3D Family Nets

Through watching a maths class, I was made aware of the importance of understanding the relationship between 2D 'nets' and their relating 3D forms. Another term for this is 'development'. This blog may prove helpful by showing students (quickly) how 3D forms can be made from a number of 2D paper nets, from the cutting stage, through to folding and finally gluing of the tabs. A video of all eight forms being made can this way be accessed HERE. The 2D nets that I used are below. Try some for yourself.

Eurasian Coot Chick

This is a continuing story about the one surviving Eurasian Coot chick at our nearby wetland. It probably survived because the other three starved before it did, plus having the most persistent 'chirp' to keep its parent feeding it. Now few months old at this stage it continues to chirp persistently (and still get food from mum/dad), and only at the end of the video does it duck its head under the water to search for something to eat from the biomass it swims over. I have collected recent videos which can be seen HERE.

Wetland Wander

This is becoming a favourite haunt during this time of physical isolation. The water foul don't seem to be worried about getting sick or future economic problems. They all seem to be quite well with the foraging amongst the reeds and under the water surface. I made another video of current action on the water. What has to be acknowledged is that three of the four Eurasian Coot chicks are no longer about. That makes me sad. The one remaining chick seems to be getting helped well enough by its parents to be thriving. The video can be viewed HERE.

Three-Quarter View Perspective Principles

The following are steps to adequately understand many of the principles that are foundational to good three-quarter view ‘visualisation’ drawing.

To establish the importance and clarity of the angular view as compared to the plan or the elevation (side view), one only needs to note the limited amount of information emanating from either view of a given form. In this ‘flat’ presentation (orthographic projection views), the form requires a combination of at least two of these views to have any hope of being understood and only then by one who has learned to read orthographic views. 

On the other hand, when the form is rotated in two directions and presented as seen from the three-quarter (or corner) view position, even the untrained observer has a full understanding of the form.

Drawing or sketching a form in the three-quarter view requires extra knowledge and much additional practice. The more correct and consistent the rules are applied the more successful the drawing presentation and the clearer the visual communication. 

Three-quarter views of the same form

Perspective Direction

One of the foundational principles is based on understanding the perspective (structural) directions. There are always three axes or three perspective directions with all forms that are drawn in the three-quarter view. Each direction concludes at a ‘vanishing point’ and all lines in that direction radiate about this point. Therefore no lines in the drawings below are parallel on the visual plane (i.e. the drawing sheet). They offer the illusion of being parallel for the viewer.

Perspective Mood

Another foundational principle is the perspective ‘mood’. It can be manipulated to suit a particular purpose.

In the first example, the perspective vanishing points have been positioned closer together in relationship to the scale or size of the form being drawn. In these cases the actual vanishing points may be included to help manage the construction of the form on the visual plane. The mood created is ‘dramatic’ because the perspective is exaggerated.

A more subtle and more realistic mood is gained with the vanishing points much further away. It takes practice and experimentation to get the desired result when vanishing points are not available to help in the drawing construction, but the object looks ‘normal’ to the eye.

Different perspective moods must never be mixed in the same drawing. 

Perspective Foreshortening

Another thing to consider is foreshortening along each axis (perspective direction). The more exaggerated perspective mood allows construction support from the three ‘horizons’ connected to the three vanishing points.

Provided the ‘surfaces’ of the drawn form are rectilinear, a diagonal can be drawn to the related horizon. The diagonal’s vanishing point on the horizon will guide the direction of other diagonals and their intersections will set up foreshortening at regular intervals. Diagonals control the foreshortening and are very useful in all forms of drawing and sketching. 

In the case of the more subtle perspective mood, geometric patterns can be used across the drawn form to create foreshortening. Diagonals are used to create a pattern on the ‘surface’ of the form being drawn and these diagonals are extended, and extra ones created, to continue the pattern. In this way the line intersections guide the foreshortening to create the correct illusion of regular spaced lines in perspective. This works well with good line control practice.

Perspective Cube

It is useful to know how to draw a convincing cube. This is harder than it looks. Practising and experimenting is encouraged until it becomes second nature.

It may be helpful to build a model cube with wire or use thin straight sticks all the same length, square joined at their ends, to form one. Such a model can be studied and photographed using different camera focal lengths to see the same cube in different perspective moods.

Rotate about the cube (or rotate the cube). Take pictures close up and then far away (on telephoto setting). Observe the negative space pattern of the cube lines and the perspective mood difference. Don’t use direct sunlight so as to avoid shadow lines.

A large sheet of white paper under and behind the cube model will optimise the observations.

Aggressive perspective  

More subtle perspective

The Cube as a Base

 

In both examples the same cube was initially used. The top example shows the cube being dissected geometrically until the desired form is reached. The example below has had additional cubes built onto the first.

