Paraline drawings include a subset of orthographic projections known as axonometric projections—the isometric, dimetric, and trimetric—as well as the entire class of oblique projections. Each type offers a slightly different viewpoint and emphasizes different aspects of the subject. As a family, however, they combine the measured precision and scalability of orthographic multiview drawings and the pictorial nature of linear perspective.
Paraline drawings communicate the three-dimensional nature of an object or spatial relationship in a single image. Hence, they are also called single-view drawings to distinguish them from the multiple and related views of plans, sections, and elevations. They can be distinguished from the other type of single-view drawing, linear perspective, by the following pictorial effects. Parallel lines, regardless of their orientation in the subject, remain parallel in the drawn view; they do not converge to vanishing points as in linear perspective—hence the term paraline. In addition, any linear measurement parallel to the three major axes can be made and drawn to a consistent scale.
Because of their pictorial nature and ease of construction, paraline drawings are appropriate for visualizing an emerging idea in three dimensions early in the design process. They are capable of fusing plan, elevation and section, and illustrating three-dimensional patterns and compositions of space. They can be cut or made transparent to see inside and through things, or expanded to illustrate the spatial relationships between the parts of a whole. They can even serve as a reasonable substitute for a bird’s-eye perspective.
Paraline views, however, lack the eye-level view and picturesque quality of linear perspectives. They present instead either an aerial view looking down on an object or scene, or a worm’s-eye view looking upward. In either case, the drawing system can be extended to include a boundless and unlocalized field of vision, unlike perspective drawings which are strictly limited in scope by the size of the visual angle. It reveals the view from an infinite set of positions rather than from a specific point in space. The viewer can move in on a portion of the drawing or move back to take in a broader vista.
Guiding the construction of all paraline drawings is the fundamental principle that lines parallel in space remain parallel in the drawn view. There are therefore three basic approaches to constructing the entire class of paraline drawings. When constructing and presenting a paraline drawing, keep in mind that paraline views are easiest to understand if vertical lines in space are also oriented vertically on the drawing surface.
Axial lines refer to those lines that are parallel to any of the three principal axes. Regardless of the approach we take in constructing a paraline drawing, we can measure dimensions and draw to scale only along axial lines. Axial lines naturally form a rectangular grid of coordinates, which we can use to find any point in three-dimensional space.
Non-axial lines refer to those lines that are not parallel to any of the three principal axes. We cannot measure dimensions along these non-axial lines, nor can we draw them to scale. To draw non-axial lines, we must first locate their end points using axial measurements, and then connect these points. Once we establish one non-axial line, however, we can draw any line parallel to that line, since parallel lines in the subject remain parallel in the drawing.
Any circle oblique to the picture plane appears as an ellipse. In order to draw such a circle in a paraline drawing, we must first draw a square that circumscribes the circle. Then we can use either of two approaches to drawing the circle within the square.
We can draw a paraline view of any curved line or surface by using offset measurements to locate the positions of significant points along the line or surface.
In order to draw a freeform shape in a paraline drawing, first construct a grid over a plan or elevation view of the shape. This grid may either be uniform or correspond to critical points in the shape. The more complex the shape, the finer the grid divisions should be. Construct the same grid in the paraline view. Next, locate the points of intersection between the grid and the freeform shape and plot these coordinates in the paraline view. Finally, we connect the transferred points in the paraline view.
Use the three cubes as a guide to draw paraline views of a cylinder, a cone, and a pyramid.
Construct a paraline drawing of the form described by the set of multiview drawings. Use the principal axes shown and double the scale.
Using the same principal axes, construct a paraline drawing of the form as it would be seen from the opposite direction.
Axonometric = axono + metric or axis-measurement. The term axonometric is often used to describe paraline drawings of oblique projections or the entire class of paraline drawings. Strictly speaking, however, axonometric projection is a form of orthographic projection in which the projectors are parallel to each other and perpendicular to the picture plane. The difference between orthographic multiview drawings and an axonometric single-view drawing is simply the orientation of the object to the picture plane.
Axonometric projection is an orthographic projection of a three-dimensional object inclined to the picture plane in such a way that its three principal axes are foreshortened. The family of axonometric projection includes isometric, dimetric, and trimetric projections. They differ according to the orientation of the three principal axes of a subject to the picture plane.
There is a significant difference between an axonometric projection and a drawing of that projection. In a true axonometric projection, the three principal axes are foreshortened to varying degrees, depending on their orientation to the picture plane. However, in an axonometric drawing, we draw the true length of one or more of these axes to exact scale. Axonometric drawings are therefore slightly larger than their corresponding axonometric projections.
