Drawing

Isometric and Axonometric Projections are fairly simple ways to approach 3D drawing. These can be used for sketches or to draw to a predetermined scale. If a scale is used, every part of the drawing can be measured with accuracy.

Unlike perspective drawing, lines in Isometric or Axonometric drawings do not converge. In fact they only go in 3 different directions. Vertical and 30 degrees left and right in Isometric Projections, vertical and 45 degrees left and right in Axonometric projections.

Often, Axonometric projections may appear to be distorted but they are very useful to show as much as possible of, for example, the inside of a room.

Although it is possible to quickly produce projection drawings using a drawing table, parallel motion and either a 30 degrees or 45 degrees angled set square, it can be even easier if a grid is placed underneath the paper. Both Isometric and Axonometric grid paper are available from art and design shops or you can make your own using a drawing table or a CAD program.

Isometric Projection: vertical, 30 degrees left, 30 degrees right

Axonometric Projection: vertical, 45 degrees left, 45 degrees right

Isometric Projection Isometric projection is one of the three forms of axonometric projection. In isometric projection the angles between the projection of the axes are equal i.e. 120ยบ. It is important to appreciate that it is the angles between the projection of the axes that are being discussed and not the true angles between the axes themselves which is always 90ยบ.

To explain the "Projection of the axes" lets take a view of a cube so that its three principal faces are visible. Lets place a transparent sheet of perspex in front of the cube and draw lines where the front edges of the cube meet at a point. The angle between adjacent edges of a cube is always 90º. After drawing the outline of the converging edges on the perspex we can measure the angles between them.

We can see that the angle between adjacent edges is greater than 90º in all three cases i.e µ>90º,ß>90º and Ø>90º.

These are the angles between the projection of the axes. Theses axes are known as the axonometric axes. If the angle between all three axes are the same then an isometric view results ( µ=ß=Ø); if two of the angles are the same then a dimetric view results (e.g.µ<>ß, ß=Ø); finally if all three angles are different a trimetric view results (i.e. µ<>ß<>Ø).

In third angle the planes of projection are in front of the object so the projection of the axonometric axes will be along the front corner of the object. The cube in the animation is in third angle as the axonometric axes intersect at its front corner.

These axes would be used to solve questions in third angle.

However, in first angle projection the planes of projection are behind the object and so the axonometric axes will intersect at the furthest back corner. First angle projection is generally preferred to third angle projection in second level schools. Where the three edges of the cube meet at the furthest corner from the observer are the axonometric axes used in first angle. The axonometric axes (isometric axes in this case) for first angle projection are shown here using a hollow cube. Compare the axonometric axes of this cube with those of the cube above.

In fact the isometric axes can be placed in any desired position so that the object will be in the position that best describes it. However, the angle between the projection of axes must always be 120º. If the object is considerably long then it is customary to place the long axis horizontally for best effect. Here are some typical positions of the isometric axes.

Dimetric projections for computer graphics and games As was the case with the isometric projection, in computer graphics some angles are preferable to others. The first dimetric projection that I propose for (tiled) computer graphics is very similar to the projection of Chinese scroll paintings. The difference is the scale of the z-axis, and the angle that it makes with the x-axis. To start with the angle, the z-axis is slanted with approximately 27 degrees (the exact angle is "arctangent(0.5)"). This is the same angle as the isometric projection for computer graphics uses. The scale is such that the width of the side view of a cube, when measured along the x-axis, is half of the width of the front face. The key phrase in the previous sentence is "when measured along the x-axis". In the two former projections, the scale factor applied to distances measured along the zaxis.

Dimetric 1:2 projection "side-view" The above projection gives a perspective that is viewed mostly from the side. I might be useful to add some depth to a side-scrolling (or "platform") game. For

board-like games, a perspective that is viewed from a greater height is more appropriate. The second proposed dimetric projection for games serves this end.

Dimetric 1:2 projection "top-view"