Looking up versus looking down: elevation and depression

0:00Here are two sightings, side by side. On the left, a person on the ground looking up at something high. On the right, a person on a balcony looking down at something on the ground. Everything the two drawings need is the same, and it is worth saying out loud what that is. There is an eye. There is a target. There is a ray joining them, and that ray is the sightline.

0:27And through the eye there is a second ray, drawn dead level, which the angle is measured against. So the honest claim is not that there are two constructions here. There is one, read twice. The names are different because of one fact about the target, and we are about to say which. Here is the whole difference. In the first picture the target is above the eye. In the second it is below it.

0:56Not the height of the observer, not the distance, not whether they are standing on the ground or on a roof. Above the level ray, the sighting is called an elevation. Below it, a depression. I built two hundred and sixteen sightings, at four station heights, six distances and nine different rises and drops. In every one the name follows the side, with no exception either way. Turn every picture, level ray and all, and not one verdict changes.

1:30Turn the points and leave the level ray where it was, and a hundred and forty four of the two hundred and sixteen change. So the word is decided by the level ray, and by nothing else in the figure. Group the sightings by which side the target is on and the name is settled. Group them by distance and it is not. Now the mistake that costs the most marks.

1:57The observer is on a balcony, looking down. The level ray still leaves the eye. It runs out from the balcony, level, over the top of the target. It does not run along the ground. Draw the ground-level ray instead and put the angle there, and look at what you have measured. You have measured an angle at a point where nobody is standing, opening against a sightline that starts somewhere else.

2:25The rule from before has not changed. The level ray goes through the eye, because that is where the instrument is. So the depression angle is the wedge between that balcony-level ray and the sightline going down. It is measured down from the level, not down from the vertical. A thirty degree depression is a shallow look, not a steep one. There is one more thing that makes depression feel like a different animal, and it is worth naming.

2:58Look at where the angle sits in each picture. In the elevation, the angle at the eye is an angle of the triangle. Two of its arms are two sides of the figure. In the depression, it is not. The level ray runs out into open air, over the target. It is not a side of anything. So the marked angle is outside the triangle, and a student who reaches straight for a ratio has nothing to reach for.

3:28That is the real difficulty in this topic. It is not that the arithmetic is harder. It is that the angle is in the wrong place. And the fix is not a new formula. The fix is to move the angle to a vertex where it can be used. Draw the level ray at the eye, up on the balcony. You already have it. Now draw a level ray at the target, down on the ground.

3:57Two level lines. Both level, so both horizontal, so they never meet. That is the only piece of geometry this borrows from outside trigonometry. They are not parallel because they look parallel in the drawing. They are parallel because both of them are level. And now look at the sightline. It runs from the eye down to the target, so it crosses both of those parallel lines. A line crossing a pair of parallels is a transversal, and a transversal makes equal angles at the two crossings.

4:35So there is a pair of angles here that must be equal, and one of them is the depression we started with. The other one is at the target, looking back up. Being equal is not enough on its own. A transversal across parallels makes several equal pairs. The one we want is the alternate pair: the two arms lie on opposite sides of the sightline. At the eye, the level ray runs out away from the building. At the target, the level ray runs back towards it.

5:11Opposite directions, opposite sides of the transversal. That is what alternate means. I checked both halves of that separately, on all hundred and ninety two sightings that are not level. The angles come out equal, and the two arms come out on opposite sides. Every time. And when the ray at the far end is turned to point the same way instead, the angles are still equal, but the arms are now on the same side.

5:42That is the corresponding pair, not the alternate one. Equal is not the whole story, and a sign test can tell them apart. So here is what the alternate step buys. The angle you drop from the roof to a point on the ground is the very angle that point raises back up to you. Depression from the top, elevation from the bottom, same number. Which means a depression problem is never a new kind of problem. It is an elevation problem entered from the other end.

