Chapter 3. Perspective
Last updated, 26 February 2020
Expressing three dimensions in two is the heart of perspective drawing. When the eye looks at something it necessarily collapses a three-dimensional world into a two-dimensional representation on the retina. You may have been told that we need two eyes to perceive a three-dimensional world. However, there are both binocular and monocular cues to depth, and the monocular cues take over in the mid to far field of view. Table 1 shows five main classes of effects.
Table 1. Depth Cues
Depth Cues Not Applicable to Painting
Binocular effects are not applicable to painting, and there are at least two of them: binocular parallax and convergence. There are also two monocular effects that are not applicable to painting: motion parallax and accommodation. All are detailed below.
Binocular parallax is the difference in retinal image from the left and right eyes. It operates in the near field (arm's-length inward) and to a lesser extent in the mid field (from arm's-length to 20ft/ 6m). The far field is anything beyond 20ft/ 6m. Despite common wisdom, depth is perceived by much more than binocular parallax.
Convergence, another binocular cue, is the felt difference in angle as the eyes pivot nasally to focus on a near object. Convergence is a near field effect. Traditional artwork cannot exploit binocular depth cues.
Motion parallax is not necessarily binocular and is operative whenever the object shifts relative to the viewer. This may be caused by motion of either the viewer or the object. Therefore, moving the head from side to side while viewing a stationary object produces motion parallax even if viewed by only one eye. Despite being a monocular depth cue, it is not useful for watercolors, though it is used in lenticular prints.
Accommodation refers to the adjustment of the eye's lens to distance. Although a monocular cue, it is not applicable to watercolors because in the ordinary case watercolors have no actual depth dimension, only an illusion of one. However, other monocular cues can be used to good effect to simulate depth.
Depth Cues Applicable to Painting
The following monocular depth cues are applicable to painting:
Image size: near objects appear larger than distant ones.
Vertical position: distant objects are closer to the horizon (occupy a higher vertical drawing position) than near objects.
Occlusion: nearer objects occlude farther ones.
Spatial frequency: in the near field, patterns are clearly discernable, as distance increases granularity is lost, devolving to texture and finally to an average color.
Aerial perspective: the intervening atmosphere absorbs and scatters light. With distance this tends to shift hue toward the sky tone and cause loss of chroma.
Comparison: experience with the size of familiar objects provides depth cues.
Light effects: light effects such as shadow, reflection, transmission, and absorption provide cues to depth.
Linear perspective: a change from parallel to converging angles simulates an object-viewer relationship that provides powerful illusions of depth.
An Example of Linear Perspective
Everyone knows that the top of a wine glass is circular. However, if you want to draw it in any view except birdseye, you must draw an ellipse, according to the rules of perspective.
All perspective is determined exclusively by the viewer's relationship to the object, and it is possible for different perspectives to occupy the same painting if objects are oriented differently with respect to the viewer. Below are examples of various perspectives; from left to right they are zero-point perspective (0PP, also called parallel projection), one-point perspective (1PP, also called central perspective), two-point perspective (2PP), and three-point perspective (3PP).
Figure 1. Various Perspectives.
Perspective types are determined by the number of vanishing points and the vanishing points are determined by the viewer's relationship to the object. In 0PP all lines of a cube are parallel; there are no vanishing points, or equivalently, all vanishing points lie at infinity. In the case of 1PP, the horizontal and vertical edges of the cube remain parallel to the viewing plane, but the edges in the depth dimension recede to a single vanishing point. In 2PP, the cube is turned with respect to the viewing plane or vice versa such that two sets of faces recede in the depth dimension to respective vanishing points at left and right. In 3PP, all three directions – length, breadth, and depth –are oblique to the viewing plane and all edges recede to one of three vanishing points. Figure 2 shows the viewing-plane-to-object relationship responsible for the various perspectives.
Figure 2 (a, b, c). Various viewing planes and resulting perspectives.
Why Study Perspective?
You may ask yourself, if I draw or paint from a picture or a natural scene, isn't the perspective already taken care of? Yes, and for this reason most introductory watercolor texts treat perspective briefly if at all. However, knowledge of perspective aids the reduction of a real scene to paper and allows one to add elements to the background of the picture; in order to do this, you will need to locate the vanishing points, know the type of perspective, and scale in the depth directions in order for the new elements to make sense. For example, the basis photograph for Manny's Pugs was taken inside a home, yet I painted them outside it. This means that the sky, grass, and clouds had to be added in perspective after the fact. Also, if you choose to draw from memory or imagination, you will need to understand the principles of perspective to render your subject with realism.
Zero-Point Perspective (0PP).
In 0PP, all parallel lines remain parallel in the length, width and depth dimensions, with no set of parallel lines converging to a vanishing point. This is only possible if the vanishing point is far away. Yet surprisingly, even in 0PP there is foreshortening in the depth dimension. Consider a cube having edge length s as shown by the figure at right. Note that a circle of radius s and centered at the top-right-front corner of the cube contacts the corners at the top-left-front and bottom-right-front corners (indicating an edge length of s) but does not contact the top-right-rear corner (indicating an edge length of less than s in the depth dimension). Yet, the cube appears to be normally proportioned, and in fact it is.
Figure 3 shows two cubes parallel to the viewing plane in 0PP but representing two different positions of the viewer (standing to the left or right of the cube, respectively). Regardless of the viewer's vantage, for cubes with one face parallel to the viewing plane the foreshortening in the depth direction ALWAYS corresponds to s/2 units in the viewing plane (Figure 3).
