6
Perspective Drawings

“Perspective” properly refers to any of various graphic techniques for depicting volumes and spatial relationships on a flat surface, such as size perspective and atmospheric perspective. The term “perspective,” however, most often brings to mind the drawing system of linear perspective. Linear perspective is a technique for 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 views of an objective reality, linear perspective offers scenes of an optical reality. It depicts how a construction or environment might appear to the eye of an observer looking in a specific direction from a particular vantage point in space.

Linear Perspective

Linear perspective is valid only for monocular vision. A perspective drawing assumes that the observer 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. Thus, linear perspective can only approximate the complex way our 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 observer.

Perspective Projection

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 observer. This convergence of sight lines differentiates perspective projection from the other two major projection systems—orthographic projection and oblique projection—in which the projectors remain parallel to each other.

Perspective Elements

Pictorial Effects of Perspective

The converging nature of sight lines in linear perspective produces certain pictorial effects. Being familiar with these pictorial effects helps us understand how lines, planes, and volumes should appear in linear perspective and how to place objects correctly in the space of a perspective drawing.

Convergence

Convergence in linear perspective refers to the apparent movement of parallel lines toward a common vanishing point as they recede.

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 observer's central axis of vision (CAV) and the subject.

Diminution of Size

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 sight lines alter the apparent size of a line or plane according to its distance from the picture plane.

Other Pictorial Effects

Perspective drawings also possess other pictorial characteristics found in multiview and paraline drawing systems.

Perspective Variables

The observer's point of view determines the pictorial effect of a perspective drawing. As this viewpoint changes—as the observer moves up or down, to the left or right, forward or back—the extent and emphasis of what the observer sees also change. In order to achieve the desired view in perspective, we should understand how to adjust the following variables.

Height of the Station Point

The height of the station point (SP) relative to an object determines whether it is seen from above, below, or within its own height.

For a normal eye-level perspective, SP is at the standing height of a person.
As SP moves up or down, the horizon line (HL) moves up or down with it.

A horizontal plane at the level of SP appears as a horizontal line.
We see the tops of horizontal surfaces that are below the level of SP and the undersides of horizontal planes that are above.
Even if not actually visible in a perspective view, the horizon line should always be drawn lightly across the drawing surface to serve as a level line of reference for the entire composition.

Distance from the Station Point to the Object

The distance from the station point (SP) to an object influences the rate of foreshortening of the object's surfaces that occurs in the perspective drawing.

As the observer's SP moves farther away from the object, the vanishing points for the object move farther apart, horizontal lines flatten out, and perspective depth is compressed.
As the observer's SP moves forward, the vanishing points for the object 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 (SP) of the perspective.

Angle of View

The orientation of the central axis of vision (CAV) and the picture plane (PP) relative to an object determines which faces of the object are visible and the degree to which they are foreshortened in perspective.

The more a plane is rotated away from PP, the more it is foreshortened in perspective.
The more frontal the plane is, the less it is foreshortened.
When a plane becomes parallel to PP, its true shape is revealed.

Digital Viewpoints

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 on this and the facing page 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.

Both one- and two-point perspective assumes a level line of sight, which results in vertical lines remaining vertical. As soon as the observer's line of sight tilts up or down, even a few degrees, the result is technically a three-point perspective.

The desire to see more of a space in a single perspective view often leads to moving the station point of the observer as far back as possible. However, one should always attempt to maintain a reasonable position for the observer within the space being represented.
Keeping the central portion of a subject or scene within a reasonable cone of vision is critical to avoiding distortion in a perspective view. Widening the angle of view to include more of a space within a perspective can easily lead to distortion of forms and exaggeration of the depth of a space.

Location of the Picture Plane

The location of the picture plane (PP) relative to an object affects only the final size of the perspective image. The closer PP is to the station point (SP), the smaller the perspective image. The farther away PP is, the larger the image. Assuming all other variables remain constant, the perspective images are identical in all respects except size.

Types of Perspective

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: one-, two-, and three-point perspectives. What distinguishes each type is simply the observer's angle of view relative to the subject. The subject does not change, only our view of it, but the change of view affects how the sets of parallel lines appear to converge in linear perspective.