Both approaches require care to make sure that additional perspective lines are travelling correctly to fit in with the existing line structure. Try to keep freehand line work straight and true.

 

Geometric construction of a Rubix Cube in perspective (below)

Perspective Circle

The human eye can overlook drawn cube construction errors, but it is more sensitive in discerning the correct shape and placement of a circle in perspective. A drawn ellipse provides the illusion a circle in perspective and there are important principles to follow for this to be convincing. The more the ellipse is in error the worse the drawing looks. The untrained observer will feel something is wrong but not know how to correct it.

Before getting into those details, it is important to understand the anatomy of an ellipse. There are many ellipse construction methods and one that helps to relate to the movement of the elliptical path is ‘scribed’ by a pencil within a string loop controlled by two focus points. All ellipses have a Major axis (greatest width) and a Minor axis (smallest width). The two focus points are found by taking half the length of the Major axis and measuring this length from the tip of the Minor axis to where it cuts the Major axis in both places.

Try hamming two small nails into a board some distance apart, make a loop of string and experiment with this ellipse drawing method using a pencil as shown in the illustration.

Ellipse Angle in Degrees

By changing the ratio of the Major and Minor axes the ‘degree angle’ of elliptical width changes. This width is identified as the ellipse ‘angle’, meaning the angle a circle has been tilted from its original horizontal position until it becomes a circle at 90°. For example, an 8° ellipse is very thin and a 30° ellipse is wider showing much more of the circle which is now tilted at 30° to the line of sight, and so on.

 

Ellipse drawing templates are usually identified by their degree angle (or numerous angles, if there is a range of different width ellipses on the one template).

Ellipses presented in a carefully planned sequence (below) can by themselves create a depth illusion. The ellipse sequence here represents a series of horizontal flat circular discs ‘staged’ with the top circle at eye level, and the others positioned under it growing wider at the next level down to create a three dimensional illusion. 

Perspective Circle Centres

A circle in perspective is a true ellipse (a true ‘oval’). This is shown below in different perspective moods. The perspective centre of the perspective circle dictates the mood. 

The centre of a circle in perspective will always be set back from the ellipse’s geometric centre (where the Major and Minor axes cross). If the perspective centre is ‘strongly’ set back, the perspective will be strongly exaggerated. This ‘circle’ has tangents on the sides where the perspective diameter passes through the perspective centre and touches the sides of the ellipse. The angle at those points on the ellipse curve indicate the perspective direction to the horizon. Thus, the perspective circle can be easily encompassed by a square in perspective, touching the ellipse in four places (tangent points) creating an illusion of a circle inside of a square on the picture plane (the drawing surface).

If the perspective centre of a circle is close to the geometric centre of the ellipse, the perspective will not be exaggerated. The perspective vanishing point may be too far off the page to be used but ‘intuitive’ control can maintain reasonable perspective direction to that far away point.

The tangents from perspective circle’s diameter line create a less aggressive perspective square that the circle fits in. Note that whatever the mood that is created, a circle in perspective is always a true geometric ellipse (true oval). 

Ellipse Direction

Cylinders are circle-ended prisms so naturally the same perspective circle conditions apply to drawn cylinders. For a right-angled cylinder to look correct on the visual plane (drawing surface) the Minor axes of the cylinder has to be in line with the centre line axis of the cylinder.

The cylinder’s centre line axis is a natural perspective direction and the Minor axis of each ellipse is in the same perspective direction in line with the cylinder’s centre line.

Whenever the Minor axis and the cylinder axes do not share the same perspective direction, the illusion is one of a cylinder that has been cut on an incline rather than at right-angles to the cylinder’s axis. If the cylinder to be drawn is meant to be a right-angled cylinder then the drawing will mislead the viewer.

 

Ellipse Degree Angle Selection

It is important to note the difference between ellipse direction and ellipse degree (both covered above). Once the ellipse direction (Minor axis) is understood and applied, consideration must then be given to the selection of ellipse angle, that is how wide to make the ellipse.

The illustration below shows how ‘awkward’ a drawing can become with the wrong ellipse degree choice even with the correct ellipse direction. In the drawing, the cube is dictating the perspective structure (mood) and the attached cylinder does not look convincing, does not look like it is attached. 

Choosing the correct ellipse angle (how wide it is) is helped by the knowledge that perspective direction lines (even a ‘vertical’ line is a perspective direction) from the ellipse centre will cross the ellipse at tangent points, and the perspective directions at these points must conform to the perspective of the rest of the drawing. If the ellipse choice is too wide or too narrow the tangent lines will not be compatible.

This lower example of the same cube and cylinder is convincing because a much wider ellipse (greater degree angle) was chosen by increasing the ellipse’s width until the perspective tangents were seen to be aligned in the same perspective mood as the rest of the structure.

