Projection Principles

Purpose

This page presents the basic principles of representing a three-dimensional (3-D) object as a two-dimensional (2-D) image through projection. These principles can be used in choosing how you represent real or virtual 3-D objects on your computer screen or on paper. The goal will always be to present the object in the most understandable, least confusing way possible.

Overview

Students will review the basic projection concepts presented on this page and explore them through a series of exercises.

Topics

Scientific Visualization: Projection Principles
Mathematics: Applied Geometry

NC Scientific and Technical Visualization Objectives

Level I:  Objective:  4.02 (F and G)  Explain the fundamental concepts of shape description.

Tools

Students will need pencils, plain tracing paper, square grid paper, and isometric grid paper. Optionally, a 12" x 12" square clear plastic (e.g., Plexiglas) and washable markers can be used to explore projection concepts. Also, 3-D modeling software such as 3-D Studio Max or TrueSpace can be used to explore different types of projections.

Teacher Background

Projections of 3-D objects is something that we experience everyday. Anytime we see a photo or drawing of a 3-D object on the computer, TV, or paper, we are seeing the object in projection. Only when we see the object 'in the flesh' are we not seeing a projected representation of it. (You could argue that even in this case, we are seeing a projection of it on our retinas in our eyes, but we won't get into that....). However, in Scientific Visualization, when we are trying to apply graphics in a conscious and precise manner, we need to develop an understanding of the different kinds of projections and the differences between them.

Key Terms

Introduction

When you represent a 3-D object on the computer or on paper, you should be aware of what type of projection method you are using in representing the object. Your goal should always be to represent the object with the You will find that these requirements are in most cases mutually exclusive. Instead you will have to decide which are the most important and which projection method will best achieve those goals.

Projection

Projection, as we refer to it here, is when you reduce the three dimensions of a real or virtual (computer) object to two dimensions. This is necessary to represent the object on a computer screen or a flat piece of paper. Different projection methods will collapse these three dimensions to two in different ways, creating differing representations of the object. One way of thinking of this process of projecting is to use projection lines to 'map' the three dimensions onto a sheet of paper (see Figure 1). You would 'see' this projection if you oriented your line of sight parallel to the projection lines. Projection lines are a way of mimicking how you might see the object with your eyes.

The orientation of these projection lines to:

will determine the type of projection it is.

Figure 1. Lines of projection from an object to the projection surface.

Parallel Projection

With parallel projection, all of the projection lines are parallel to each other. The parallel projection lines means that edges that are parallel on the real object are also parallel in the projection. This allows for the least amount of distortion of features within the object. It also allows for, under some circumstances, measurements of the object to be taken off the projection. With most types of parallel projection, the projection lines are perpendicular to the projection surface. Different types of parallel projections are created by orienting the object differently relative to the projection surface and, thus, the lines of projection.  Parallel projections include multiview and pictorial projections.

Multiview Projection

With a multiview projection, the projection lines are oriented parallel to one of the principle axes of the object. This is the case in Figure 1. Notice that the projection lines are following a large number of edges on the object. These edges define one of the principle axes which, in turn, defines one of the primary dimensions of the object. By projecting along one of the primary dimensions, this dimension is collapsed completely into the projection. Another way of stating this is that when you orient a principle axis perpendicular to a projection surface, it is not seen at all. Notice in the projection, all of the features represented along that primary dimension are completely missing on the projection. How do you determine what are the principle axes? Objects don't come predefined this way, so you end up having to decide what orientation of three mutually perpendicular (orthogonal) axes follows a majority of the key edges of the object. These axes can also be thought of as the Cartesian coordinate axes, X, Y, and Z. One way of thinking of this is imagine putting the object in the smallest box possible. How is the object oriented in the box? The corners of the box now represent the primary axes relative to the object.

Multiview projection gets its name because only two dimensions of the object are shown in each projection. These two principle dimensions displayed are shown in true size, there is no distortion. This is the case because these principle axes are both parallel to the surface it is being projected onto. If you are going to describe all three dimensions of the object, you must have two or more (multi) views. Multiple multiview projections brought together into a single drawing is the standard format for technical drawings used in engineering and architecture. Figure 2 shows the three most common multiview projections:

Figure 2. The most common parallel projections.

The 3-D arrow shows the direction you would view the object to see this particular multiview projection. Note that each of these views is parallel to one of the principle axes. In addition to the three multiview projections, there is also  another parallel projection (the isometric pictorial)  that will be discussed later. The next set of figures show what these different multiviews would look like if viewed along the 3-D arrows:

Figure 3. Front View
 
 
 
 

Figure 4. Side view
 
 
 
 

Figure 5. Top view.

Pictorial Projection

Pictorial projection, unlike multiview projection, is designed to allow the viewer to see all three primary dimensions of the object in the projection. The degree to which a dimension gets 'collapsed' in the projection depends on the orientation of the line of sight relative to the object. Whereas a multiview is designed to focus in on only two of the three dimensions of the object, a pictorial provides a holistic view of the object. The tradeoff is that a multiview allows, in general, a more undistorted view of the features in the two dimensions displayed while lacking a holistic view of the object (thus needing multiple views to fully describe the object).

