Ground line drawings usually start with a blueprint or set of plans for the primary form (diagram, right). The plan image is critical for scaling the width and depth of the perspective image. The elevations are critical for exterior details and contrasting surface features of the front and side faces. deriving measurements in the ground line method The first step is the projection of vanishing points and primary form measurements onto the horizontal and vertical measurement lines (diagram, above). The station point is established in relation to a plan of the primary form, using the plan scale to determine the object distance. For example, if the scale of the plan is 1/4" = 1', and the form will be viewed from 50 feet, then the station point is located 12.5" from the near edge of the plan outline. This object distance and orientation is usually judged by drafting a large isosceles triangle with an internal apex angle equal to the desired circle of view; the object is then moved toward the apex of this triangle until it just touches both sides, while turning the object so that the plan orientation gives the desired direction of view. The horizontal measurement line is drawn perpendicular to the line of sight (center of the cone of vision), typically so that the measurement line passes through a front, prominent feature of the plan. Divergent sightlines are drawn from the station point through the principal features of the plan, and their intersections with the horizontal measurement line are marked and labeled, along with the location of the 2PP horizontal vanishing points and the line of sight (the "sight point"). The corresponding vertical measurement line is projected by parallel lines from the elevations of the primary form, including its intersection with the ground line. deriving measurements in the ground line method Next, the horizon line and ground line are located in the image area to produce the desired image composition, specifically the height of the viewpoint to the primary form, its distance from the viewer, and the angle of view in relation to the ground plane (diagram, above). The horizontal measure line is located below the image area, parallel to the horizon and ground lines, with the sight point vertically aligned with the center of the image area. Vertical lines are used to project the horizontal plan and vanishing point measurements onto the ground line. The vertical measurement line is located on the ground line. Its measure points were not projected from the station point, therefore they must be projected from the horizontal vanishing points forward or backward from the ground line, and these projected measurements carried by horizontal lines into the image of the primary form. Note that the vertical measurement line can be freely shifted along the ground line, if convenient; the horizontal measurement line, because it was scaled to a specific location of the station point in relation to the primary form, must be exactly aligned with the sight point and cannot be moved. There are many interesting methods of scaling and projection with the ground line method that I omit in this brief outline. To conclude, I mention four principal shortcomings to the ground line method: •procedural geometry: The ground line framework is basically a scaffold built expressly to deploy specific procedures for the construction of an image; it has a weak connection to perceptual geometry. In particular, it is more difficult to deduce the construction principles for a new geometry, because important basic concepts (such as measure points or vanishing point rotation) are typically excluded. The better ground line tutorials discuss what to do if you discover that the perspective view is not optimally oriented, or is incorrectly scaled, after the drawing has been started; those kinds of problems result from the narrowly focused, piecemeal nature of the ground line method. •errorful, inflexible scaling method: The ground line method of scaling an image, which requires the plan be oriented to scale and then projected by diverging lines onto a horizontal measure line, is only practicable for large objects seen from a relatively close vantage point; otherwise the method becomes inaccurate. It also requires greater precision, as drawing errors are magnified by the expanding lines. If the decision is made to change the angle of view on the primary form, the entire scaling operation must be repeated. In the 90° circle of view framework, scaling to measure points compresses measurement errors, and a new rotation of the vanishing points allows the same measure bar to be used from different perspective views. •complex line construction: All the ground line demonstrations that I have seen require a large number of construction lines — at a minimum, the projection onto the measurement lines, the projection of the measurements across the image area, and the joining of vanishing lines. •adapted to drafting tools: The ground line framework is adapted in various ways to technical drafting tools, and when used with those tools it is very efficient. But this means the framework is less effective as a general, artist oriented perspective model. For these reasons, and in particular when linear perspective is taught to photographers and painters, I strongly prefer 90° circle of view method. who has a 12 foot table? Unfortunately it is fairly common to start with the primary form in an orientation that puts the two vp's inconveniently far apart. In the previous cube construction example, assuming a 10 foot circle of view, the cube is oriented so that the two vp's would about 11 feet apart — one 