Depth by Diagonals: method II

As discussed last week, the current section of the book Successful Drawing which I am reviewing, deals with some of the drafting rules for determining the depth of an object by casting diagonal lines.  Last week we covered the first method, and this posting then will explain Loomis’ second method which is outlined in the text.

The use of the bisecting vertical

This method is very similar to the one outlined in the previous posting, but here, Loomis introduces a few more ‘controls’ to make certain that the placement of receding points are accurate.  He advocates for the use of a bisecting vertical to be placed along the horizon line, to keep everything accurate.  I find this method to be very useful for drawing buildings, with repeating surface features, such as windows, facade details, or columns on ancient buildings.  Let’s get into it.

  1. First, let us consider that you are trying to draw a building, with a series of columns located in regular intervals along the side of it, going off into the distance.  First, create the near corner height of the plane you wish to decorate, and sight the V.P.per2
  2. Now, the artist must place the ending of the facade plane at what Loomis calls the optional depth.  This is the most unscientific part of his instruction, because it leaves the artist to place the second line by eyeball.  He uses the word optional not in the sense that one may draw it or not draw it, but instead Loomis means where the line is drawn is optional, i.e. it is up to you.  Once the visual depth chosen pleases the artist’s eye, he has created a section. The second line drawn is called the Terminus Line, as it delineates where the section terminates.  This section is now easily cloned in perspective.per3
  3. Now, the next step is the technical part of the procedure.  The artist casts lines diagonally through the section, starting at the near top and bottom corners and extending through their diagonally opposing corners. Following that, a vertical line must be drawn bisecting the horizon perpendicularly, through the V.P.  The diagonal lines must extend far enough to reach the point where the new vertical line delineates.  Mark the points where the diagonals cross the vertical.  Let us call the new high and low points, the Vertical Points.per6
  4. This X of diagonal lines shows the centre of the section, and sets the artist up to clone the depth for the following section.per7
  5. To find the depth of the second section, the artist again casts new diagonals, but this time he must start from the top and bottom of the newly drawn Terminus Line of the established section.  These lines are cast diagonally up and down to both the high and low Vertical Points,  This will create another X within the Lines of Perspective.  per9
  6. A new Terminus Line is drawn vertically where the diagonals cross the Perspective Lines, and a second section is established.per10
  7. Repeat from step 5, casting diagonals to the Vertical Points.  The sections will be drawn in their proper visual depth as per the laws of perspective. per11
  8. These sections could delineate where a repeating surface feature is found on a building, such as a column or any regular repeating element, including doors, windows and other architectural features.  This method is even useful for drawing a series of parked cars along the sidewalk, since many cars are relatively the same length, the sections can represent the cars’ body length.  Any anomalies of vehicular type can be adjusted for as needed. per12

This concludes our posting of Understanding Loomis.  Next week’s post will again unpack another of Andrew Loomis’ methods of finding depth by diagonals.  Stay tuned.

Finding Equal Spacing using Diagonals

In this posting of Understanding Loomis, I will explain the primary method Andrew Loomis outlines in what he calls ‘depth by diagonals’.  This is a rule of drafting which accurately represents depth in a subject matter featuring a repeated pattern, or a repeated spacing of surface feature.  This particular treatment is useful for illustrating a sidewalk receding into the distance, or a tiled floor going off down a hall, or a train track with ties crossing the rails, traveling away to a vanishing point.  It is a very useful mechanic to know.

To emphasize why knowing this mechanic of perspective is important, I will describe a scenario where such knowledge would save the artist painstaking hours of ineffective trial and error, or unconvincing guesswork.  Consider; you as an illustrator have been given the task of drawing a rural road, where small trees and shrubs are growing along the side. Also standing along the road at regular intervals of 30 feet, are a series of 6 telephone poles on the shoulder.  The image is to be drawn with the vanishing point of the road in the centre of the horizon, as if the viewer were standing at the head of the road, looking off to where it fades into the distance.

Drawing the simple one-point perspective of the road is no problem, and the shrubs/trees needn’t be precisely drawn on account of their being organic and irregular in nature, hence their placement is not a problem either.  The telephone poles on the other hand are man-made, and erected with a constant distance of 30 feet between each of the pole bases.  To be able to accurately place each pole as they recede into distance, is a baffling task to those unaware of technical perspective.  Many artists would rely of approximate placement of the poles, by ‘eye-balling’ the spacing.  This may be acceptable in some cases, but where approximation is not acceptable, a technical solution is necessary.