An isometric projection is an axonometric projection of a three-dimensional object inclined to the picture plane in such a way that the three principal axes make equal angles with the picture plane and are equally foreshortened.
To better visualize this, construct an isometric projection of a cube in the following manner.
In developing an isometric projection of a cube, we find that the three principal axes appear 120° apart on the picture plane and are foreshortened to 0.816 of their true length. The diagonal of the cube, being perpendicular to the picture plane, is seen as a point and the three visible faces are equivalent in shape and proportion.
Instead of developing an isometric projection from a set of plan, elevation, and auxiliary views, it is common practice to construct an isometric drawing in a more direct manner. First, we establish the direction of the three principal axes. Since they are 120° apart on the picture plane, if we draw one axis vertically, the other two axes make a 30° angle with a horizontal on the drawing surface.
To save time, we disregard the normal foreshortening of the principal axes. Instead, we lay out the true lengths of all lines parallel to the three principal axes and draw them to the same scale. Thus, an isometric drawing is always slightly larger than an isometric projection of the same subject.
An isometric drawing establishes a lower angle of view than a plan oblique and gives equal emphasis to the three major sets of planes. It preserves the relative proportions of the subject and is not subject to the distortion inherent in oblique views. Isometric drawings of forms based on the square, however, can create an optical illusion and be subject to multiple interpretations. This ambiguity results from the alignment of lines in the foreground with those in the background. In such cases, a dimetric or oblique might be a better choice.
Construct an isometric drawing of the construction described in the paraline view.
Construct an isometric drawing of the structure described by the set of multiview drawings.
Construct an isometric drawing of the object as it would be seen from the direction indicated.
A dimetric projection is an axonometric projection of a three-dimensional object inclined to the picture plane in such a way that two of its principal axes are equally foreshortened and the third appears longer or shorter than the other two.
To better visualize this, construct a dimetric projection of a cube in the following manner.
In developing a dimetric projection of a cube, we find that an infinite number of views and pictorial effects are possible. A series of symmetrical views develops as the cube rotates about a horizontal axis. Another series of asymmetrical views emerges as the cube rotates about a vertical axis. Depending on the orientation of the cube to the picture plane, a dimetric view can either emphasize one major set of planes while subordinating the other two, or emphasize two major sets of planes equally while subordinating the third.
A dimetric is a paraline drawing of a dimetric projection, having all lines parallel to two of the principal axes drawn to true length at the same scale, and lines parallel to the third either elongated or foreshortened.
As with isometric drawings, we usually construct dimetric drawings in a direct manner. We first establish the direction of the three principal axes. Assuming one principal axis remains vertical, we can lay out the angles of the two horizontal axes in several ways. While these angles do not correspond exactly with the angles that result from dimetric projection, they are convenient to use when drafting with 30°/60° and 45°/45° triangles.
We can now lay out the lengths of all lines parallel to the three principal axes. Two of the three principal axes make the same angle with the picture plane. We draw lines parallel to these two axes at the same scale, and lines parallel to the third at a proportionately greater or smaller scale. The circled numbers indicate the whole and fractional scales at which we draw the three principal axes in each dimetric view.
The use of two scales and odd angles make dimetric drawings slightly more difficult to construct than isometric drawings. On the other hand, they offer a flexibility of viewpoint that can overcome some of the pictorial defects of isometric drawings. A dimetric view can emphasize one or two of the major sets of planes as well as provide a clearer depiction of 45° lines and surfaces.
A trimetric projection is an axonometric projection of a three-dimensional object inclined to the picture plane in such a way that all three principal axes are foreshortened at a different rate.
A trimetric is a paraline drawing of a trimetric projection, showing all three principal axes foreshortened at a different rate and therefore drawn at different scales. Trimetrics naturally emphasize one major set of planes over the other two. We rarely use trimetrics because what they reveal does not justify their complex construction. Isometric and dimetric views are simpler to construct and just as satisfactory for most purposes.
Oblique projection is one of three major types of projection drawing. The images that emerge from oblique projections belong to the pictorial family of paraline drawings but are distinct from the isometric and dimetric views that develop from orthographic projection. In oblique projection, a principal face or set of planes in the object is oriented parallel to the picture plane as in orthographic multiview drawing, but the image is transmitted by means of parallel projectors oriented at any angle other than 90° to the picture plane.