Figure 3. Foreshortening in 0PP
So if you want to properly draw a cube in 0PP having one face parallel to the viewing plane, always offset the rear square by half the distance of the front square for the depth dimension. We prove this is so after introducing 1PP.
One-Point Perspective (1PP)
Railroad tracks vanishing toward a single point in the distance are an example of one-point perspective. In this case, the railroad ties are parallel to the viewing plane. (If they are not, you are in 2PP.) Drawing the rails is simple enough: draw the rails emanating from a vanishing point toward yourself (foreground). But what about the railroad ties running in the horizontal direction? Since these are parallel to the viewing plane, the line segments remain horizontal and parallel. Since they are bound by the rails, they shrink as they recede. But how does one scale the distance between each tie since that distance is continually diminishing as it recedes toward the vanishing point? One way is to "eyeball" the distance and just gradually reduce it as the railroad tracks recede to the background. But there is an exact way to scale this distance. Not only must the rails recede to a vanishing point, but so must any imaginary line through any two similar points. For the purpose of scaling, the diagonals between any two railroad ties provide convenient vanishing points as shown in the next figure.
Use of Diagonal Vanishing Points
For a rectangular layout there are two diagonals and therefore, two diagonal vanishing points (DVPs): one to the left, and one to the right. Only one is required to scale the railroad ties, but I have included both for reference. These are constructed by using a single railroad tie as a reference. I have colored my reference tie in red. Next, I decide on the spacing between railroad ties. I have placed the next railroad tie toward the foreground at a pleasing distance from the reference tie. The diagonal from that tie to the reference tie (red vanishing line) is extended into the background. The intersection of this diagonal with the horizon fixes a DVP. (One can just as easily pick the DVP first which then determines the spacing between ties.) The intersection of the right crossing diagonal with the horizon fixes the right DVP and likewise for the left. Once either DVP is determined, all other railroad ties may be scaled by it; one may just as easily use the right DVP as the left. Select the one that is most convenient.
Method of Diagonals
It is also possible that one or both DVPs (and possibly the VP) may actually be off the paper as is the case for Figure 1. One may scale from the vanishing points wherever they happen to be. However, internal diagonals may prove more convenient. Since the diagonals of any rectangle must cross at the centerline even in the depth dimension, it may be used to scale regular elements forward or backward as shown at left and it works for all perspectives, 1PP, 2PP, or 3PP.
Step 1, a set of diagonals is used to locate the center of the rectangular element, and a centerline is projected from there to the vanishing point, or at least to the midpoint of the next horizontal member. This midpoint can just as easily be found by measurement as by projection.
Step 2, once located, a diagonal is drawn through the midpoint of the member to intersect the opposite vanishing line. This determines the position of the next element.
Step 3, the method is repeated in seriatim until all the horizontal members are located.
Internal diagonals may be more convenient than DVPs for scaling in the depth dimension if the DVPs lie off the page.
All three-dimensional objects may be scaled with reference to the three-dimensional space surrounding them, even if the shapes themselves are not cubes or are otherwise irregular. The simplest space-containing object is the regular hexahedron (cube). Since we can scale any cube in perspective, we can scale any object. This is one of several reasons that tutorials use cubes to illustrate perspective.
[Note, when I use the term orthogonal, I mean that all edges meet at 90°. Angles that are not 90° are known as oblique. When I use the term regular, I mean that all edges and angles are identical.]
Figure 4 highlights the essential features of one-point perspective.
Figure 4. Cubes in One-Point Perspective.
In 1PP, the DVPs will always be equidistant from the VP and will define the diameter of a 90° circle-of-view (COV), which is generally considered the extent of the visual field. However, a 60° COV results in figures with less distortion. We can always scale the COV from the 90° COV using the formula COV(a)/COV(90°) = tan (a/2), where a is the solid angle of view defining a cone of vision. Thus, the 60° COV = 57.7% of the 90° COV; i.e., tan (60°/2)=0.577. Despite being in proper perspective, any object beyond the 90° COV appears distorted because it lies outside the periphery of normal vision and is therefore unfamiliar. All COVs lie where the cone of vision and the viewing plane intersect. Thus, the COV lies on the viewing plane. Whichever way you turn your head, the viewing plane follows. The image at the viewing plane is what we want to transfer to our drawing. The 90° COV is the intersection of a 90° solid angle emanating from our eye (the 90° cone of vision) with the viewing plane; the 60° COV is the intersection of the viewing plane with a 60° solid angle emanating from our eye (the 60° cone of vision); etc. The 90° COV is tangent to the ground at the ground line (green) directly below the VP. The horizon is the diameter for the 90° COV and is parallel to the ground plane and represents eye level.
About Horizon and Ground
The horizon runs horizontally in either direction from the VP and is always parallel with the ground plane. The horizon is actually the eyeline of the viewer (technically a plane projecting in the distance direction from eye level). It may differ from the natural horizon where the sky meets the sea or land. For example, if you look down into a canyon, your line of sight may not even include the sky. It is always the horizon as defined by the viewer's eyeline that is of importance in perspective drawing. Similarly, the natural ground may be irregular, rocky, or slanted, but the artist's ground line is always parallel to the viewer's horizon and tangent to the 90° COV; i.e., horizon and ground are technical terms whose meanings differ from their colloquial use and are defined exclusively by the viewer's orientation to the scene, not the actual scene in itself.