This then is how it is done.

Depth by Diagonal

(For the sake of these perspective lessons, crude sample drawings which were done in Photoshop in a matter of a few seconds, will be used forthwith.  Proper, elegant drawings will be used as examples where actual drawing matters are being considered)
  1. First represent the road and the bushes in one point perspective.  Bushes and trees are organic, and can be represented by eye without difficulty.  The rules of perspective still apply, but the artist can be a little sloppy in representing their relative size, since trees and shrubs are not regular.road1
  2. Next place your first pole.  Use the relative size of the trees around it for its height. road2
  3. Next find the Vanishing point of the road on the horizon line.road3
  4.  Now, draw a diagonal line from the near corner of one of the Lines of Perspective, through the base of the first pole to a secondary point on the horizon line. Let us call this new point on the horizon Point One. road5
  5. Next, draw a horizontal line (i.e. parallel to the horizon line – natch!) from the base of the pole, crossing through both Lines of Perspective.road6
  6. From where this horizontal line meets the opposite Line of Perspective, draw another diagonal line to the secondary point you drew in step 4, which we called Point One.road7
  7. Cast lines to the actual VP from the top and bottom of the telephone pole, and pay attention to where the bottom line crosses the diagonal which you drew in step six.road8
  8. At the crossing point, draw your second telephone pole.  The top line will show you how tall to make it.  road9
  9. Repeat the process, and erase your guidelines.  The end result will show the poles moving off in correct perspective, spaced evenly apart.  road10
  10. Work in reverse  to fill in any posts which should appear before the initial one you drew.road11

The Uses of  Depth by Diagonal

Shown below is a sample of the nuts and bolts of the Depth by Diagonal mechanic.  This example is shown stripped of any context in order to help the student imagine the uses of this rule of perspective.  Some ways I have used this method of representing perspective in drawing comics are as follows:

  • representing regular sized plates of steel, clad on the hull of a ship
  • drawing the separation lines of ceiling panels on a t-bar drop-ceiling in an office
  • sidewalks, with contraction joints showing in the concrete
  • patterned carpet expanses
  • windows, or other architectural features on buildings
  • any regularly spaced series of object, such as telephone poles, the cars of a train, the ties between train track rails, the broken lines on a road which indicate a passing lane etc.


Next week, we will examine a variation on this method of finding Depth by Diagonals.  I hope you will join me.

Drawing the Perspectival Ellipse, or the ‘Laying Circle’

To conclude the written section on perspective, Andrew Loomis gives us some rules to remember as well as a few reminders of some familiar points he has elaborated on in previous sections.  Consider these to be Loomis Axioms, as they are each gems in their own right.  In rapid succession, here are the highlights from the end of the written section on perspective:

  • To know is to save time, and the more you lean on your ‘crutches’ the more your strength will ebb
  • be more concerned with geometric shape and light/dark than with little muscular ‘lumps and bumps’ when figure drawing
  • photos with several light sources -which is the norm for most modern photography- defy every principle of good drawing
  • art chooses to elevate a subject above the multiplicity of nature
  • since we as humans cannot do otherwise than live with nature and her laws, art cannot either
  • deeper knowledge of reality will bring the artist added power
  • Caravaggio had it harder than we do, and he was better
  • use the wrist for your strokes, not your fingers
  • avoid scratchy, small, thin-lined strokes for greys and blacks

Perspective the Artist Should Know

The following section consists of several examples of concrete perspective rules.    These are concepts derived from drafting, and methods of solving perspective problems which are not well-known amongst artists unfamiliar with the mathematics of representing reality.

A few postings ago, I explained the methodology of drawing what I termed ‘the Laying Square’, that is a square drawn in a specific perspective so that it appears to lay on the ground.  I also mentioned how Andrew Loomis outlines the principle of using a rectangular prism as a temporary enclosing box, around whatever subject the artist is trying to represent.  As a rectangular prism is easy to imagine in perspective, this temporary shape drawn in place of the actual subject, helps the artist visualize the mass and perspective of what he is actually trying to draw*.  We shall proceed then, from this point.

*Please refer to the previous 2 postings for elaborations on this concept.