3.2 feet to the left of the dv, and the other 7.7 feet to the right. This isn't very convenient for a drafting table. If you have a 12 foot table, push pins and lots of string (or the specialized drafting equipiment that rescales vanishing point locations within a small work area), you can work out the geometry of a cube at any size, no problem. If you're lacking the table, you can lay the support on any large bare surface, for example a clean kitchen floor or concrete patio, and work there — using tape instead of pins to hold the string. If those alternatives don't appeal to you, then you can rescale the drawing. The basic geometry of the vp's works exactly the same no matter how big or small the circle of view is assumed to be. So just get a large sheet of paper, draw the 90° circle of view to a conveniently small size (20cm works well), work out the vp's and perspective drawing in that format, make a careful outline drawing in perspective, then transfer the drawing to the painting support, enlarging it as you make the transfer. You can control the enlargement by squaring the diagram or by using a surface projector, adjusted so that the size of the image matches the length and location of a reference vertical (front vertical edge) marked in the right place on the support. When one or both vp's are really far from the drawing surface, it's possible to calculate the relative sizes of edges and angles in a drawing without ever anchoring the vp's with string or ruler: you just need to work out a few key measurements on a calculator, and you need to know the exact distance of the two vanishing points from the principal point (dv), which is found either with a careful rotation around the viewpoint, or by multiplying the radius of the 90° circle of view by the tangent of the vanishing point angle to the direction of view. method for scaling new lines without vanishing points The diagram shows all the points required for these calculations. The procedure is simple once you understand it: go through the instructions slowly and carefully, and you should have no trouble. (Caution: use a metric or engineering ruler for these tasks.) There are two situations: the anchor line is either entirely above (or below) the horizon line, or it straddles the horizon line. Start with the straddled line (right side of the diagram). The key measurements you must know in advance are: (1) the length from dv (the direction of view) to c (the vanishing point), (2) the length of the anchor line above (A 1 to A 2) and below (A 2 to A 3) the horizon line, and (3) the distance from the direction of view to the anchor line (d to a). Once again, the triangular proportions provides the frame of reference. First, by subtracting the distance dv-a from the distance dv-c, you determine the distance ac from the anchor line to the vanishing point. (If the anchor line is on the opposite side of dv from the vanishing point, you add the two distances to get ac.) Now you want to insert a new line N 1 to N 3 which has the same height in depth as the anchor line. First, determine how far to the left or right of the anchor line the new line should be placed: this defines ab, which gives bc when subtracted from ac. Then the ratio bc/ac tells you the length of the new line N 2-N 3 in relation to the length of line A 2-A 3, and the length of the new line N 1-N 2 in relation to the length of line A 1-A 2. For example, if bc/ac equals 0.80, and the upper part of the anchor line A 1-A 2 is 2cm long, then the upper part of the new line N 1-N 2 will be 0.80 * 2.0 = 1.6cm long. Repeat for the line segment below the horizon line (N 2-N 3). If the new line is closer to the direction of view d than the anchor line, or on the opposite side of d from the anchor line, then you would add ba to ac. In that case the ratio bc/ac will be greater than 1.0, and the new line will be correspondingly larger. If the line is entirely above (or below) the horizon line (left side of the diagram), then the ratio bc/ac is applied to the length A 1-A 3 to get the top end of the new line, and to the length A 2-A 3 to get the bottom. How do you define the crucial distance dc (from the direction of view to a vanishing point) in the first place? The easiest method is to use my vanishing point calculator to get the measurements of the vp's and mp's, and adjust the viewing distance to the object and your angle of view until you get the proportions that seem desirable. Or, as described above, you can reduce the circle of view to a workable size, use the method for rotating the vanishing points to determine the locations of vp 1 and vp 2, measure the distance from these to dv on the diagram, then scale those distances back to life size. Unfortunately this method, even after you get the hang of it, still forces you into a lot of poking of a pocket calculator, and is hopelessly tedious and prone to error if many lines must be inserted in your drawing. The ultimate solution is to generate a recession grid for the distant vanishing point, and use this grid to determine the perspective reduction for any verticals in the drawing. using a recession grid for distant vanishing points First work out the angles and distances of your point of view within a reduced (20cm) circle of view drawn on a large sheet of paper. Carefully measure with a metric ruler the distances from dv to the two vanishing points vp, the diagonal vanishing point dvp and the two measure points mp, then multiply these by 15 to get them in the same scale as the 3 meter circle of view (the scale of the perspective drawing). Locate the horizon line, dv and the two measure points on your