Scaling the Unit Measure in the Depth Direction for 1PP
Figure 4 shows a unit metric on the viewing plane in red. Since the vertical and horizontal directions are are parallel to the viewing plane in 1PP, vertical and horizontal rulers have identical measure. To map the unit measure to a parallel plane in the depth dimension, we may use a DVP.
For example, a line from the right DVP to the left side of the bottom unit reference crosses the VP line on the bottom left of the cube. Therefore, this distance must also be one unit in length, but foreshortened for receding distance. Rays emanating from the DVPs and VP scale the unit cube anywhere (for example, to the right and farther away as shown in Figure 4).
Where to put the DVPs
How do we know where to put the DVPs? They may be placed anywhere equidistant from the VP. Like the VP, DVPs are arbitrary because they define the relationship between the artist and the object, and the artist can always change this relationship by moving toward or away from the vanishing point. However, once the viewing position is fixed, the DVPs determine the degree of foreshortening in the depth dimension.
0PP as a degenerate case of 1PP
We are now ready to consider why 0PP always results in depth dimension scaling equivalent to half the object measure. Figure 5 shows the 1PP setup.
[The only reason I venture into mathland is because it's the only way to rigorously prove that "if you want to draw a cube properly in 0PP, always offset the rear square by half the vertical distance of the front square." If you trust that this is true, you can skip the math.]
Figure 5. 0PP as a degenerate case of 1PP with the VP far away.
Like all perspective setups, this is a two-dimensional figure. It accurately represents a 2D projection of 3D reality, which is why the illusion is so convincing. However, all the labeled dimensions are metrics in the 2D viewing plane.
The VP and DVP are separated by an arbitrary distance n. We place a cube of edge length s such that its also n units in the foreground. Now if s/n is small (in other words, the VP is far away in terms of cube dimensions) then the lines running to the VP will be approximately parallel (red lines shown). They will be exactly parallel if the cube is infinitely far from the vanishing point (that is, n approaches infinity or equivalently, s/n approaches 0). To keep the analysis perfectly general, we make provision to slide the cube an arbitrary distance to the left or right between the VP and either DVP. We do this by dividing n according in two arbitrary portions, p and 1-p, where p is between 0 and 1. This changes the planar angle a of the edge that recedes in the projection of the depth direction (see next figure).
[The actual formula for the angle is a=atan(1/p), that is, the angle is a function of p, or equivalently, sliding the cube from left to right with respect to the VP will change the cube's angles. But you probably didn't need an equation to tell you that.]
The cube of Figure 5 is enlarged and annotated in the figure at left. If we know x and y then we know how much to offset the rear square from the front square in order to draw the illusion of a properly proportioned cube in 0PP. The similar triangles in this figure and Figure 5 (red and blue) give y/x=n/(n[1-p]) and y/(s-x)=n/(n[1-p]+s) from which we may obtain x=s[1-p]/(2+s/n) and y=s/(2+s/n). Focusing on the equation for y, note that as s/n goes to zero the equation reduces to y=s/2 regardless of the value of p or x, which does not even appear in the formula. Therefore, so long as the cube is sufficiently far away from the vanishing point (i.e., we are in 0PP), the vertical offset of the rear square from the front is always s/2 units. How far to slide the rear square left or right is a function of p because if s/n vanishes then x=(s/2)(1-p). But p is arbitrary, and up to the artist. So change angle a as you please. No matter, it is proved: for proper 0PP perspective the rear square is always offset by half the distance of the front square corresponding to the depth dimension.
Orthogonal Rotations of 0PP
Although 0PP is derived in this section as a devolved case of 1PP, a 0PP figure can be rotated in 90° increments or reflected to give new orientations because the artist has freedom to decide which direction is the depth direction, though, multiple viewpoints cannot be combined in the same drawing unless curious depictions are the goal. My point in the figure at left is merely to show that even in 0PP multiple orientations are available to the artist.
Construction of Inclined Planes in 1PP
Staircases, ramps, things on hills, etcetera, may be constructed in 1PP. However, since the plane is inclined with respect to the ground plane, it will not share the same VP, but rather a displaced VP. The easiest way to construct an inclined plane in 1PP is to first construct a hexahedron such that it envelops the inclined plane and key points such as corners and edges. With the key points placed in proper perspective, construction of the surrounded inclined plane is straightforward.
Rotation of an Object in 1PP
Now, suppose we wish to rotate an object. If a cube is turned at any angle along the floor, it will change its relation to the viewer. Since two axes will now be oblique to the viewing plane, the cube will no longer be defined by a single VP in 1PP but by two VPs in 2PP.
Nonetheless, the rotated cube can be drawn even in 1PP by first constructing a surrounding hexahedron in 1PP sufficient to contain the rotated cube (rose). To do this, I use elevation views, others use the plan view. These views are known as orthographic projections. They show the object from any three convenient but mutually orthogonal directions: for example, top-down (a.k.a., a plan view, that is a bird's eye view), west looking east (side elevation view), and north looking south (front elevation view). In the case of the rotated cube shown, the front and side elevations are identical. Since the front view is in the viewing plane it does not foreshorten, so we begin our construction with it.
The Hard Way to Rotate Figures in 1PP
The DVP has the ability to take any measure parallel with the viewing plane and to map it to a vanishing line in the depth direction. In our case, we have used the DVP in conjunction with a horizontal ruler to project these measures into the depth dimension. For clarity, these construction lines are omitted, but the reader may verify that a line from the left DVP to any mark on the horizontal metric will intersect the vanishing line at the tick marks shown on the depth ruler.