6:18Now, that equality has a reason, and I want to be sure the reason is doing the work. It is the levelness. Both rays are level, so they are parallel, so the alternate angles are equal. So I tilted the ray at the far end and asked again, five different tilts, across every sighting. Nine hundred and sixty readings, and the equality failed at every single one. Not one accident.

6:48That is the whole content of the step. Level, therefore parallel, therefore equal. So the working starts by moving the angle, and the figure is worth drawing twice. Once with the angle where it is marked, at the eye. And once with it moved to the far vertex, where the sightline meets the target. In that second drawing the angle sits inside a right triangle, with the drop facing it and the horizontal run touching it.

7:21So the tangent of the angle is the drop over the run, and now there is something to compute. The station also has an angle inside the triangle, between the plumb line and the sightline. That is a different angle. The two are complementary, and their squared cosines add to one, at all hundred and ninety two. They are the same number in only sixteen of them, which are the forty five degree ones.

7:50So using the marked angle where the triangle's own angle belongs turns the ratio upside down, in a hundred and seventy six cases out of a hundred and ninety two. That is the commonest wrong start in the whole topic, and it is one relabelling away from being right. A worked case. You are standing on a bridge, three metres above the banks of a river. You look down at the near bank and read a depression of forty five degrees.

8:23You look down at the far bank, on the other side, and read thirty. How wide is the river? Swap both angles first. The near bank raises forty five back at you. The far bank raises thirty. Now there are two right triangles sharing the same three metre vertical. On the forty five side, the tangent is one, so the horizontal run is three metres, the same as the drop.

8:52The triangle is isosceles, so that branch needed no table at all. On the thirty side, the run is three divided by the tangent of thirty, which is three root three. So the river is three plus three root three metres across. That is a shade under eight point two. Notice that the smaller angle went with the further bank. A shallower look reaches further, and that is worth carrying.

9:21A harder one, with two buildings. You are on the roof of the tall one. You sight the top of a shorter building at a depression of thirty, and its foot at forty five. The short building is eight metres. How tall is yours, and how far apart are they? And now find the rectangle, because it is the only way the eight metres gets in. Two level lines, at your roof and at the short building's top. Two plumb lines, the two buildings.

9:53That is a rectangle, so opposite sides are equal, and the piece of your building above the short one's roof is the only unknown drop. The gap equals the horizontal side, and the short building's height is the piece of your plumb line below the corner. The short building never gets a triangle of its own. It enters as a segment of yours. Work it through and the drop is four root three plus four, so your building is four times three-plus-root-three, a shade under eighteen point nine three metres.

10:30And the gap is the same number, because the forty five degree sighting made that triangle isosceles too. Three more, all the same skeleton. A lighthouse seventy five metres tall, with two ships on the same side, one behind the other, at depressions of forty five and thirty. Swap both. The near ship is seventy five metres out, the far one seventy five root three. The gap is seventy five root three minus seventy five, which is a shade under fifty four point nine one metres.

11:08From a seven metre rooftop, a cable tower's top is at sixty degrees up and its foot at forty five degrees down. The depression gives the horizontal distance, which is seven metres, and then the elevation gives seven root three above the roof. The tower is seven plus seven root three, a shade under nineteen point one three. And a car approaching a tower at a steady speed, seen first at thirty degrees down and six seconds later at sixty.

11:42That one is the nicest, because the tower's height never appears in the answer. I ran it symbolically at twelve different tower heights, and every one gave the same three seconds. The height cancels out of the answer entirely. Elevation and depression are one construction, read twice. The sightline, and the level ray through the eye. Which word applies is decided by one fact: whether the target is above the eye or below it.

12:15The level ray always leaves the eye, even when the eye is on a balcony and the ground is a long way down. In a depression the marked angle sits outside the triangle, which is why it has to be moved before anything can be computed. Moving it is the alternate angle step, and it is legal because both level rays are level, so they are parallel. The depression from the high point to the low one equals the elevation from the low point back to the high one.

12:50Every depression problem is an elevation problem you have not relabelled yet.