Front, top and side views, and what each one loses

0:00A solid, and a flat sheet of paper. The job is to get the solid onto the paper so completely that somebody who has never seen it could build it. The trouble is obvious the moment you try. The solid has three directions in it and the paper has two, so one of them has to go. So the honest question is not how to avoid losing a direction. It is: which one, and what does that cost?

0:29Answer that precisely and you can lose a direction on purpose, three times over, and lose nothing at all. Start with one point and one plane. Drop a line from the point straight down onto the plane and mark where it lands. That is the projection of the point. Straight down has to be said carefully. It means the drop makes a right angle with every line you could draw through the landing place inside the plane.

1:01Every line, not one line - that is the whole definition. Here is a point in the plane that is not the landing place. The drop to it makes a perfectly good right angle with one direction in the plane, and none at all with the other. Check one line and you would have called this the projection. Check every line and only one point survives. Now a whole segment. Drop both ends and join the two landing places: that is its projection.

1:35The question is how long it is compared with the segment itself. From the upper end, run one extra line across to meet the other drop. Four corners, and they make a rectangle. They have to. The two drops are both perpendicular to the plane, so they are parallel to each other; and the piece joining the landing places lies in the plane. Two pairs of parallel sides, and a right angle where a drop meets the plane.

2:06Opposite sides of a rectangle are equal, so that extra line is exactly as long as the projection - the shadow, carried up off the plane and laid alongside the segment itself. And now look at what the rectangle left behind. A triangle: the segment as one side, the copied projection as another, and a right angle between the second and the third. In a right triangle the longest side is the one opposite the right angle, and that is the segment.

2:42So the projection is a shorter side of a triangle whose longest side is the segment. It cannot be longer. Ever. Here the segment is five, the projection three, and the drop of four makes up the difference. And they are equal exactly when the triangle collapses - which happens exactly when both ends are the same height above the plane. So a projection never stretches anything, and it keeps a length only when that length was already parallel to the plane. Everything else here follows from that one sentence.

3:21Take a square of side three and hold it parallel to the plane. Every side is parallel to the plane, so every side keeps its length. The shadow is the same square, area nine. Now tilt it. Different sides lean by different amounts, so they shrink by different amounts, and the shadow is a parallelogram: two different side lengths, no right angle, area six. Tilt it another way and the shadow is a rhombus.

3:53Stand it edge-on and the shadow is a segment. A whole direction gone, and no area at all. One square, and the answer to what shape it projects to is a list. This is also why a regular shape need not stay regular: regular means the sides agree, and projection treats different sides differently. One thing does survive, and it makes all of this drawable. Take two parallel lines. They have the same direction, and projection does the same thing to a direction wherever it finds it.

4:30So the two shadows stay parallel - or the direction was the one being thrown away, and both collapse to points together. There is no third outcome. A parallelogram has two pairs of parallel sides, and both pairs stay parallel. So it can project to a leaning parallelogram, a rectangle, a rhombus, or flat to a segment. It can never project to a trapezium. Not from any angle. Cut one out and turn it about in sunlight as long as you like.

5:06Here are three segments: one eight long, one ten, one seventeen. They differ only in how high they climb. From above, height is thrown away, so all three arrive as the same shadow, eight long. One picture, three different objects, and nothing in the picture that could tell you which. The same happens with solids. Three blocks, same height, same width, each deeper than the last: four cubes, then eight, then sixteen.

5:37From the front, depth is the direction being thrown away, so all three give the same small square. One view is evidence about a solid. It is never a description of one. So use more than one. Put a plane in front of the object, another underneath it, and a third off to one side. The shadow on the plane in front is the front view; the one underneath is the top view; the one at the side is the side view.

6:09Each keeps two directions and throws one away, and they do not throw away the same one. The front view keeps length and height. The top view keeps length and depth. The side view keeps depth and height. So a block that measures three by four by five gives three rectangles: three by five, three by four, and four by five. Every side of it named twice. Named twice, and that is the part worth stopping on.

6:41Each direction is kept by two of the views and dropped by the third, which means two views already carry all three between them. The third view is not there because the first two left a direction out. It is there because two shadows can still leave the object ambiguous. Go back to the three blocks. Their front views are identical. Their top views are not: two by one, two by two, two by four. Nor their side views: one by two, two by two, four by two.

7:16The view that could not tell them apart is the one that had thrown away the direction they differ in. You can watch a projection happen. Almost. Put an object in front of a wall and shine a torch at it, square on. The shadow is the right sort of shape, but too big, because the rays leave the torch and fan out. With the object two steps off the wall and the torch four, the shadow is twice the size of the projection.

7:47Walk the torch back to six and it is one and a half times. Ten, a quarter bigger. Twenty, a ninth bigger. At a thousand steps it is bigger by less than a four-hundredth. It never becomes the projection. It gets as close as you ask. And sunlight falling square on a wall casts shadows nobody could tell from projections. It is the distance that does it, not the brightness.

8:17Three views, three directions, nothing left out. So do they pin the solid down? Usually. Not always. And the smallest counterexample fits inside a box two cubes on a side. Three cubes, arranged like this. Front view, top view, side view. Now add a fourth cube, here. The front view does not change. The top view does not change. The side view does not change. Three cubes and four cubes, with all three views identical.

8:51And it is not luck. That fourth cube sits where all three views were already showing a square, put there by the cubes around it. It casts nothing new, because everything it could cast was already being cast. Here is a stack of cubes, stepped, with ten of them facing you. How many cubes are there? You cannot tell. Tuck one in behind and the front view is exactly the same, though the stack now has eleven.

9:22That is not a flaw. A projection throws a direction away, and what it throws away is gone. So it comes down to one exchange, made three times. You give up a direction. In return you get a flat picture you can measure, in which lengths never grew, parallel stayed parallel, and anything lying flat came through untouched. Three of those, chosen so nothing is lost between them, and somebody can build your solid from the page.

9:54Not because the drawing shows everything - because you know exactly what each view left out.