Now that we have shown how a horizontal measure can be rotated into the depth dimension, we shall now use this strategy to rotate a cube (or any object) within a surrounding hexahedron. Since the hexahedron will remain unrotated, it remains in 1PP. Notwithstanding, I do NOT recommend that you use this procedure, because I will show you an easier way momentarily, but bear with me for now.
Figure 6. Rotating a Figure in 1PP the Hard Way.
In Step 1, we place the front projection along the horizontal ruler (see boxed numeral 1). We then draw vanishing lines from it to the VP (rose lines) to indicate where the left-front and right-rear corners of the rotated cube will go. That leaves the left rear and right front corners, which require a side-view projection (Step 2).
For Step 2, we have placed a side-view projection at the origin along the horizontal ruler (rose lines, see boxed numeral 2). It appears a bit distorted because it is outside the 90° COV, but it is correct. We then use parallel lines (dotted rose) to project the side view on to the reference cube. Parallel projections give us the left-rear and right-front corners of the cube.
In Step 3, we simply connect the places where our metrics intersect our reference hexahedron to locate our remaining corners, see boxed numeral 3. Now that the corners of the rotated cube are known, we connect them to form the rotated cube (heavy black lines). Note that the vanishing lines from the cube will now intersect the horizon in two places (indicated by the gray lines and arrows going to two vanishing points, one of which is off the page to the right. The rotated cube is in 2PP. If other cubes or hexahedrons share the same orientation, they will use the same two vanishing points.
We could also do this with only one small parallel projection if we just use a DVP to rotate the image 90° directly in the hexahedron. Or, we may simply determine the coordinates of the rotated cube and place the vertices directly using the rulers. Thus we see that the choice of 1PP or 2PP is determined by the viewer-object relation. If we were to tilt the rotated cube upward (or move our head downward) then all three axes would be oblique to the viewing plane and the cube would be in 3PP. In other words, if an artist chooses to draw in 1PP, the object space has only one direction oblique to the viewing plane. If an artist chooses 2PP, two directions will be oblique to the viewing plane. Finally, in 3PP, all three directions of the object space are oblique to the drawing plane. Again, it is the viewer-object relation that determines the type of perspective, not the object itself. Once a cube is placed, all other figures parallel to it will share the same vanishing points.
The Easy Way to Rotate Figures in 1PP (Use 2PP)
Rather than go through the tedium of the above approach, we may think of 1PP as a degenerate case of 2PP where one of the VPs is located in the same position as the the VP in 1PP, and the other VP is located at infinity.
Figure 7. 1PP as a degenerate case of 2PP
Rules for 2PP
At the bottom of the figure is the Pivot Point (PP). Here are some facts about 2PP.
ALL 2PPs preserve a 90° angle between VPs.
The vertex of the 90° angle is ALWAYS located at the PP.
The PP is ALWAYS located on the 90° COV.
For the case of Figure 7 we have rotated our VPs clockwise, and VP2 is located at +∞. We could have rotated counterclockwise, putting VP2 in place of VP1 and placing VP1 to the left at –∞. No matter, vanishing lines to the off-page VP remain parallel to the Horizon and Ground Lines; i.e., with these rotations, 2PP degenerates to 1PP. Therefore, in 1PP we can remove the designation VP1 or VP2 and simply use a single VP.
The Hard Way to Construct Measuring Points for 2PP
Figure 8. 1PP and 2PP figures in the same scene.
In Figure 8, I have colored red the face or edge of the cubes that touch the viewing plane. These are the only places where the horizontal and vertical measures remain at their unit scale. All other locations recede into the depth direction and must therefore be scaled by the depth ruler or equivalently, by the respective DVPs. For the 1PP case, the horizontal rule may be scaled to the depth dimension using its single VP. It is also possible to do this in 2PP.
Figure 9. 2PP figures sharing common VPs and DVPs.
We focus now on the rotated cube and any faces or cubes parallel to it (Figure 9). These will all share the same set of VPs and DVPs. The vanishing lines are now colored light blue (dashed), while the diagonal vanishing lines are now colored dark red (dash-dot). From these two sets of lines we can determine any new parallel cube or face. In the case shown, only one edge of one cube intersects the viewing plane (heavy green bar). This is the only edge that has a unit measure in the viewing plane and thus, does not need to be scaled for depth. In some cases, it would be useful to map this metric to other spaces in the drawing. In 1PP the horizontal ruler is scaled directly by the VP. However, in 2PP, the horizontal ruler cannot be scaled in the depth direction with either vanishing point because both VPs are oblique to it. In that case, we need measuring points (MPs) that will do this.
We determine MPs by first using a DVP to scale the unit dimension from the ruler toward the vanishing point. Then we draw lines from the horizontal ruler through the scaled unit dimensions and extend them to the horizon (these are the two black lines shown in Figure 9). The intersection of the measuring lines with the horizon determine two MPs, one associated with the left VP (this is the rightmost MP) and another associated with the right VP (this is the leftmost MP). This provides an alternative method of scaling the unit dimension in the depth direction (gray lines) that may be more convenient. [In fact, there is an easier way to construct the MPs without the need to determine DVPs. We delay this discussion until after Figure 10.] We thus have three ways of scaling in the depth dimension: DVPs, internal diagonals, and MPs. Use whichever is most convenient for the case at hand. Just remember that in 2PP each direction scales differently unless the VPs are equidistant.
