What Is Orthographic Projection? How to Draw an Object in Three Views

According to Wikipedia, orthographic projection uses several two-dimensional images — the views — to represent a three-dimensional object. There’s no single drawing with depth rendered in perspective. Instead there are several flat drawings, one for each viewing direction, and together they’re enough to reconstruct the object’s shape.

The standard projection planes number three: horizontal, vertical, and profile, each set at a right angle to the other two. According to tecnologiaduepuntozero.it, these three planes together form what’s called Monge’s trihedron. Each plane receives a different view: the top view looks straight down and lands on the horizontal plane; the front view looks straight ahead and lands on the vertical plane; the side view comes from looking at the object from the left or right and lands on the profile plane. Gaspard Monge’s original version of the method used only two planes — horizontal and vertical, giving the top and front views — and the profile plane for the side view came later.

Where the three views sit on the page isn’t a free choice. Two different conventions produce the same information but place it in opposite spots. Most of Europe, including Italy, uses first-angle projection. According to Studenti.it, the main view goes in the centre of the sheet and every other view sits on the side opposite the face it shows: the view taken from the object’s right side is drawn to the left of the main view, the one from the left side to the right, the view from below above, and the view from above below. One way to picture it: imagine the object inside a box with glass walls standing in for the projection planes. What an observer sees from the right gets drawn on the opposite wall — on the left.

The United States uses third-angle projection instead. Here the projection plane sits between the viewer and the object, not behind the object as in first-angle projection, and it’s imagined as transparent. Looking at the object from the right now puts that view to the right of the main view — the opposite of what first-angle projection does.

For the same view, the two conventions place it on opposite sides of the sheet.

Practical example: take a simple rectangular box, with a wider face at the front and a narrower one at the side. In first-angle projection, the front face goes in the centre of the sheet as the front view. The top face, seen from above, is drawn below the front view — that’s the top view. The right face, seen from the right, is drawn to the left of the front view — that’s the side view. Edges covered by the body of the solid, such as the vertical edge at the back that disappears when you look at the object head-on, are drawn dashed in every view where they stay hidden; edges you can see directly are drawn solid. For a curved solid such as a cylinder, the edges are replaced by the lines that generate the surface — the rims of the circular bases and the lines running along the side.

Anyone who has worked through solid geometry will recognise the same box that shows up in volume formulas; orthographic projection just looks at that shape from another angle, the drafting table’s rather than the calculator’s. The same goes for rectangular prism volume: its flat, right-angled faces make the three views unusually easy to read, since each face projects onto one of the three planes with no distortion.

The two-plane method is also known as Monge’s projection. According to Treccani, French mathematician Gaspard Monge (1746-1818) built it on the projection of an object onto two planes at right angles to each other — a horizontal plane and a vertical one. Fold one plane onto the other and you can reconstruct the shape and measurements of the figure in space, and the process works in reverse too. Monge published the method in “Géométrie descriptive” (1794-95), a record of lectures he gave at the École normale, and it still underpins geometric drawing today.

Curved surfaces, like those of solids of revolution, follow the same general rules when projected: there are no edges to trace, but the drawing still marks a line of contour, and still distinguishes the parts a viewer can see from the parts hidden behind the solid. Anyone who wants to check a projection with calculation, say by working out the angle at which a slanted face meets a plane, finds the tools for that in trigonometry.