support. Now draw a vertical line on the lefthand side of the drawing, anywhere that is convenient — the line should be farther to the left than any major form in the drawing area. Mark off increments on this line above and below the horizon line using any convenient interval of measurement. Next, draw a similar line on the righthand side of the drawing, again putting it far enough to the right so that it won't obstruct any major forms in the drawing. Now you want to find a reduced scale of measurement for this righthand line to represent perspective recession from the lefthand line toward the vanishing point. You already know how to do this: treat the lefthand line as the anchor line, figure out the distance from this line to the vanishing point (ac), then the distance from the righthand line to the vanishing point (bc), and finally the ratio bc/ac. This is the reduction in the scale of measurement required for the righthand line. For example, if the intervals on the lefthand line are in inches, and the ratio bc/ac is 0.80, then the intervals on the righthand line are in 0.80 inches. When you have intervals marked on both vertical lines, connect the corresponding points to make a recession grid of converging lines (parallel lines in perspective). These lines show you the slope of any horizontal converging to the distant vanishing point. You can either draw the horizontals along an inscribed recession line (as in the base of the building in the diagram), or draw horizontals between and roughly parallel to any two lines (as in the top of the tower of the building in the diagram). This grid is especially convenient if you must work out the perspective recession for many repetitive or similar lines, for example the windows, columns, cornices and ledges on the front of a building. vp spacing from an object drawing Why not just say ... heck with it, I'll just draw the cube at whatever size fits the drawing, at whatever angles look good to me, and let the vanishing points fall where they may? You can do this, especially in a freehand drawing of the object from life. In that situation the principles of linear perspective guide you to look at the edges and faces and proportional sizes of the parts, and to draw these elements more accurately in relation to their fixed vanishing points. This "imagined" perspective context is useful because you can introduce expressive distortions to the perspective facts, controlling by eye how obvious or subtle they appear. However, if you are drawing an imaginary or remembered form from scratch, such as that cute little cottage you saw on yesterday's hike, then your placement of the vanishing points can go badly astray without the visual example in front of you. And once you have drawn the primary form, you still have to draw everything else to match its vanishing points, direction of view and horizon line. The most common drawing fault is placing the vanishing points too close together. The informal recommendation is simply to put the vanishing points very far apart ... no, farther than that ... keep going ... — with the idea that inaccuracies in widely spaced vanishing points are harder to see. Let's start with a perspective constant: the distance between 2PP vanishing points depends on the viewing distance to the object. The closer an object is to our view, the closer together its two perspective points will be in relation to the object size. This has a very powerful effect on the perspective view, as is apparent in these four cubes of exactly the same vertical size drawn as they would appear at increasing viewing distances. 2PP cube seen from four different distances a cubic box 2 meters high seen from (left to right) 3 meters, 6 meters, 12 meters and 24 meters The shape of the cube alone tells us a lot about its distance from us. The flattening in the "far" cube (24 meters, at right) is what we'd expect to see in binoculars or a telescopic lens, while the bulging in the "near" cube (3 meters, at left) mimics a wide angle lens. This "near" cube resembles many badly done perspective drawings, because the cube is too large relative to the vp's. So the perspective problem is to find a vanishing point separation that matches the apparent distance to the object we want to represent. And this is a problem that the circle of view framework is designed to solve. Fortunately, if we start with an acceptable 2PP drawing of the front sides of the primary form, we can reconstruct the 90° circle of view from the object drawing using the semicircle of Thales construction. The circle of view then can be used to locate the vanishing points. using a semicircle of Thales to find the 90° circle of view Let's take as our starting point a rough, freehand drawing of a cubic box 2 meters high viewed from about 6 meters. We made this drawing on the back of an envelope in the field, and now we want to build a more finished drawing upon it. First, extend the front edges of the primary form on either side until they meet in two vanishing points, vp 1 and vp 2. Connect these points with a straight line, which is the vanishing line for the primary form; if the form is level and upright to the ground, such as a building, then this is also the horizon line. This is the point to make any esthetic corrections. For example, if this is the horizon line, it should be level. If it is not, redraw it level and relocate the vanishing points on it by moving them vertically up or down. Then redraw the vanishing lines from these points back to the object drawing. Next, find the midpoint M on the horizon line between the two vanishing points, using a ruler or the line bisection method. Then