Whatever perspective we use, ALL VANISHING POINTS MUST LIE ON THE HORIZON. Why? Because we only have two eyes and two points define a line; thus, two eyes define an eyeline. As already noted, it is this eyeline which forms the horizon in perspective drawing.
Two-Point Perspective, in Detail
Consider once again the railroad tracks that introduced us to 1PP. Do you notice anything different about these compared to the ones in the 1PP section? If you look carefully, you will see that the railroad ties in this perspective drawing are not horizontal – that is, both the rails and the ties have different vanishing points. Therefore, the tracks are oblique to the picture plane in two directions and the figure is in 2PP. Two-point perspective is the go-to perspective for architectural drawings. Why? A number of reasons.
Buildings are sited on level ground. Thus, the structure and many of its features will be parallel to the ground plane. At the same time, it is generally desirable to reveal two sides of the building from street level. This requires rotation of the viewing plane with respect to the building . Rotation of an object on the ground plane is the defining characteristic of 2PP (see Figure 2b).
Buildings and houses are typically viewed such that the vertical direction is aligned with the viewing plane, again making 2PP the obvious choice.
2PP has enormous flexibility. It can accommodate interior and exterior views of any objects parallel to the ground plane in any rotation or for multiple rotations within the same drawing.
For all these reasons, 2PP is the preferred perspective for architecture. However, this is would NOT be the case for anything that requires the viewer in a 2PP scene to tilt his head upward or downward – for example, skyscrapers or towers viewed from street level or in aerial views. These will require 3PP. [Remember, the viewing plane is always normal to the viewer. If you tilt your head upward or downward, the viewing plane moves with you leading to the orientation of Figure 2c.] However, 2PP can be used even for these objects if the viewer is sufficiently distant.
Figure 10 shows the basic layout of 2PP. Let's look at each of its elements.
Figure 10. Railroad tracks in 2PP.
Familiar elements. As usual, we show
the 90° COV (dotted black),
the horizon (solid blue), and
the ground line (solid green) with tick marks to reference a convenient horizontal unit metric;
the VPs are placed where the horizon intersects the 90° COV.
Construction of MPs (The Easy Way). The MPs are labeled. Rather than construct them from a series of internal diagonals and subsequent regress from the measure line to the horizon, there is a more direct approach. Each of the shorter legs of the right triangle may be pivoted about each respective VP until it intersects the horizon (red dotted lines). This determines each MP directly. Rather than label them left and right, I have chosen to subscript them so you can more clearly see the association. MP1 is used to measure depth scaled by VP1, and MP2 is used to measure depth scaled by VP2. For the case at hand, the distance between ties is scaled by VP1 and the distance between rails is scaled by VP2.
Tie placement. The intersection of the measuring lines from MP1 and the vanishing lines from VP1 determines the location of each tie (gray lines).
More About the Pivot Point (PP). I have labeled the Pivot Point (PP) on the lower right. This point can be anywhere on the 90° COV. However, I have chosen to place my pivot point at the vertex of the right angle of a 30°-60°-90° triangle whose hypotenuse is the horizon line of the 90° COV. Architects typically organize their drawings using either a 30°-60°-90° or a 45° right triangle (i.e., 45°-45°-90°). However, no matter what acute angles are used, the pivot point is always the vertex of a 90°angle and the horizon is always the hypotenuse. Figure 12a shows the set up for 45°-45°-90° layout.
0PP as a Degenerate Case of 2PP (Rotation in 0PP)
It is also possible to rotate an object in 0PP for which we may consider 0PP to be a degenerate case of 2PP. Figure 11 shows the arrangement.
Figure 11. 0PP as a degenerate case of 2PP (Rotation in 0PP).
Figure 11 shows the usual 2PP case. As before, if the edge of the cube (s) is small with respect to the distance to the horizon (n) then the vanishing lines from the cube to each respective VP are nearly parallel. The cube is rotated by angle a and intersects the horizon at two VPs spaced apart by distance d. The ratio of n/d=sina cosa. Therefore, as d increases without limit, so will n, but the ratio n/d will remain constant. To see what will happen as d increases without bound, we magnify the cube with special attention to the triangle x-y-s circled in red (figure at right).
From similar triangles it is possible to derive expressions for the vertical (y) and horizontal (x) offsets of the cube. Note that as d increases without bound (or equivalently, as s becomes small with respect to d) then s/d vanishes and the expressions for x and y simplify to x = s cos a and y = s sin a. But these are merely the horizontal and vertical expressions of a square rotated by angle a. In fact, since all vanishing lines must be parallel in 0PP, the bottom and top faces of the cube must merely be rotated squares. Since the vertical distance of a cube in 2PP does not vanish, the top and bottom squares must be separated by distance s. Thus, to draw a rotated cube in 0PP with edges parallel to the viewing plane, merely rotate the base square by any desired angle a and offset an identical square by distance s. Does this violate our earlier rule for 0PP as a degenerate case of 1PP? No. In the former case, the cube had faces parallel to the viewing plane and so was a degenerate 1PP case. Here, we have edges parallel to the viewing plane which makes it a degenerate 2PP case.
Drawing 3D Objects in 2PP from Orthogonal Projections
We shall now place a schoolhouse in 2PP from plan and elevation drawings. Since the schoolhouse has a long and a short side, the 30°/60° view is preferred, but for the sake of variety, we shall use a 45° view. To the right are elevation and plan views of the schoolhouse, which we would like to layout in 2PP.