Selection Examples

These principles for circles in perspective are essential knowledge for good convincing drawing and sketching in the three-quarter view.

In the examples below, the ellipses in the first two of each row are in error while the third is the best with the most convincing ellipse direction and degree angle.

In the first of each set the ellipse direction is in error and in the second the direction is correct but the ellipse degree is not correctly chosen.

The error is usually obvious and is avoided through the principles of correct ellipse use.

Radius Curved Edges

Many modern product applications have radiussed edges so it is important to know how to apply good ellipse positioning principles when drawing these forms.

When three radiussed edges meet at a corner they create a spherical surface. This is shown in the picture plane as a true circle. Note that the Minor axis of each ellipse in the example follows a perspective direction and there are three perspective directions involved (often identified as X, Y and Z axes).

The Double-Curved Surface

To properly draw an S-bend pipe, preparation for the centre line path must follow the same ellipse positioning principles described above. The Minor axes of both ellipses control the ellipse direction and the ellipse degree angle (how wide it is) is governed by the tangent points where a perspective line going the other way crosses the ellipse curves.

Once the centre line of the pipe’s path is established, a series of spheres can be drawn around points along this line. These are repeating circles in the drawing. These repeating spheres represent a ‘stop motion’ sequence of the movement of a sphere along the centre line. If it is too difficult to do this freehand a compass can be used with the compass pin placed at intervals along the centre line for each scribed circle.

A line can be drawn along the edge of these spheres to represent the visual side of the pipe. The edge must follow the guiding spheres as far as practicable and where an edge becomes a surface it must be ended neatly and intuitively. 

 

The use of multiple spheres works in the same way with a flexible pipe (follow the centre line) or a closed circle centre line such as a ‘donut’ tube (centre points on the ellipse).

The Sphere

When the sphere itself is the object to be drawn, it is constructed as a true circle on the visual plane (the drawing surface). When other forms are added to the sphere, care and understanding is needed to create a convincing illusion.

A hole, cylinder or cone attached to a sphere will involve the use of ellipses (illusions of circles) with their Minor axes aiming toward the centre of the sphere (assuming the ‘drilling’ and ‘attaching’ are right-angled to the spherical surface).

Once the ellipse direction (Minor axis) is in place, the ellipse degree angle (ellipse width) is dealt with. The closer to the edge of the sphere visually the thinner the ellipse degree will be.

 If the ‘drilling’ and ‘attaching’ is not at right-angles to the spherical surface, creating complex joins and shapes, difficult plotting and sectioning will be required this will not be explained in this very basic text.

This ‘chain’ of joined ellipses from top to bottom (illusion below of a ring of circles in perspective) shows that each ellipse has its own perspective direction to the vanishing point. Note also that these ellipses are the same distance from the perspective vanishing point, so each ellipse will have the same degree angle (same ellipse width) irrespective of its size.

Practical Applications

Practice is the single most important ingredient to all good perspective drawing and sketching. The principles above help to create a professional ‘edge’ in the process.

The ability to work with complex shapes requires a competent grasp of three-quarter view drawing principles. It allows and promotes good visualisation in the designing process because many sides of the drawn form are seen at the same time.

Design Sketching
A simple black ball-point pen is a useful medium for this type of drawing. There is a variety of ‘greys’ that can be generated with varying pressures, and finishing with a strong black line. The lines cannot be erased and that can be an advantage.

Using a Hard Edge

These principles offer a foundational understanding for all freehand and constructed perspective drawing. If a hard and sharp edge is required, the tools needed are a number of pen thicknesses, a straight edge (clear ruler is best) and a collection of ellipse templates to cater for all positions that a perspective circle might be in. The better template sets provide ellipse angles from 10° to 80° (in 5° steps) and in sizes from 5mm to 100mm measured along the Major axis, and these are expensive.

A compromise is inevitable if the set is limited to a few sizes and degree angles so it may be better for the sake of practice and creative flow to rely mostly on freehand, but note that a good quality rendering of a perspective drawing does need the use of the hard edge and it is not very professional to mix hard edge lines with freehand lines. It’s one or the other.

Summary
Whatever the approach, hard edge or soft, the same principles apply. There is the perspective mood to consider and to keep this mood consistent over the whole drawing. A perspective circle is rendered as a true geometric ellipse (true oval). Ellipses create illusions of cylinders and cones, and their Minor axes travel in a structural or perspective direction.

 The width of each ellipse (the angle in degrees) will vary depending on the position of the perspective circle. A perspective sphere is rendered as true circle in a drawing and perspective circles on the spherical surface are drawn as ellipses with their Minor axes pointing to the sphere’s centre. A succession of spheres along a centre line help shape the surface of such things as hoses, pipes and donut tubes.

The application of these principles lead to convincing three-quarter view drawing and opens the door to more adventurous 3D visualisation. Good luck.