Drawing the Ellipse

Being able to draw an ellipse is very important.  The ellipse is a fancy word for an oval, and the oval is how we perceive circles in perspective.  There have been many instances where I’ve needed to draw a circle correctly in perspective in my professional art career; I can’t seem to get away from them! So many things from the man-made world are circular.  Here are a few I have already needed to draw in my career as a comic artist.

  • tires of cars
  • dinner plates and mouths of cups
  • the bases of columns
  • warrior shields
  • flying saucers(!)

There are certainly countless more examples.  Since there are so many circular things which an illustrator may be called upon to draw in perspective, it is essential to understand how it is done.

If any object can be conceived of as being bounded within a rectangular prism, then the drawing of the Laying Square is essential to begin with, in order to draw an ellipse.  Once the Laying Square is drawn correctly to the horizon, and in the correct perspective to your other scenic objects, the next step is to divide the Laying Square.

In the following examples, I will use a regular square, not drawn in perspective. This is done so that the steps I am outlining are in their easiest form to comprehend.  The rules are the same for a Laying Square, though it may be slightly harder for some to draw.

Step one: Draw your Laying Square in the perspective you want. (here represented in flat-on perspective)

drawn square Step two: find the centre of the square by drawing diagonal lines from each of the 4 corners to their opposite. This will create an X sign.

corner to corner

Step three: Draw lines from the centre of each of the sides, crossing through the middle point of the square, to a point on the middle of the opposing side. This will create a + sign.

side to side

Step four: Using the side to side lines as a guide, draw a new square, turned at 45 degrees to the original square, inside of it.

divide with lines

Here is what you should have with the square actually in perspective.loomis laying square

Step 5: Mark points A and B as shown in the example on  the side/corner of the 2 squares you have drawn.  You will need to do this for each of the sides/corners.loomis laying square points ab

Step 6: Mark the halfway point on the line between point A and point B.

loomis laying square points ab halfway

Step 6: Draw an arc just shy of the halfway point, spanning from the two corners of the inner, 45 degree square.

loomis laying circle

Step 7: Do this for each of the sides of the inner, 45 degree square. This will produce an ellipse, which will be the accurate representation of a circle were it laying in the perspective you have chosen.  We can call this the Laying Circle, or a perspectival ellipse.

loomis laying circle Red

This is how the artist can accurately draw columnar shapes in consistent perspective with other shapes in a picture.  All that is needed to be done is to draw verticals from the sides of the Laying Circle, and a secondary Laying Circle (derived from a Laying Square) as the cap.loomis laying circle column

Beyond this example, Andrew Loomis indicates to us a few facts about drawing the Laying Square and hence the Laying Circle.  He states that when drawing small objects (thus small Laying Squares), it is best to place the two Vanishing Points far apart from one another.  If the object you are drawing is big (thus the Laying Square will be big), place the two Vanishing Points closer together.  If one doesn’t adjust the V.Ps this way, distortion will occur in the drafting.  He concludes by saying that your eye will see the error right away, and most artists will make this “closer/further apart” adjustment naturally.

My upcoming several posts of Understanding Loomis will each concern a single one of Loomis’ perspective examples, as well as my explanation.

See you in seven.



Concerning Rendering of the Geometric Shape

My last post introduced the reader to Andrew Loomis’ concept of imagining any object primarily as a geometric shape, prior to attempting to draw it.  After one is able to conceptualize subject matter as a series of three-dimensional geometric shapes, Loomis reminds the reader that:

Every plane must have its relative values correctly rendered, or the entire image will fail to convince.

It is appropriate at this time to therfore indicate how one correctly renders the five essential, three-dimensional geometric shapes.  To do this we will perforce need to leave Andrew Loomis for a short while.

The essential shapes, if the reader recalls, are

  1. the Cylinder
  2. the Cone
  3. the Sphere
  4. the Torus
  5. the Rectangular Prism

All other geometric shapes are derivatives of these primaries.