In Figure 12a, we begin by drawing our 90° COV and finding the VPs, MPs, PP, and horizon and ground lines. Then we place the side and front elevation views on the ground line, joined at the southwest corner (green vertical bar). It is only this SW corner that touches the viewing plane. All other features must be scaled using the respective VPs and MPs. For now, we will ignore the chimney, roof, and windows and just focus on placing the walls (blue lines).
FIgure 12a. Transferring elevation drawings into 2PP.
We draw the vanishing lines from VP1 and VP2 to the corner of the house that is in the viewing plane (green bar). Then, we use the respective measuring points to scale the elevation views to the vanishing lines as shown above. Now, using the same procedure, we lay in the windows and doors. To keep clutter to a minimum, the vanishing and measuring lines are shown only for a particular window and door (highlighted in red).
FIgure 12b. Transferring elevation drawings into 2PP.
As always, the VP controls the recession in the depth dimension while the MP controls the projection from the viewing plane onto the vanishing plane. Portions of the roof and chimney are in planes that are neither parallel to the ground nor the walls. Therefore, we must project from some surface parallel to known planes. Let’s start with the chimney.
FIgure 12c. Transferring elevation drawings into 2PP.
Chimney
Using the side and front elevation views, project the chimney downward to the ground line (light blue dotted lines).
At the point where these intersect the ground plane, extend vanishing lines from the ground line projections to the respective vanishing points. The intersection of these lines projects the top view of the chimney onto the floor (thick light blue square on ground plane).
From there, project upward (dotted blue lines) to the top of the chimney as determined by vanishing lines from VP1 to the top of the chimney on the front elevation view.
Define the point where the chimney penetrates the roof by projecting vanishing lines from VP1 to those points on the front elevation view.
Roof
A vanishing line from the gable of the front elevation view to VP1 determines the top roofline.
The roof must terminate in the vertical planes at the front and back of the house. To find these, project a vertical line upward from the northwest and southwest corners of the house. These must terminate at the vanishing line from VP1.
Project a vanishing line from VP2 to these verticals.. The top roofline must terminate where VP1 and VP2 intersect. Figure 12d shows the perspective view along with the orthogonal views.
FIgure 12d. Finished 2PP drawing and orthogonal views.
Rotation in 2PP, the Easy Way
Now suppose we have an identical school building that is rotated 15°. As a result, we will not be able to use the existing vanishing and measuring points. We could use the procedure we used earlier to rotate an object in 1PP by constructing a hexahedron to surround the rotated object. But this just resulted in a 2PP layout, and in retrospect, it would have been easier to simply have used 2PP from the beginning. We have a similar situation here. We could go through the tedium of constructing a surrounding hexahedron that uses the existing VPs, and mark key points on it to obtain the rotated layout. However, there is a much easier way.
Recall that VPs in 2PP must form a 90° angle at the PP. Therefore, the easy way to rotate a figure 15° (or any angle), is simply to rotate the vanishing lines about the PP (Figure 13). To minimize clutter and tedium, I will just show the outlines of the rotated building (red). For reference, I have placed the existing schoolhouse we constructed (blue) using the previous 45° perspective coordinates (heavy blue lines). I have also left the original VPs in place (labeled in blue).
FIgure 13. Rotated buildings in 2PP.
The first step is to determine new projective coordinates for the rotated system. To do this, we rotate the vanishing lines by 15° at the PP.
Since the included angle must remain 90°, the new vanishing lines (pink) are at 15° and 75° at the horizon. Their intersection defines two new vanishing points (VP'1 and VP'2).
These in turn allow construction of to new measuring points (MP’2 and MP’2). This is most conveniently defined by rotating each vanishing line from the PP to the horizon as shown (dotted pink arcs).
Now that we have defined our new projective coordinate system, we need to decide where to place the rotated object. We do this by placing the SW corner of the new building (green bar) at a desired location.
Using the measuring points, we project measuring lines from them toward the ground line to find where along the ground line to place our elevation drawings. Since we have chosen an anchor point that is not on the viewing plane, the side and front elevations are separated. (The SW corners in the elevation drawings are emphasized with heavy black lines to show the correspondence.) Equivalently, we could have scaled the elevations to the parallel plane touching the new building, in which case they elevation views would not be separated.
We now construct the rest of the measuring lines using the respective measuring points and elevation drawings (light blue lines). Note that the MPs project our building dimensions (horizontal and vertical) from the viewing plane onto the perspective planes.
Once we map the distance onto the perspective planes, we use the vanishing lines to connect the key points. This defines the red building. I have also shown the hidden corners for clarity (dashed black lines).
At right is the drawing with the vanishing and measuring lines gone. The buildings are now at a 15° mutual angle and are in proper perspective. If the buildings were parallel or at right angles to one another as often happens in the typical grid layout of a city, we could have constructed all buildings with the same VPs. However, a rotation of rectangular objects by any angle except a multiple of 90° requires determination of new VPs and MPs as given in Figure 13. In real scenes multiple vanishing points are the rule and will always be required for objects that are oblique to one another.
What About Curves?
Now, one may ask, okay great – now I can draw stuff with straight lines, but what about curves? Well, if you can draw the space surrounding a curve, you can draw the curve (or any object for that matter) by mapping key points onto an outline of the surrounding space. I’ll show you. When a circle is projected onto an oblique plane, it becomes an ellipse (Figure 14), but the projection is foreshortened in the vanishing plane so the horizontal dividing line is the vanishing line, not a horizontal line with respect to the viewing plane. Moreover, the vertical dividing line does not split the ellipse evenly in the oblique figure for the same reason – the plane of projection is oblique to the viewing plane.