One-Point Perspective

If we view a cube with our central axis of vision (CAV) perpendicular to one of its faces, all of the cube's vertical lines are parallel with the picture plane (PP) and remain vertical. The horizontal lines that are parallel with PP and perpendicular to CAV also remain horizontal. The lines that are parallel with CAV, however, will appear to converge at the center of vision (C). This is the one point referred to in one-point perspective.

Two-Point Perspective

If we shift our point of view so that we view the same cube obliquely, but keep our central axis of vision (CAV) horizontal, then the cube's vertical lines will remain vertical. The two sets of horizontal lines, however, are now oblique to the picture plane (PP) 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.

Three-Point Perspective

If we lift one corner of the cube off the ground plane (GP), or if we tilt our central axis of vision (CAV) to look down or up at the cube, then all three sets of parallel lines will be oblique to the picture plane (PP) and will 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.

One-Point Perspective

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 (PP), 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 PP and parallel with the central axis of vision (CAV). All lines parallel to CAV converge on the horizon line (HL) at the center of vision (C). This is the particular vanishing point referred to in one-point perspective.

The one-point perspective system is particularly effective in depicting the interior of a spatial volume because the display of five 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. 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.

Diagonal Point Method

The diagonal point method for constructing a one-point perspective uses the geometry of a 45° right triangle and the principles of convergence to make depth measurements in perspective.

The technique involves establishing one side of a 45° right triangle in or parallel to the picture plane (PP) so that we can use it as a measuring line (ML). Along this side (OA), we measure a length equal to the desired perspective depth.
Through endpoint O of this length, we draw the perpendicular side that converges at the center of vision (C).
From the other endpoint A, we draw the hypotenuse that converges at the vanishing point for lines making a 45° angle with the picture plane (PP).
This diagonal marks off a perspective depth (OB) that is equal to length OA.

One-Point Perspective Grid

We can use the diagonal point method to construct a one-point perspective grid easily. 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 to regulate the position and size of objects within the space.

One-Point Perspective Grid

Plan Setup

Before beginning the construction of any perspective, we should first determine the desired point of view: What do we wish to illustrate in the perspective view and why?
After we determine the space we are going to illustrate, we next establish the station point (SP) and the central axis of vision (CAV) in the plan view.
Because this is a one-point perspective, CAV should be parallel to one major axis of the space and perpendicular to the other.
We locate SP within the space but far enough back that the majority of the space lies within a 60° cone of vision.
SP and CAV should be located off-center to avoid constructing a static, symmetrical perspective image.
For ease of construction, we can locate PP coincidental with a major plane perpendicular to CAV.

Diagonal Points

If we draw a 45° line from the station point (SP) in a plan view of the perspective setup, it will intersect the picture plane (PP) at the vanishing point for that diagonal and all lines parallel to it. We call this vanishing point a diagonal point (DP).
There is one DP for horizontal diagonal lines receding to the left (DPL), and another for horizontal diagonal lines receding to the right (DPR).
Both diagonal points lie on the horizon line (HL), equidistant from the center of vision (C). From the geometry of the 45° right triangle, we know that the distance from each DP to C is equal to the distance from SP to C in the plan setup.
Note that if we move each DP toward C, this is equivalent to the observer moving closer to PP. If we shift each DP farther away from C, the observer also moves farther away from PP.

We can lay a piece of tracing paper over this perspective grid and draw in the major architectural elements of the space. With the same grid, we can also locate the positions and relative sizes of other elements within the space, such as furniture and lighting fixtures.

We transfer measurements only along axial lines.
For circles in perspective.
It's good practice to include people in our perspectives to indicate the function and scale of the space.

Perspective-plan views—one-point perspectives of interior spaces viewed from above—can be effective in illustrating small, highly detailed rooms.
When drawing a one-point perspective of a space, we notice that the observer's eye-level, equivalent to the height of the horizon line (HL) above the ground line (GL), as well as the location of the observer's center of vision (C), will determine which planes defining the space will be emphasized in the perspective view.
The perspective drawing below uses the perspective grid shown on the facing page. Note that, particularly in interior views, properly cropped foreground elements can enhance the feeling that one is in a space rather than on the outside looking in. The center of vision (C) is closer to the left-hand wall so that the bending of the space to the right can be visualized. The change in scale between the right-hand shelving and patio doors beyond, and a similar change between the foreground table and the window seat beyond, serve to emphasize the depth of the perspective.