Rendering the Shapes

Each of the above 5 shapes show their dimensionality and mass in predictable and easily memorized  patterns of rendering. To help the student, the patterns of value can be remembered as “value shapes”, in that the lights and darks will take on a recognizable geometric form when laying upon the larger three-dimensional shape being represented.  These rules should be memorized in order for the artist to properly represent the shapes. (each of the rendering examples are from this site by Bill Martin)

  • The Cylinder:  rendering the cylinder is done in bands of value, laying lengthwise along the body of the shape; i.e.  if the cylinder is sitting upright, as does a soup can in your cupboard, the bands of value will be vertical.  Should the cylinder be laying down, as does a felled tree or a log, the bands of value will run horizontally (or at whichever angle the log lays at). Below is a representation of a cylinder, as well as an example of how the bands of value will fall.  The “value shape” is essentially rectangular.
  •  The Cone:  when rendering the cone, the values take on a basically triangular shape, with the vertex of each  triangular “value shape” corresponding to the vertex of the cone.  The bases of the “value shapes” are slightly curved to follow the curvature of the cone’s base.


  • The Sphere: to render light and dark on the sphere, the artist must look to create two sections of “value shapes”.  Primarily, an oval shape, which will form on the section of the sphere where the highlight is, and the corresponding darker values in crescent shapes, nearer to the edges of the sphere
  • The Torus:  the torus is somewhat complicated in that it’s “value shape” bears the qualities of both the cylinder and the sphere.  To render the torus, some sections of the “value shape” need to be treated as one does the cylinder; i.e. it will be basically rectangular, whereas other sections of the shape is more like the sphere’s crescent.  The rectangular, band-like aspect of the torus’ “value shape” are found within the length , whereas the crescent aspects  are found at the extreme right and left of the shape, relative to the viewer.
  • The Rectangular Prism:  this shape, though likely being the most familiar to the artist, is actually the most complex to render.  This is because of the necessary lack of curvature on the Rectangular Prism’s 6 planes.  The flatness of the planes necessitates a gradation effect to be represented on their faces. The Rectangular Prism therefore has no “value shape” to look for when rendering its planes; each of the shape’s sides will feature a gradation. The key factor to remember when rendering the Rectangular Prism is that the gradation on the planes are in opposition to one another, i.e. that the darkest value of any plane will meet on an edge  with the lightest value of the adjacent plane. Correspondingly, the values of each other section of plane will meet on an edge with their opposite value on adjacent planes.


Next time, we will return to Andrew Loomis and his final considerations of geometric shapes, their use in figure drawing, and his closing thoughts to this section of the book, Successful Drawing.

Of interest to my readers, I would like to inform you that I will be now posting new entries for Understanding Loomis every Sunday, in order to complete my synthesis of Andrew Loomis’ writing in a more timely manner.

See you in 7 days.

Imagining the Geometric Shape

Previously, I went at some lengths to explain a concrete method of drawing a laying square at any perspective.  The reason this was done was in order to establish a groundwork for the student to begin to visualize the square as the base of a cube. A cube can me made from the laying square by drawing vertical lines up from the corners, and by adding an identical laying square as the lid.  Alter the height and width dimensions, and one can easily draw various Rectangular Prisms in perspective.

The Rectangular Prism in 3-Dimensional space is the primary shape Loomis encourages us to become familiar with from any angle.  The reason being, is that this shape will enclose any other shape which exists. Even the perfect sphere will fit in the perfect cube.  Some shapes are long, some tall, many are irregular; but when one visualizes a Rectangular Prism bounding what one is trying to draw around the object, the artist takes the first step to being able to draw that object with mass and perspective.

If one does not use a beginning framework of Rectangular Prisms to work out the perspective of the various objects within a composition,  the shortcomings will be evident to viewers on account of, again, ‘Intelligent Perception’.  Stated differently, there is no way to “fake” correct perspective, so it does one best to work it out properly at the beginning.

Setting up a a geometric shape as a shorthand for an organic shaped object has further uses beyond perspective and understanding mass.  An additional use is in aid of the artist seeking to establish values.  When an artist is trying to work out his values (lights and darks) within a picture, it is much easier -and more appealing visually- if the gradation of light to dark is done in a series of planes, rather than rendered as one continuous blend.  The ability to see form as a series of interconnected geometric shapes is essential to rendering transition of values in planes.  This treatment of light and dark creates an aesthetic which the photograph cannot duplicate, and thus, it remains the domain of visual artists.