Oblique drawings show the true shape of planes parallel to the picture plane. Onto this frontal view, top and side views are attached and projected back into the depth of the drawing. This yields a three-dimensional image that represents what we know rather than how we see. It depicts an objective reality that corresponds more closely to the picture in the mind’s eye than the retinal image of linear perspective. It represents a mental map of the world that combines plan and elevational views into a single expression.
The ease with which we can construct an oblique drawing has a powerful appeal. If we orient a principal face of an object parallel to the picture plane, its shape remains true and we can draw it more easily. Thus, oblique views are especially convenient for representing an object that has a curvilinear, irregular, or complicated face.
While oblique projection can suggest the solidity of a three-dimensional object and produce a powerful illusion of space, it also allows the composition of lines to remain on the surface as a flat pattern. This can lead to optical illusions and therefore ambiguity in the reading of an oblique drawing.
Oblique projection represents a three-dimensional object by extending parallel projectors at some angle other than 90° to the picture plane. We usually orient a principal face of the object parallel to the picture plane so that we can draw it to exact scale and represent its shape and proportion accurately. We can therefore construct an oblique drawing directly from an orthographic projection of that face.
There are two rules that minimize distortion and make an oblique drawing easier to construct.
While an oblique drawing naturally emphasizes the planes that are parallel to the picture plane, planes perpendicular to the picture plane usually appear foreshortened in the drawing. The apparent size and shape of receding planes depend on the angle at which the principal axes perpendicular to the picture plane recede into the depth of the drawing. By varying this angle, we can emphasize one of the sets of receding planes over the other or show them to be of equal importance.
The angle that the oblique projectors make with the picture plane determines the lengths of the receding axial lines in an oblique drawing. If the projectors are at a 45° angle to the picture plane, the receding lines will be projected in their true length. At other angles, the projectors will cause the receding lines to appear either longer or shorter than their true length. In practice, we can lay out and draw the receding lines of an oblique drawing to their true lengths or at a reduced scale to offset the appearance of distortion.
The term cavalier derives from the past use of this projection system in drawing fortifications. In cavalier projection, the projectors form a 45° angle with the picture plane. We can therefore draw the receding axial lines at the same scale as the lines parallel to the picture plane.
While the use of a single scale for all three principal axes greatly simplifies the construction of an oblique drawing, the lengths of receding lines can sometimes appear too long. To offset the appearance of distortion, we can foreshorten the receding lines by drawing their lengths at the same reduced scale, usually 2⁄3 to 3⁄4 of their true length.
The term cabinet comes from its use in the furniture industry. In cabinet projection, a three-dimensional object is represented by an oblique drawing having all lines parallel to the picture plane drawn to exact scale and receding lines reduced to half-scale. Cabinet drawings suffer from a major pictorial defect—the length of receding lines can sometimes appear too short.
In architectural graphics, the two major types of oblique drawings are elevation obliques and plan obliques. Most of the examples on the previous two pages are elevation obliques.
An elevation oblique orients a principal vertical face parallel to the picture plane and therefore reveals its true shape and size. We can therefore construct an elevation oblique directly from an elevation view of the principal face. This face should be the longest, the most significant, or the most complex facade of the subject.
From significant points in the elevation view, we project the receding lines back at the desired angle into the depth of the drawing. In drafting with triangles, we typically use a 30°, 45°, or 60° angle for the receding lines. In freehand sketching, we need not be as precise, but once we establish an angle for the receding lines, we should apply it consistently.
Remember that the angle we use for the receding lines alters the apparent size and shape of the receding planes. By varying the angle, the horizontal and vertical sets of receding planes can receive different degrees of emphasis. In all cases, the primary emphasis remains on the vertical faces parallel to the picture plane.
Construct two series of elevation obliques of the building form described in the set of multiview drawings. In the first series, draw the lines parallel to the receding axis at full scale but vary their direction—draw the receding lines first at 30° to the horizontal, then at 45° to the horizontal, and finally at 60° to the horizontal.
In the second series, draw the receding axis at 45° to the horizontal but vary its scale—draw the lines parallel to the receding axis first at three-quarter scale, then at two-third scale, and finally at one-half scale.
Compare the pictorial effects of the various elevation obliques. Do any of the elevation obliques appear to be too deep? Do any appear too shallow? Which sets of receding planes does each elevation oblique emphasize?
A plan oblique orients a horizontal plane or plan view parallel to the picture plane and therefore reveals its true shape and size. We usually rotate the plan view so that both sets of vertical planes appear in the oblique view. Rotating the plan offers a wide array of possible views in which the two sets of vertical planes can receive different degrees of emphasis. In all cases, however, plan obliques offer a higher point of view into an interior than isometric drawings and the primary emphasis remains on the horizontal set of planes.
