The sun from above

One figure: a compass card seen from straight overhead, with you at the centre and north at the top. Height above the horizon runs inward, so the rim is the horizon and the middle of the card is straight up. A curve crosses the card for the chosen day's whole track, dashed where it runs below the horizon through the twilight bands, with the sun drawn on it at the chosen time and a shadow laid out from the centre away from it. Three fainter rails behind it are the two solstices and the equinox. Underneath, two lines say in words where the sun stands and when it rises, peaks and sets.

You are the dot in the middle and north is at the top, so this is the view a bird has of you. The sun's bearing is the direction it lies in, read clockwise from north, and on this card that is simply the direction it is drawn in. How high it stands is how far in from the rim it sits: on the horizon it touches the rim, at 45° up it is halfway to the middle, and only a sun straight overhead sits on the dot with you.

The curve is the whole of the chosen day. Follow it from the point on the rim where the sun comes up, round and inward as the morning climbs, past its closest approach to the centre at solar noon, then out and down to the point on the rim where it goes down. The dashed tails carry on below the horizon, through the three twilight bands, which is where the sun still lights the sky without being in it. Hover anywhere along the curve for the time and the bearing there, click to send the day to that moment, or drag the sun itself along it. Focus the sun and the arrow keys walk it a quarter hour at a time. Run a day sends it round the whole track on its own, which is the quickest way to see how far along the rim a whole day of bearings actually spreads.

The straight line out from the middle is a shadow. A shadow points exactly away from the sun, always, so its direction is the sun's bearing turned half round, and its length is the height of whatever is casting it divided by the tangent of the sun's altitude. At 45° up a shadow matches the thing that throws it and every degree lower stretches it, which is why the low sun of a winter afternoon throws the longest ones of the year. Seen from straight above an upright thing is a point, so the line drawn here is the shadow against the height rather than against the ground, and the metres are lettered at the end of it. Change the height and the lettering moves and the line does not: the direction is the sun's alone.

The rim is a compass card, and the sixteen names around it are the points of the compass. The angle the sun is drawn at is its azimuth, its bearing clockwise from north: east is 90°, south 180°, west 270°. How far in from the rim it sits is its altitude, its height above the horizon. Those two together are the sun's horizontal coordinates, which is everything you need in order to point at it. The bands just inside the rim are the three twilights, civil to −6°, nautical to −12° and astronomical to −18°, and they sit outside the horizon rather than inside it because a sun below the horizon is further away than a sun on it.

The three faint rails behind the day's curve are the solstices and the equinox, the sun's track at its northernmost, its southernmost and over the equator. What the date control moves is the sun's declination, how far north or south of the equator the sun itself stands, and it runs from 23.4° one way to 23.4° the other because that is the tilt of the Earth. So today's curve never leaves the space between the two solstice rails, wherever you stand, and the two points where it meets the rim walk along the rim with it. Only on the equinox does the sun come up due east and go down due west, and that is true everywhere on Earth on the same day.

Far enough north or south and the curve stops meeting the rim at all. Inside it and the sun never sets, that day, and the card shows a closed loop that never reaches the horizon: a midnight sun. Outside it and the sun never rises, the whole curve sitting beyond the rim in the twilight bands, which is a polar night. The figure draws what is there in both cases and the lines underneath say which one it is, rather than printing a sunrise for a day that has none.

The bearing answers on the calculator draw this same card underneath themselves, set to the place, the date and the moment of whatever was asked about, which is why an answer about a direction carries a link over here. Ask for which way is the sun in zurich and the card you get is the one under that answer, at full size and with the controls open. The place names and the clock times stay the calculator's: this figure knows where the sun is, and only the calculator knows what you asked.

The same sun drawn edge on is at the day ring. The other two figures are the clock board and the size map.