The Male Figure rendered in planes

Drawing the ‘Laying Square’

Technical Perspective Continued

Last posting quickly outlined the principles of one point perspective.  The key points included:

  1. Every picture has a horizon
  2. An object’s relationship to the horizon creates the illusion of a perspectival view of the scene to the viewer
  3. The vanishing point is the point where a viewer’s sight is limited to

The previous posting used a simplified fir tree as the subject.  It is true that every picture has a horizon and involves the vanishing point, but it must be mentioned that natural phenomena, such as mountains, rivers, trees and bushes, are all very forgiving regarding perspective.  These forms are organic and do not involve straight lines, and furthermore, viewers will accept a very wide range of sizes and shapes in their representation.  This aspect of the inanimate, natural phenomena makes landscape painting and drawing much easier than works which involve architecture, humans or animals.  Traditionally, landscape painting has been considered the third lowest form of painting on account of the relative simplicity of the pursuit.

This posting will outline the fundamentals which lead to understanding how to draw geometric shapes in perspective.

As mentioned before, the initial step to drawing a geometric shape in perspective is mastering the ability to draw a perfect square, appearing as if it were laying flat on an imaginary surface.  This challenge is essential for revealing how an artist begins to create the illusion of space within a drawing.

Andrew Loomis does not break this task down for us, and many artists will be able to do this, merely with their eyes.  I personally find it very easy to draw a perfect square laying on an illusionary ground on any angle.  This is because I can visualize the correct placement and degree of the angles from the years of experience which I have integrated.

For the sake of the artists who do not find this easy, I have devised a simple method of finding the correct angles needed to draw a laying square from any position of perspective.

The Lesson

  • Draw a horizon, and keep it towards the top third of your paper.  Every drawing has a horizon.  perspective 1.jpg


  • This time, we will use 2 Vanishing Points.  Place them far apart.  Putting the V.P.’s too close together will create distortions in your perfect laying square.  The left hand V.P. will be “a”, the right hand V.P. will be “b”.perspective 2.jpg


  • Now, we draw a line at any angle from V.P. ‘a’, such that it crosses an imagined point between your two V.P.’s.  This line we will call the First Line.

perspective 3.jpg


  • Now we will repeat the previous step, but this time using a different angle of trajectory.  We will call this line, Second Line.perspective 4.jpg


  • Now, for the next step, we measure 3cm from V.P. ‘a’, down the horizon line.  This point we lightly mark for reference.perspective 5.jpg


  • Using the 3cm mark as the starting point, measure the distance to the First Line, in a perpendicular to the horizon.  This will create a right triangle between the horizon, the First Line and the perpendicular measured.  In my drawing, the distance happens to be 2.5 cm.  This is not a set measurement, and it will always be different depending on the angle which the First Line was drawn at.perspective 7.jpg

perspective 8.jpg


  • The next step is to duplicate those points with exact measurements starting with V.P. ‘b’.  This will create an equal, but reversed right triangle with V.P. ‘b’ as the first vertex. perspective 9.jpg


  • From V.P. ‘b’, draw the Third Line, using the newly drawn low vertex point as the angular guide. Draw this Third Line such that it crosses through the First and Second Lines.  The crossing points with the First and Second Lines we will call Crossing ‘x’ and Crossing ‘y’. perspective 10.jpg


  • Measure the distance between Crossing ‘x’ and Crossing ‘y’.  My drawing happens to be 4.6 cm.  This measurement is relative to the angles of the First and Third lines, and is therefore not a constant measurement.perspective 11.jpg


  • Now, we measure that same distance up the First Line, from Crossing ‘y’, towards V.P. ‘a’.  Mark the point.  Again, my measurement of 4.6 cm is particular to my drawing.perspective 12.jpg


  • Draw Fourth Line back from that point to V.P. ‘b’ to complete the perfect laying square.

perspective 13.jpg


A Quick Way

There is a quick way to skip essentially from step 2 to step 8, but it involves having a tool with a set 90 degree to trace from (such as a drafting triangle or a t-square).  I will outline it here for those with such tools.

  •  Draw horizon.  Choose vanishing points

quick 1.jpg

  • Measure length between the two V.P.’s, and half that number.  Measure that distance  over from one of the V.P.’s to find the mid point between them.

quick 2.jpg

  • Drop a 90 degree line down from the centre point, perpendicular to the horizon. Call this point the Centre Vertexquick 3.jpg
  • Draw First and Third Lines from that point to either V.P.’s.quick 4.jpg


Drawing the Second and Fourth lines will work the same way if you choose a point above or below the Centre Vertex to create different angles.


