Understanding Graphical Projections: Perspective, Oblique, and Orthographic

Here is a solid block. It has width, it has depth, and it has height: three directions at once, and six flat faces. And here is the difficulty. A page is flat. It has two directions to spend, and the block has three, so one of those three cannot be drawn as it truly is. Every drawing system in this lecture is a different answer to that. So look down on the whole business from above. The page stands edge on, as this pale vertical line. The object sits over here, and the drawing will appear where the rays cross it. Suppose every ray runs to one single point, an eye out at the front. From each corner of the object, a straight ray to that point, and a mark where it crosses. Between the two marks is the drawing, and its height is h prime. Now compare the two heights. The object stands this tall, and its drawing on the page is well under half of that, because rays that converge shrink whatever is far away. That shrinking is perspective, and it is exactly what your own eye does. Now change one thing only. Take the eye away, and let every ray run parallel to its neighbours instead of gathering to a point. Nothing converges now, so nothing shrinks. The drawing grows until it is exactly as tall as the object itself, and a ruler laid on the page measures the real thing. So there are two great families of projection, and every drawing ever made belongs to one of them. Rays that meet at a point, or rays that never meet at all. The converging family has just one member, and it is called perspective. The parallel family splits in two, by the angle at which the rays strike the page. At a slant, the drawing is called oblique. Square on, it is orthographic, and that includes the axonometric views, of which isometric is the one you have certainly seen. Perspective, oblique, orthographic. One of the three shows the world as an eye receives it. The other two show something no eye can ever see, and that is exactly why they are what you build from.

Start with the converging family. One eye, one page, one post. The eye is here, and the page stands at a fixed distance in front of it, which I will call f. The post stands out at a distance Z, and it is h tall. Now the two rays. One from its top to the eye, one from its foot to the eye. Each crosses the page on the way, and between the two crossings is the drawn image, of height h prime. Now the geometry does the work. The big triangle from the eye out to the post, and the small one from the eye to the page, share an apex and a direction, so they are similar. Similar triangles give a proportion: h prime over h equals f over Z. Rearrange that, and the drawn height is f times h over Z. There is the whole of perspective in a single line: what you draw is inversely proportional to how far away the thing is. Put numbers on that. At one page distance out, the drawing is life size. At twice the distance, half size. At four times out, a quarter. And here it is happening. The post walks out, and its mark on the page closes down. Further still, and smaller again. Bring it back in, and the mark swells. Now the famous consequence. Take two rails that really are parallel, and really are the same distance apart at every step of the way. Their images do not stay apart. Every ray to the eye pulls them closer, and on the page they meet, at a single point on the horizon. That is the vanishing point, and nothing whatever in the world stands there. Now lay sleepers across the track, equally spaced, one every metre. Out in the world every gap is the same. On the page they crowd together as they go back, and the last of them are almost on top of each other. So a perspective drawing is faithful to the eye and useless to a ruler. Two marks of the same length on the page can be a metre apart in the world, or a hundred, and the drawing will not tell you which. Which is why perspective belongs to painting, to photography, to film, and to the camera in every video game: all the places where the job is to look right. It is not where you go when the job is to be built.

Cross into the parallel family. Here is the page again, edge on, and the object behind it, and this time the rays arrive at a slant. Set the front face of the object flat against the page. Those points are already on it, so they draw themselves: same size, same shape, same angles, exactly. Now follow the rear face. Its rays are slanted, so they land further along the page, and every one of its points is shifted by the very same amount, because the rays never converge. That shift is the depth of the drawing, and the slant angle alone decides how big it is. Near and far are drawn at one single scale. Here is what that gives you on a cube. The front face is a true square: right angles, equal sides. Measure it on the page and you have measured the object. The depth runs off at forty five degrees, the same length from every corner, and the far face is a second true square, parallel to the first. Drawn at full depth, this is cavalier oblique, and every single edge on the page is a true length. But look at it: it seems to swell as it goes back, because your eye expects far things to be smaller, and this drawing refuses. So draughtsmen halve the depth. That is cabinet oblique: the front face still exactly true, the receding edges at half scale, and the solid suddenly looks like a solid. The price is easy to state. Only the face lying in the page can be measured directly, and only lines parallel to that face keep their length. Everything in the depth direction is a convention rather than a measurement. It is cheap to draw by hand and it reads at a glance, which is why oblique filled workshop notebooks and furniture drawings for two hundred years. The cabinet convention is named after cabinets.

One change is left to make. Here are the slanted rays again, with the front face true on the page, and the rear face shifted along it. Now swing the rays square to the page, and watch the shift close up. There it goes: the rear face lands exactly on top of the front face, and the depth of the object has gone out of the drawing altogether. This is orthographic projection, and losing the depth is not a defect. What is left on the page is one face, dead true: true lengths, true angles, and every one of them measurable. But one view like this cannot describe a solid. A thin plate and a long block give exactly the same front view, so the answer is to draw more than one of them. So here is the standard set of three. The front view, straight on. Above it the top view, looking down from over the block. Beside it the side view, looking in from the right. And the three are locked together. Width is shared by the front and the top, so those two line up along here. Height is shared by the front and the side, so they line up across here. Between them, three flat drawings hold every dimension of the block, each one at true size. Nothing here looks like the object, and that is the whole trade. The drawing has stopped being a picture and become a measurement. This is the language that workshop drawings and building drawings are written in. There is a middle way, and you have seen it a thousand times. Keep the rays parallel and square to the page, exactly as they are now, but turn the block itself, so that three faces show at once. Now it reads as a solid again. But look along its edges: the ones that are parallel on the block are still parallel on the page. Nothing converges, so this is still a parallel projection, and it can still be measured. Choose the tilt so that the three axes are treated alike, and you have the isometric view. On the page those axes stand a hundred and twenty degrees apart, and all three are shortened by the very same factor, about zero point eight one six. That is the pair of properties that made it famous. It looks like a solid, and one ruler measures all three directions at once. Which is why isometric is the language of assembly instructions, of pipe runs, of patent drawings, and of a whole generation of video games.

So which of these is real? There is one honest answer, and then a twist. Perspective is what an eye receives, because an eye has a single pupil, and every ray that reaches it converges there. A photograph is a perspective projection whether the photographer wanted one or not. Now the twist. Carry the eye backwards, and leave the object and the page exactly where they are. The rays flatten out. The image on the page grows, and the near and far parts of the object are treated more and more alike. Take that to its end. Put the eye infinitely far away, and the rays are exactly parallel. The shrinking stops altogether, and the mark on the page is precisely the size of the object. So a parallel projection is not a mistake about vision. It is the view from an infinite distance, which no eye can ever occupy. That is why an oblique or an isometric drawing always looks very slightly impossible. It is impossible on purpose. Then the useful question is not which one is real. It is what each one throws away, and who can afford to lose it. Perspective keeps the look of the world and throws measurement away. For a painting, a photograph, a film, or the camera in a game engine, that is exactly the right bargain. Oblique keeps one true face and throws the rest away. It is the quickest useful pictorial drawing there is, which is why it filled notebooks and furniture drawings for two hundred years. Orthographic throws the picture away entirely and keeps every size. Nothing is ever built from a perspective drawing. Buildings and machine parts are built from orthographic views. And isometric sits between the two. One scale for all three axes, a solid that reads as a solid, and a drawing you can still measure: assembly guides, pipe runs, and a great deal of game art. So there is the whole map, and one question decides it. What is the drawing for? If it must look right, let the rays converge. If it must be measured, keep them parallel, and then choose what stays true: one face for oblique, one view at a time for orthographic, all three axes alike for isometric.