Axonometric Pictorial Projections

When parallel projection is used to create a view showing all three dimensions of an object, this is called an axonometric pictorial projection. Axonometric projections are classified according to the orientation of the principle axes relative to the projected surface. This orientation of axes determines how much each principle dimension is distorted. When a principle axis is oriented at something other than parallel to the projected surface, then the lengths of features in that dimension are shown shorter than their true length. This is called foreshortening. The closer the axis comes to being perpendicular to the projection surface, the more foreshortened it becomes, until it finally collapses to zero length.

The most common type of axonometric projection is called an isometric pictorial projection. With this pictorial, all three principle axes are oriented at the same angle to the projection plane, creating an equal amount of foreshortening in all three dimensions. Figure 6 shows an example of an isometric projection. Notice that the three principle axes overdrawn on the object make an angle of 120 degrees to each other on the projected surface. In the real object, these axes would actually be 90 degrees to each other.

Figure 6. Isometric pictorial projection with the three principle axes highlighted.

Figure 7 shows another axonometric projection. In this case none of the three axes are oriented the same relative to the projection surface. They also make different angles relative to each other on the projection surface. This type of projection is called a trimetric pictorial projection. Trimetric projections are considered more realistic than isometric projections. Since there are no restrictions on the orientation of the principle axes relative to the projection surface, you can orient the object to emphasize one or two dimensions over others. Just be careful not to foreshorten (thus distort) one dimension too much. If a dimension becomes too foreshortened, it is probably better to simply show it as a multiview rather than a pictorial.

Figure 7. Trimetric pictorial projection

It is important to remember that all axonometric pictorials are parallel projections. This means that edges that are parallel on the object are also parallel in the pictorial (Figure 8).
 
 

Figure 8. Parallel edges on the object are also parallel in an axonometric projection

Perspective Projection

If parallel projection provides the least of amount of distortion, why would you want to do anything else? In fact, parallel projection does not do a very good job of mimicking how we see the real world around us. When looking around us, objects that are farther away look as though they are smaller. Similarly, single objects that span a great distance, such as roads or railroad tracks, look as though parallel edges are getting closer together as they recede into the distance. Finally, when an object gets to a theoretical 'far point', they disappear all together. This happens at what we call the horizon line. We mimic this effect by allowing edges that are parallel on the object to converge as they move towards the theoretical horizon line on the projection surface. This technique uses perspective projection, which has lines of site which are not parallel to each other nor perpendicular to the projection surface. The rate at which parallel edges converge is called the perspective angle (Figure 9). This angle is determined by the distance an imaginary viewer is from the object being represented.

Parallel project mimics the case were the 'viewer' is infinitely far away from the object. In this case, the perspective angle is zero and the lines of projection are parallel. As the viewer gets nearer to object, the angle increases and the rate of convergence of edges grows (Figure 9).

Figure 9. Medium angle perspective projection. Notice that compared to figure 8, the parallel edges are beginning to converge to the right, left, and down. (This is very difficult to see in the projection of a small object but is much more obvious in the drawing of a large object such as an apartment building.)

As the angle increases, the degree of convergence and distortion increases (Figure 10). The perspective angle you choose will be determined by the size of the object and the distance of your imaginary viewer you want to simulate.

Figure 10. Wide angle perspective projection. Notice that compared to figures 8 and 9, the parallel edges are strongly converging to the right, left, and down.

Summary

So what type of projection should you use? Here are some general rules of thumb:

If the object is essentially two-dimensional in nature (e.g., a leaf, a snowflake) or you want to show certain features in their true size and shape, use a multiview.

Note that there are always exceptions to these rules. Ultimately, the choice of projection depends on the capabilities of your software/hand tools and what best conveys your message you are trying to communicate.

Student Assignment Sheet

1. Collect images of objects from magazines, newspapers, etc. or take your own images with a camera. Collect a range of sizes of objects, ranging from buildings and bridges to cups and pencils. Tape tracing paper over the images. 2. If you have a piece of Plexiglas, set a small object on a counter and place the Plexiglas between you and the object. Move around the object until you think you've identified the principle axes of the object. Orient the Plexiglas perpendicular to these axes and sketch the object on the Plexiglas with washable marker. These are multiviews of the object. Now orient the Plexiglas to create pictorial projections that capture key features. Sketch these on the Plexiglas

3. Do what you did in Exercise 2, but don't use the Plexiglas. Simply sketch the objects on plane paper as you see them.

4. Repeat Exercise 2, but now use square grid paper for creating the multiviews. Pay attention to preserving the proportion of features and the parallelism of edges. Now create an isometric sketch using isometric grid paper. Tracing paper can be taped over an isometric grid to preserve grid paper.

5. Go outside and sketch your school building from 30' away from the building and from as far away as you can easily get (no more than 500'). Orient yourself so that you are looking at the same corner of the building at both locations. Compare the degree of convergence in the two images. Try and identify the convergence points at the horizon in both sketches.

6. Use a 3-D modeling software package to recreate the object used in the figures in this lesson. Create the following projections:

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