Section Perspectives

The section perspective combines the scaled attributes of a section drawing and the pictorial depth of a perspective drawing. It therefore is able to illustrate both the constructional aspects of a design as well as the quality of the spaces formed by the structure.

Proceed by using the diagonal point method to construct the one-point perspective.

In design drawing, one should remember to emphasize the form of the interior and exterior spaces that are cut through rather than the construction details of the structure itself.

Two-Point Perspective

The two-point perspective system assumes that the observer's central axis of vision (CAV) is horizontal and the picture plane (PP) is vertical. The principal vertical axis is parallel to PP, and all lines parallel to it remain vertical and parallel in the perspective drawing. The two principal horizontal axes, however, are oblique to PP. All lines parallel to these axes therefore appear to converge to two vanishing points on the horizon line (HL), one set to the left and the other to the right. These are the two points referred to in two-point perspective.

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 par-ticularly effective in illustrating the three-dimensional form of objects in space ranging in scale from a chair to the massing of a building.

The pictorial effect of a two-point perspective varies with the spectator's angle of view. The orientation of the two horizontal axes to PP determines how much we will see of the two major sets of vertical planes and the degree to which they are foreshortened in perspective.
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.

Measuring Point Method

The following is a method for constructing a two-point perspective grid utilizing measuring points. As with the construction of a one-point perspective, you should first establish the observer's point of view. Determine what you wish to illustrate. Look toward the most significant areas and try to visualize from your plan drawing what will be seen in the foreground, middleground, and background. Review the perspective variables on pages 115–120.

Plan Setup

At a convenient scale, construct a plan diagram of the perspective setup to determine the desired angle of view.
Lay out the major baselines of the space.

Establish the station point (SP) and the observer's central axis of vision (CAV), being careful that most of what you wish to illustrate lies within a 60° cone of vision.
Locate the picture plane (PP) perpendicular to CAV. It is usually convenient to have PP intersect a major vertical element of the space so that it can be used as a vertical measuring line (VML).
Locate the left and right vanishing points (VPL and VPR). Remember that the vanishing point for any set of parallel lines is that point at which a line drawn from SP, parallel to the set, intersects PP.

Measuring Points

A measuring point (MP) is a vanishing point for a set of parallel lines used to transfer true dimensions along a measuring line (ML) to a line receding in perspective. The diagonal point in one-point perspective is one example of such a measuring point.

In two-point perspective, you can establish two measuring points (MPL and MPR) for transferring dimensions along the ground line (GL) to the two major horizontal baselines that are receding in perspective.

Include vanishing points for secondary lines that might be useful in constructing your perspective. For example, if you have a series of parallel diagonals in your design, establish their vanishing point as well.

Two-Point Perspective Grid

Constructing the Perspective Grid

Draw the horizon line (HL) and ground line (GL) at any convenient scale. This scale need not be the same as the scale of the plan setup.
At the same scale, transfer the positions of the major left and right vanishing points (VPL and VPR) and the left and right measuring points (MPL and MPR) from the plan setup.

Along GL, lay out equal increments of measurement to scale. The unit of measurement typically is one foot; we can use smaller or larger increments, however, depending on the scale of the drawing and the amount of detail desired in the perspective view.
Establish the position of a vertical measuring line (VML) from the plan setup and lay out the same equal increments of measurement.
From VPL and VPR , draw baselines through the intersection of VML and GL.
Transfer the units of measurements on GL to the left baseline in perspective by drawing lines to MPR. Transfer scale measurements on GL to the right baseline by drawing lines to MPL. These are construction lines used only to transfer scaled measurements along GL to the major horizontal baselines in perspective.
A fractional measuring point can be used to reduce the length of measurements along GL. For example, you can use ¹/₂ MPR to transfer a 5-foot measurement to a point 10 feet beyond the picture plane along the left baseline.

From the major left and right vanishing points (VPL and VPR), draw lines through the transferred measurements along the major horizontal baselines in perspective.
The result is a perspective grid of one-foot squares on the floor or ground plane. When one-foot squares become too small to draw accurately, use two-foot or four-foot squares instead.
From VPL and VPR, draw lines through the scaled measurements along VML to establish a similar vertical grid.
Over this perspective grid, you can lay tracing paper and draw 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 facilitates the plotting of points in three-dimensional space, regulates the perspective width, height, and depth of objects, and guides the drawing of lines in proper perspective.

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. Therefore, you can 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.