Figure 14. Mapping Curves in 2PP.
Nonetheless, these projected figures appear reasonable to us because our visual apparatus adjusts to take account of the distant oblique plane in which they supposedly reside. This is exactly how they should and would appear because we have followed the rules of perspective to create the illusion of distance and obliqueness. However, we must remind ourselves that this IS an illusion – there is no depth dimension. There is only a flat piece of paper and all the drawn objects reside on it. The illusion works because the figures are distorted and foreshortened in a way that would be identical to the retinal image those objects would create if they resided on a distant oblique plane.
When we place them all on the same plane, the illusion is broken and the distortion is better appreciated. Out of context their relative proportions look odd. With no perceptual cues for depth, the figures appear much smaller and severely distorted.
Three Point Perspective (3PP)
Okay, it has finally come to this – everything is oblique to the viewing plane. Now what? Airline pilots speak of roll, pitch, and yaw to describe the attitude of their aircraft. If your standing body were an airplane pointed toward the sky, roll would be turning of your head (and the viewing plane) to the left or right by whatever means (by pivoting a foot, turning through the waist, turning your head or even glancing left); rolling our head left and right is the way we gesture "no". Pitching our head back and forth is the way we gesture "yes". Finally, yaw would be a tilt of the head toward either shoulder, a gesture we sometimes use as if to say "Huh?", but one more usually associated with confused cocker spaniels. Any one of these movements will take a scene from 1PP to 2PP. Any two of these movements will take a scene from 1PP to 3PP. Thus, if one rotates (rolls) the head or body to face a building such that its lines are oblique to the viewing plane, we have gone from 1PP to 2PP. If this is combined with a pitching of the head to gaze at the top of the building, we have moved from 2PP to 3PP. What about adding yaw? Will we move to some mysterious fourth dimension? No. Artists live in a three-dimensional world despite their best efforts; adding yaw will move things around a bit, but real scenes can always be reduced to at most 3PP.
Most recognize that 3PP is the usual perspective for views of canyons and skyscrapers. But, perhaps more surprisingly, 3PP is also required for many still lifes because the viewing plane is oblique to the object in these cases. Since still lifes typically showcase nearby objects, some vanishing points tend to be quite distant from the drawing (Figure 15).
Figure 15. "Contain the Strain." A still life in 3PP.
3PP is more complex than 2PP, especially when scaling and rotating objects. Let us begin with 2PP and then cause some roll, pitch, and yaw and see what we get.
Figure 16. Oblique faces in 2PP leading to 3PP.
Figure 16 shows a cube in 2PP with an inscribed cuboctahedron. This looks like quite a trick, but we have just connected the midpoints of the cube's edges. Now, if we look at the vanishing lines of the square faces (which are mutually orthogonal but rolled, pitched, and yawed with respect to the viewing plane) and trace their vanishing points we find we have three of them, not two. Whether we like it or not, we have just entered the world of 3PP because three directions are now oblique to the viewing plane, each with their own VP and MPs.
General Procedure for Constructing VPs and MPs in 3PP
Place three VPs at desired locations and connect one to another with horizon lines (heavy blue lines in Figure 17 below), each similar to the horizon line of 2PP. This forms a horizon triangle (heavy blue) triangle.
Outside the horizon triangle, draw semicircles whose diameter is the length of each horizon line (dotted blue half circles). So far, this is just the 2PP procedure repeated thrice.
Now, from each vanishing point, draw a line perpendicular to the opposing horizon line and intersecting its semicircle. This construction is usually done with the right angle of a drafter’s triangle, or if the work is large, a carpenter’s square. These three lines will meet inside the horizon line triangle at a single point.
Draw chords from each VP to each intersection point on the respective semicircles. These make three right triangles with their hypotenuses as horizon lines (heavy blue lines).
Construct an arc from each right angle to its respective horizon line using each VP as the pivot point. This gives each horizon line two MPs, for a total of six measure points (dashed red arcs).
This gives the following diagram. For clarity, I've called out each MP as MP(ab), where a is the VP of the arc's pivot point and b is the VP the arc pivots toward.
Figure 17. Construction of MPs in 3PP
General Procedure for Finding the COVs
With 2PP, the 90° COV was merely a circle whose diameter was the distance between VPs. However, this will not work for 3PP because distance is foreshortened in all three directions. The center of all COVs lies at the intersection of the perpendiculars. To find the 90° COV in 3PP requires the following procedure.
Pick the largest semicircle to begin and extend it to a full circle (green dotted circle in Figure 18). Actually, any semicircle will do, but the largest one tends to give the most accuracy.
Measure the distance of the perpendicular from the vanishing line to the extended circle (solid red line). This will become the radius for a new circle (red dash dot line).
Construct the new circle.
Draw a line segment through the intersection of the perpendiculars that is parallel to the longest vanishing line and whose ends terminate on the new circle (heavy black line). This is a chord to the new circle but a diameter to the 90° COV.
Draw a circle centered at the intersection of the perpendiculars and having the same diameter as the chord from step 4. This is the 90° COV. All COVs are centered at the intersection of perpendiculars, but their diameters are scaled by the tangent of the half angle. In other words, the diameter of the 60° COV is just tan(60°/2) = tan(30°) ≈ 0.577 times the 90° COV.
Figure 18. Construction of COVs in 3PP.
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The Five Essentials for Constructing the COV
Figure 19, expresses the essence of what we have done.