In drafting with triangles, we rotate the plan 30°, 45°, or 60° from a horizontal on the drawing surface. In freehand sketching, we need not be as precise, but once we establish the angle of rotation, we should apply it consistently. We should remember that the angle we use determines the apparent size and shape of the vertical planes.
Once the plan is rotated to the desired angle, we draw the receding lines as verticals on the drawing surface. We can draw these verticals at the same scale as the plan view or foreshorten them if their lengths appear exaggerated.
Construct two pairs of plan obliques of the building form described in the set of multiview drawings. In the first series, draw the lines parallel to the vertical axis at full scale but rotate the plan view 30° clockwise about point A, then 45° clockwise about point A, and finally 60° clockwise about point A.
In the second pair, rotate the plan view in the same way, but draw the lines parallel to the vertical axis at three-quarter scale.
Compare the pictorial effects of the various plan obliques. Do any of the plan obliques appear to be too tall? Do any appear too short? Which sets of vertical planes does each plan oblique emphasize?
Even though a paraline drawing always presents either an aerial view or a worm’s-eye view of a subject, we can construct a paraline view in any of several ways to reveal more than the exterior form and configuration of a design. These techniques allow us to gain visual access to the interior of a spatial composition or the hidden portions of a complex construction. We categorize these techniques into phantom views, cutaway views, and expanded views.
A phantom view is a drawing having one or more parts made transparent to permit representation of internal information otherwise hidden from our view. This strategy effectively allows us to unveil an interior space or construction without removing any of its bounding planes or encompassing elements. Thus we are able to simultaneously see the whole composition as well as its internal structure and arrangement.
We use a phantom line to represent the transparency of a part, an alternative position of a moving part, the relative position of an absent part, or a repeated detail or feature. A phantom line is a broken line consisting of relatively long segments separated by two short dashes or dots. In practice, phantom lines may also consist of dashed, dotted, or even delicately drawn lines. The graphic description should include the thickness or volume of the transparent part as well as any details that may exist within its boundaries.
A cutaway view is a drawing having an outer section or layer removed to reveal an interior space or an internal construction. This strategy can also effectively manifest the relation of an interior to the exterior environment.
The simplest method for creating a cutaway view is to remove an outer or bounding layer of a composition or construction. For example, removing a roof, ceiling, or wall allows us to look down and see into an interior space. Removing a floor permits a view up into a space.
We can remove a larger section by slicing through the heart of a composition. When a composition exhibits bilateral symmetry, we can make this cut along the axis and indicate the footprint or plan view of the part removed. In a similar fashion, we can create a cutaway view of a radially symmetrical composition by slicing through the center and removing a quadrant or similar pie-shaped portion.
In order to reveal a more complex composition, the cut may follow a three-dimensional route. In this case, the trajectory of the cut should clarify the nature of the internal organization and arrangement and be clearly articulated by a contrast in line weights or tonal values.
Even though a portion is removed in a cutaway view, its presence can remain in the drawing if we delineate its outer boundaries with a dotted, dashed, or delicate line. Indicating the external form of what is removed helps the viewer retain a sense of the whole.
While a paraline is a single-view drawing useful in displaying three-dimensional relationships, a series of paraline views can effectively explain processes and phenomena that occur in time or across space. A progression of paraline drawings can explain a sequence of assembly or the stages of a construction, with each view successively building upon the preceding one.
The portions removed from a drawing may not disappear but merely shift to new positions in space, developing into what we call an expanded or exploded view. An expanded view shows the individual components of a construction or assembly separately but indicates their proper relation to each other and to the whole. The finished drawing appears to be an explosion frozen at a point in time when the relationships between the parts are most clear.
The displacement of the parts should be in the order and direction in which they fit together. For axial compositions, the expansion occurs either along the axis or perpendicular to it. For rectangular compositions, the parts relocate along or parallel to the principal axes. In all cases, we indicate the relationships between the parts to each other and to the whole with dotted, dashed, or delicately drawn lines.
Expanded views are extremely useful in describing the details, layering, or sequence of a construction assembly. At a larger scale, expanded views can effectively illustrate vertical relationships in buildings as well as horizontal connections across space. In clarifying spatial relationships and organizations through displacement, expanded views can simultaneously combine the revealing aspects of phantom and cutaway views.