Now with a laying perfect square in perspective, it is a simple thing to draw the perfect cube from the groundwork we have lain.   We will complete this 3 Dimensional cube in the next posting.


Technical Perspective

Perspective an artist needs to know

Now begins the more technical aspect of Successful Drawing.  In the following postings, I will still begin with a short written section, but the meat of the post will consist of scanned images of my versions of Loomis’ drawings.  The drawings are direct copies of his works and lessons, and they were done to help me learn.  I encourage you to do the lessons as well, and draw the samples -even if you feel you understand the concept.  There is a truth revealed when you get actually down to drawing the exercises that is greater than an intellectual understanding only.

Loomis begins by encouraging us to be diligent and to practice.  He tells us frankly:

The difficulties of not knowing are always much greater than the effort of learning.

A note about supplies:

Loomis advises us to proceed in the following way:

Mark Making media:  He suggests that you use a pencil, but is quick to add that drawing is drawing in any medium, be it chalk, charcoal, crayons etc.  He says to select the one you like best, but to avoid hard media or inks, as well as to avoid overly dark pencils as they are hard to erase.

Erasers:  the kneaded eraser is best

Paper:  large pads of layout bond paper are mentioned, but he is not specific -it does not matter, as long as the paper is not too thin and transparent.

Loomis adds at this point that in terms of technical use of the pencil to avoid scratchy, small and thin-lined strokes.  He says such marks look amateurish and fussy.

In addition to the above mentioned, there is a need of a fairly large drawing table, a t-square and a right triangle in order to do the perspective exercises.

The Exercises

Since the  3 dimensional block is the primary shape everything fits into, Loomis begins with drawing the square.  Drawing a perfect square is the first step to learning how to draw form in perspective.

Exercise I: the perfect square

  1. A perfect square has each of its sides of equal length. Draw one.  To find the centre of a square or a rectangle, draw lines from opposing corners.  To divide the square into quarters, draw a horizontal and a perpendicular through the centre point which the diagonals have revealed.  From this many things will develop.bisecting squares.jpg
  2. To draw the 3-dimensional square in perspective (the cube) one must start by being able to draw the perfectly proportional square laying flat on an illusory plane created by the artist.  This is easily done, and uses 2 Point Perspective, but there are a few things you need to know before that.

Exercise II: creating the illusion of space

  • Looking at a blank paper is potential for space, but without anything for the viewer to comprehend, no illusion is created.  Furthermore, to attempt to create the illusion of space and to misapply these fundamental laws of perspective will alert a viewer’s Intelligent Perception of falsity.  Here is a blank paper and no illusion of space is apparent.                                                                                                                                                                       nothing.jpg                                                                                                                                                                                                                                   
  •  The addition of a simple horizon line is the first step to creating the sense of space.  Every drawing has a horizon whether the viewer can see it or not.  The horizon line could also be considered the point where the viewer’s eye level rests.

a horizon.jpg

  • If one adds a simple recognizable object such as a tree, the illusion of space is developed further.  By moving the trees closer to and further away from the horizon, the viewer’s eye level is changed by the artist. Changing the objects’ relation to the horizon gives the viewer the sense of flying above the trees or looking from below the trees.
  •  By making the trees smaller or larger, the artist creates the illusion of being nearer or further away from the subject matter.
  • The final step to creating realistic simple perspective is the addition of a Vanishing Point to the horizon line.  This point establishes where the artist wants the viewer  to be situated outside of the picture.  That is, when a Vanishing Point is properly used, the artist can show the viewer the desired perspective of his image

    The Vanishing Point (V.P.) represents the place where the viewer’s power of sight is limited to.  To understand the term ‘vanishing’, one must consider a scenario where a series of identical trees were lined up next to one another in a perfect row.  The closest tree would appear the biggest, and the furthest tree away would get so small, it would vanish.  The top and bottom lines drawn straight to the V.P. show how much smaller the artist needs to make the duplicate objects.  Without regarding the top and bottom height lines, the artist will fail to create the illusion of depth.  This is called One Point Perspective.

Now you are prepared for Two Point Perspective, which is the key to creating the 3-Dimensional cube, situated in space.