Figure 19. Summary of COV Construction
Essentially, we have defined the 90° COV by determining five things.
the intersection of the perpendiculars (P, determines the center of all COVs)
the intersection of a perpendicular with a side, e.g., from VP1 to L23 (Q)
the parallel projection (of L23 through P)
the circular arc (of diameter L23 centered at Q)
the diameter of the 90° COV (formed by the intersections of the parallel projection and the arc.)
The Use of Measure Points in 3PP
In 2PP, MPs are used to transfer a measure from the ground line (or any line parallel to the horizon) to a vanishing line. In 3PP there are three horizon lines. In clockwise order they are: one that runs between VP1 and VP2, another between VP2 and VP3, and a third running between VP3 and VP1. Each horizon line hosts two measure points. The MPs are used to transfer any measure from a line parallel to their horizon line to the vanishing lines that run to their VPs. For example, MP21 will transfer a measure from a metric parallel to the VP1-VP2 horizon to vanishing lines terminating at VP2, while MP12 will do the same for vanishing lines terminating at VP1.
Figure 20. Use of a measure star in 3PP to construct a cube.
The green measure star (Figure 20) contains the unit dimensions we wish to map to the vanishing lines and it resides on the viewing plane. The only place that the star touches the cube is the top front-facing corner because that is the only place that the cube touches the viewing plane. We show all six MPs, and six spokes on the measure star, but half of each are redundant because we can define the edges of the cube as few as three spokes and three MPs. Notice that the spokes are parallel to the horizon lines hosting their respective MPs. For example MP12 and MP21 attach to the spoke parallel with the horizon line running between VP1 and VP2. The cube can be scaled to any point in space using the MPs and VPs. Figure 21 shows how this is done by mapping additional measure points. To cut down on the already formidable clutter, we use only one MP on each vanishing line and a three-spoked measure star to sufficiently map the perspective.
Figure 21. Placing cubes in 3PP Space.
Note that as the cube dimensions grow larger in relation to the vanishing point distances (i.e., as the s/d ratio increases) the perspective becomes more pronounced and as this ratio goes to zero, the perspective devolves to 0pp. The average object/distance (s/d) ratio for Figure 21 is about 1/6. Figure 22 shows the effect of increasing or decreasing this ratio. Note that as s/d = 1/∞ = 0 all lines become parallel.
Figure 22. Effect of average s/d ratio decreasing in 3PP.
Figure 22 shows that as the vanishing point grows more distant (s/d ~ 0) 3PP devolves to 0PP. Figure 22 reports an average s/d (hence the overbar) because the s/d for each perspective direction is different owing to the non-uniform distance of each VP from the perspective center. As d becomes larger the COVs must also grow, and in the limit all COVs are infinitely large. Thus, any perspective (i.e., 1PP, 2PP, or 3PP) can devolve to 0PP if the vanishing points are sufficiently far away as measured by the characteristic length of the object space (i.e., an s/d ratio approaching 0). So, although the rightmost figure is in 0PP, all sides have dissimilar lengths owing to rotation and tilt of the figure with respect to the viewing plane. Also note that the 0PP projection is ambiguous and could represent an interior or exterior corner and may invert from one to the other with prolonged gaze. By contrast, the use of perspective compels the corner to be an exterior one for the left and middle figures.
In summary, if 0PP is devolved from 1PP, one plane of the cubic space will intersect the viewing plane or a plane parallel to the viewing plane (Figure 3). If 0PP is devolved from 2PP, one edge of cubic space will intersect the viewing plane or a plane parallel to the viewing plane (see the righthand figure after Figure 11). If 0PP is devolved from 3PP, one vertex of the cubic space will intersect the viewing plane or a plane parallel to it but edge lengths may differ in each direction (Figure 22, right). In all cases for 0PP parallel lines remain parallel with no vanishing points by definition. However, perspective drawings of any order tend to be more realistic than parallel projections.
The 60° 3PP layout
The analog to the 45°-45°-90° layout in 2PP is the special case of VPs subtending 60° angles in a 3PP layout. In the 45°-45°-90° layout a unit cube is viewed edge on (Figure 2b) so that both faces recede from the viewing plane symmetrically. Analogically, in the 60° 3PP layout, the viewing plane is aligned with the corner of a unit cube (Figure 2c). The 60° layout is the only time COVs are centered in 3PP space (Figure 23) and all VPs are equidistant.
Figure 23. The 60° 3PP layout.
To Infinity and Beyond
So how many kinds of perspective are there? WIth all deference to Buzz Lightyear, no matter how complex the shape, the surrounding space can only be three dimensional at most. This is why we only need to study up to 3PP, because those are all the dimensions we have and we can always project complex shapes onto a line, rectangle, or hexahedron. Although there may be many many more than three VPs in real scenes owing to different oblique orientations of many objects, no better than 3PP is required to handle them all.
What about reflections in a water drop or your face reflected in a spoon? There are curvilinear perspectives of higher orders than 3PP. However, due to physics and optics, linear perspective remains valid and it can handle even these complex cases. The key for applying linear perspective to curved objects is to think about the surrounding space rather than the curved object itself. In three dimensions, one would draw a reference hexahedron to represent the space and then map key points on the curved surface to the hexahedron and project them according to the rules of linear perspective. Notwithstanding, there is often no reason to bother with such tedium for these complex shapes and the artist may just freehand the shapes in place. After all, unlike cubes and circles, there are no standard spoons or waterdrops so it is difficult for any viewer to insist that the curvature of a spoon or water droplet is different than what the artist has represented unless the errors are gross.