Even a simple line drawing of a paraline view induces a powerful sensation of space. This is due not only to the depth cue of overlap, but also our perception of parallelograms as rectangles occupying space. We can enhance the perceived depth of a paraline drawing by contrasting line weights or tonal values.
We use a hierarchy of line weights to distinguish between spatial edges, planar corners, and surface lines.
In order to separate planes in space, clarify their different orientations, and especially to distinguish between the horizontal and the vertical, we can use contrasting tonal values, textures, or patterns. The most important distinction to establish is the orthogonal relationship between horizontal and vertical planes. Applying a tonal value to the horizontal planes in a paraline view not only establishes a visual base for the drawing but also aids in defining the shape and orientation of the vertical planes.
The grouping and layering functions of 2D drawing and 3D modeling and CAD programs give us the ability to more easily create the different types of paraline views. By organizing elements and assemblies of a three-dimensional construction into separate groups or layers, we can selectively control their location, visibility, and appearance.
Illustrated is a paraline view of the Hirabayashi Residence in Yamada, Japan, designed by Tadao Ando in 1975. First draw the paraline view with a single line weight. Then use a hierarchy of line weights to differentiate spatial edges, planar corners, and surface lines.
Remember that line weight is not simply a matter of density. Rather, we rely on contrasting line thicknesses to distinguish one line weight from another.
For more practice in articulating spatial edges, planar corners, and surface lines, apply a hierarchy of line weights to any of the paraline views executed in Exercises 7.4 through 7.8.
The casting of shade and shadows in a paraline drawing enhances our perception of the three-dimensional nature of volumes and masses and articulates their spatial relationships. In addition, the tonal values used in rendering shades and shadows can help differentiate between vertical, horizontal, and sloping planes. For the basic concepts and terminology of shade and shadows, refer back to Chapter 6.
It is convenient to visualize the three-dimensional relationships between light rays, shade lines, and cast shadows in paraline views because they are pictorial in nature and display the three major spatial axes simultaneously. In addition, parallel light rays and their bearing directions remain parallel in a paraline drawing.
In order to construct shade and shadows, it is necessary to assume a source and direction of light. Deciding on a direction of light is a problem in composition as well as communication. Remember that cast shadows should clarify rather than confuse the nature of the forms and their spatial relationships. The lower the angle of light, the deeper the shadows; the steeper the angle, the shallower the shadows. In any case, the resulting shadow patterns should not conceal more than they reveal about the forms being depicted.
Occasionally it may be desirable to determine the actual conditions of light, shade, and shadow. For example, when studying the effects of solar radiation and shadow patterns on thermal comfort and energy conservation, it is necessary to construct shades and shadows using the actual sun angles for specific times and dates of the year.
For ease of construction, the bearing direction of the light rays is often parallel with the picture plane and emanates from either the observer’s left or right. Consequently, the altitude of the light rays appears true in the drawing and their bearing direction remains horizontal. While the desired depth of shadows should determine the altitude of the light rays, we often use 30°, 45°, or 60° angles because of their convenience when drafting with 45° and 30°/60° triangles.
We can construct a rectangular prism to discern the direction of shadows cast by vertical and horizontal lines parallel to the major axes of a paraline drawing. From the apex of a vertical shade line, draw the direction of the light rays to meet the shadow cast by the line on a horizontal surface in the bearing direction of the light. This is the volumetric diagonal of the prism. Then construct the remaining edges of the prism parallel to the major axes of the paraline drawing.
Each of the upper horizontal edges casts onto a vertical face to which it is perpendicular a shadow in the direction of the diagonal of the face. Each casts onto a parallel vertical face a shadow that is parallel to itself.
3D modeling software typically includes the ability to specify the direction of sunlight according to the hour of the day and the time of the year, and to cast shade and shadows automatically in both paraline and perspective views. This feature can be especially useful in the schematic design phase to study the form of a building or the massing of a building complex on a site and to evaluate the impact of the shadows they cast on adjacent buildings and outdoor areas.
The digital technique for determining what surfaces are in shade and the shapes of the shadows cast in a three-dimensional image or scene is referred to as ray casting. While efficient and useful for preliminary design studies, ray casting does not take into account the way the light rays from an illuminating source are absorbed, reflected, or refracted by the surfaces of forms and spaces.
Construct the shade and shadows for the structure described in the paraline view below. Assume the parallel light rays of the sun have an altitude of 45° and a bearing direction to the right and parallel to the picture plane.
For more practice, assume the same direction of light rays and construct shade and shadows for the structure described in Exercise 7.4.