Then elevate each of the corners to its perspective height using either a vertical grid or the known height of the horizon line (HL) above the ground line (GL).
Complete the object by drawing its upper edges, using the principles of convergence and the grid lines to guide their direction.
Remember to transfer all measurements only along axial lines.

Two-Point Perspective Drawings

These three perspectives use the perspective grid shown on the preceding page. In each case, however, the height of the observer's station point (SP) above the ground plane (GP) has been selected to portray a specific point of view, and the scale of the grid has been altered to suit the scale of the structure.

This interior perspective also uses the grid shown on page 134. Note that the left vanishing point (VPL) lies within the drawing, enabling three sides of the space to be shown and a greater sense of enclosure to be felt. Because VPL lies within the drawing, greater emphasis is placed on the right-hand portion of the space. If the left-hand side of the space is to be emphasized, use a reverse image of the grid.

Perspective Measurements

The combined effects of convergence and diminishing size make it more difficult to establish and draw measurements in linear perspective than in the other two drawing systems. But there are techniques we can use to determine the relative heights, widths, and depths of objects in the pictorial space of a perspective drawing.

Digital Measurements

Perspective measurements are not a major issue in 3D-modeling programs because the software uses mathematical formulas to process the three-dimensional data we have already entered.

Measuring Height and Width

In linear perspective, any line in the picture plane (PP) 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 (ML) to scale dimensions in a perspective 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 ground line (GL) is one example of a horizontal measuring line.

Measuring Depth

Measuring perspective depth is more difficult than gauging heights and widths in linear perspective. 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.

Subdividing Depth Measurements

There are two methods for subdividing depth measurements in linear perspective: the method of diagonals and the method of triangles.

Method of Diagonals

In any projection system, we can subdivide a 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 geometric 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 to subdivide its receding edges into a series of unequal segments, its forward edge must be parallel to the picture plane (PP) so that it can be used as a measuring line (ML).

Method of Triangles

Because any line parallel to the picture plane (PP) can be subdivided proportionately to scale, we can use such a parallel line as a measuring line (ML) to subdivide any intersecting line into equal or unequal parts.

Extending a Depth Measurement

If the forward edge of a rectangular plane is parallel to the picture plane (PP), we can extend and duplicate its depth in perspective.

From this point, we 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 foreshortened in perspective.
We can repeat this procedure as often as necessary to produce the desired number of equal spaces in the depth of a perspective drawing.
Note that it is usually better to subdivide a larger measurement into equal parts than it is to multiply a smaller measurement to arrive at a larger whole. The reason for this is that, in the latter procedure, even minute errors can accumulate and become visible in the overall measurement.

Inclined Lines

Once we are familiar with how lines parallel to the three principal axes of an object converge in linear perspective, we can use this rectilinear geometry as the basis for drawing perspective views of inclined lines, circles, and irregular shapes.

Inclined lines parallel to the picture plane (PP) retain their orientation but diminish in size according to their distance from the spectator. If perpendicular or oblique to PP, however, an inclined set of lines will appear to converge at a vanishing point above or below the horizon line (HL).

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 HL. If the set rises upward as it recedes, its vanishing point will be above HL; if it falls as it recedes, it will appear to converge below HL.

A more precise method for determining the vanishing point for an inclined set of parallel lines is as follows:

Stairs

Drawing stairs in perspective is easiest when we can determine the vanishing point for the inclined lines that connect the stair nosings.

Circles

The circle is the essential basis for drawing cylindrical objects, arches, and other circular forms.

Note that the major axis of the ellipse representing the circle in perspective is not coincident with the geometric diameter of the circle.
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 in the mind's eye 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.

Reflections

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 image of the object being reflected. For example, if an object is resting directly on a 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.

Any reflecting planar surface parallel to one of the three major axes extends the perspective system of the subject. Therefore, the major sets of parallel lines in the reflection appear to converge to the same vanishing points as do the corresponding sets of lines in the subject.

When the subject is in front of or above a reflecting surface, first reflect its distance to the reflecting surface, then draw its mirror image. The plane of the reflecting surface should appear to be halfway between the subject and its reflected image. For example, the waterline establishes the horizontal reflecting plane. Point o lies in this plane.
Therefore, oa = oa' and ab = a'b'.
Reflections of lines perpendicular to the reflecting surface extend the original lines.

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 on the previous page.