Objects appear at the same distance in back of or below the reflecting surface as they are in front of or above the surface.
Any reflecting plane surface parallel to one of the three major sets of parallel lines continues the perspective system of the subject. Therefore, the three major sets of lines in the reflection appear in the same perspective as the lines in the subject, remaining parallel and converging to corresponding vanishing points.
If sitting directly on the reflecting surface, the reflected image is a direct, inverted copy of the original. Thus, in a perspective view of the reflection, the reflected image follows the same perspective system of lines already established for the original image. If the object being reflected is at some distance from the reflecting surface, then the reflection can reveal normally hidden aspects of the object. First reflect the distance from the object to the reflecting surface, then draw the mirror image of the object. The plane of the reflecting surface should appear to be halfway between the object and its reflected image.
Oblique lines not parallel to the reflecting surface slant at an equal but opposite angle in the reflection.
When drawing a perspective of an interior space having a mirrored surface on one or more of its major planes, we extend the perspective system in the manner described above. Sightlines reflect off a mirrored surface at an angle equal to the angle of incidence. Each reflection therefore doubles the apparent dimension of the space in a direction perpendicular to the mirrored surface. A reflection of a reflection will quadruple the apparent size of the space.