draw a semicircle around the midpoint M from one vanishing point to the other. This is the semicircle of Thales. The useful geometrical fact is that the 90° angle of a right triangle must lie on a semicircle, if the diameter of the semicircle is also the hypoteneuse of the right triangle. This right angle corner is of course the viewpoint that we use to rotate the 2PP vanishing points. This viewpoint must lie on the semicircle of Thales. But where? To find it, we have to locate the direction of view. This is a somewhat arbitrary decision, but usually the dv is located on the horizon line somewhere around the front edge or center of the form. From the dv, extend a line perpendicular to the horizon line up to the semicircle of Thales, which locates the folded viewpoint. The distance from dv to the rotated viewpoint is the radius of the 90° circle of view. I claimed that the cube in this example was cube 2 meters high viewed from 6 meters. Let's check. The direction of view (dv) is located about 3/4ths up the front edge, so the viewing height is about 1.5 meters above the ground. As the horizon line is always at the same level as the viewpoint, this corresponds to our standing height on level ground when viewing the cube. By definition, this 1.5 meters is also the radius of the 90° circle of view, and is also the implied viewing distance to the finished image. In the drawing, the vertical of the cube is 16% of the diameter of the circle of view, or 48cm; this is the drawing size. So we have the viewing distance (150cm), drawing size (48cm) and object size (200cm). With formula 3, we find that the object distance from the viewpoint must be 3.2 times the object size, or 6.3 meters. Thus, from a rough but accurate perspective drawing, we have reconstructed the location of the vanishing points and the circle of view. We now have the framework to insert accurately details of the primary object, and to add objects around it in the same perspective space. The spacing of the vanishing points in relation to the drawing size is not merely "good enough," but represents the spatial relationships we intend to portray. where is the center of projection? The methods just described can also be used to locate all the perspective elements implied by a finished painting. This is a problem of more interest to art historians than to artists, but I will describe the methods here for both 1PP and 2PP paintings. Here again are the perspective elements that we need to identify in the approximate order we locate them: •Median Line. This is parallel to the side edges of a rectangular image format, or perpendicular to the floor when the painting is correctly hung. The median line is nearly always through the center of the image format. •Horizon Line. This is usually parallel to the bottom or top edge of a rectangular or square image format, or parallel to the floor when the painting is correctly hung, at the height of the eyes of standing figures (for a standing artist and viewer), or in similar proportion to windows, tables, walls between floors and ceilings, and so forth. The horizon line is rarely through the center of the image format. •Direction of View. This is at the intersection of the horizon and median lines. •Distance Points. In 1PP perspective these can be found as the diagonals of any square element receding to the dv, commonly the floor tiles of Renaissance paintings or frescos. •Vanishing Points. In a 2PP painting these are located from the edges of any suitable rectalinear (right angled) object in the image, of convenient clarity and size. •Circle of View. The radius of the circle of view is determined by the distance points in central perspective, or by the method of the semicircle of Thales in two point perspective. •Center of Projection. The implied correct perspective location for viewing the painting (the perspective center of projection) is at a distance equal to the radius of the circle of view along a line perpendicular to the direction of view dv. 1PP Construction. A straightforward example is provided by Raphael's Philosophy, his first fresco decoration on a large wall of the Vatican chambers. The image I can provide is drastically small (the original is a 27 feet wide), but a large format reproduction is available in most Renaissance art textbooks. 1PP reconstruction of the center of projection By examination we conclude that the fresco is done in central perspective, which means we are looking for the direction of view (dv) and the distance points or diagonal vanishing points (dvp's). These give us the center and radius of the circle of view and the implied center of projection. The orthogonals necessary to find dv are found in the receding barrel vaulted passageway at the center of the image. I chose the square edged capitals of the columns along both sides, which define two orthogonals (red lines) intersecting between the figures of Plato and Aristotle. Since the composition is in central perspective I know the horizon line is level and the median line perpendicular to it, so I go ahead and draw these in (blue lines) from the dv, extending the horizon line far off the picture to one side. The obvious place to prospect for diagonals is in the tiled floor at the foot of the fresco. However, the visual angle on these floor tiles is rather small, making them hard to see clearly; also, I may be wrong to think they are true squares. So I take a second diagonal from the diagonal corners of the capital of the front rectilinear column, which conventionally would be square in classical architecture. These diagonals (orange) intersect the horizon line in close agreement (thank you, Raphael!), so I conclude I really have found the diagonal vanishing point. A circle with center at dv through the dvp defines the 90° circle of view, so the distance between dv and dvp is also the viewing distance to the painting. The width of the painting is 27 feet, so by calculation the center of projection should be about 31.5 feet directly in front of dv. I haven't been to the Vatican chamber where this fresco is located, but the photographs I have examined suggest the center of projection is not in a practicable viewing position. (It is approximately 10 feet above the floor and several feet on the other side of the opposite wall!) The horizon line also lets me locate the implicit location of the painter in terms of the room represented in the fresco, which is roughly at the height of the white robed figure to the right of the dv (if the artist is assumed to be standing), or at the height of the central figures at the top level (if the artist is assumed to be sitting). From the dvp I can also determine the circle of view of the painting. Drawing a straight line from dvp to the top of the fresco defines a 20° angle. So the barrel vault boundary of the fresco represents a 40° circle of view. 2PP Construction. In the case of two point perspective, the necessary elements are the same, except that you must start by finding a rectangular form or forms that will reveal at least two vanishing lines for each of the two vanishing points. This object does not have to be square, but it must contain a right angle at the intersection of two sides. These vanishing points in turn determine all the other perspective elements. The intersection of each pair of lines defines a vanishing point, and the two vanishing points define the horizon line. Orthogonals, if visible, will point to the direction of view on the horizon line; if there are no orthogonals, then the median line of the format can be used to locate the dv. The final step is to reconstruct the right triangle whose hypotenuse is the horizon line between the two vp's and whose right angle lies on the median line drawn from dv perpendicular to the horizon line. This is easiest to find simply by dragging the right angle of a drafting triangle, or the corner of a large sheet of paper, up the median line until both sides can be aligned over the two vanishing points: the right angle corner is then on the folded vanishing point on the circle of view. Or, we can use the semicircle of Thales method. 2PP reconstruction of the center of projection In this painting by Edward Hopper, I ignore the suggestion of 3PP in the slightly upward flaring sides of the house and tilt of the telephone pole (we'll come back to this later). The diagonals from the eaves and base trim of the windows (orange) are a little sloppy, but my best guess puts their intersections (and the horizon line) at the bottom of the picture. (This construction fundamentally determines everything else, so it should be done as carefully as possible, using as many vanishing lines as you can find.) The median line and direction of view (dv) are arbitrarily located on the midline of the painting. To find the folded viewpoint on the circle of view, move the right angle corner of a drafting triangle up the median line until the two edges lie on both vanishing points: the right corner is then on the circle of view; or use the semicircle of Thales method by bisecting the distance between the two vanishing points. We discover several things from this construction: • Hopper's eye level (the horizon line) is level with the sidewalk in front of the house, which implies that the house is at the top of a hill and the artist was downhill from the house when painting — how far downhill depends on whether he was seated or standing. • The radius of the circle of view is approximately 1.8 times the width of this 50cm wide painting, which locates the center of projection (viewing distance) about 35 inches from the painting surface; for best effect the painting should also be hung slightly high, with the bottom edge at eye level of an average sized viewer. • By taking the largest circle around dv to the edge of the image, then measuring the angle at the folded viewpoint (25°) defined by the radius of this circle, we determine that the image is enclosed by a 50° circle of view, which creates the slightly bulging appearance of the front angle of the building. • However, the outward flaring in the sides of the building is directly contrary to the perspective geometry: viewed from below, a building's sides (and edges parallel to them, such as the telephone pole) should seem to converge toward a third vanishing point far above the horizon line, not below it. The fact that they flare outward as the rise above the viewer is a perspective distortion explicitly introduced for its dramatic effect, as it gives the old pile a characterful dynamism. The Renaissance artist and his modernist colleague have kept their art well within the commonly recommended 60° circle of view but have allowed some perspective distortion for dramatic or esthetic impact. Creatively "adjusting" perspective distortion has been one of the subtleties of painting for almost six centuries. 3PP Construction. Finally, it is even possible to identify the center of projection and circle of view in a 3PP perspective drawing, provided that all three vanishing points can be established from edges or lines within the image. The method for doing this is complex, but is explained on the next page as the perspective sketch method of 3PP construction. N E X T : Three Point Perspective Last revised 07.I.2015 • © 2015 Bruce MacEvoy |