You may be wondering if the azimuth and altitude values (particularly the altitude) displayed in TPE relate to the centre of the sun or moon, or to some other point on the disc. This article explains what you'll find where.
As a general principle, the displayed azimuth and altitude always pertain to the centre of the disc, as shown below:
If you read an az/alt from the app, it will be for the centre of the disc.
Rise and Set
However, the centre of the disc not always what is used to calculate the time or position of the sun or moon. In particular, sun- or moonrise is defined as when the upper limb of the disc is first visible above the ideal horizon, corrected for atmospheric refraction. And sun- or moonset is the moment when the upper limb disappears from the horizon. (See Sunrise (etc.) is incorrect for further details.)
When TPE calculates the time for sunrise, for example, the required altitude of the sun is not 0°, but 0° minus the semi-diameter of the sun, ~0.26°, and minus the effect of atmospheric refraction, ~0.57°, which serves to "lift" the sun and moon in the sky, especially around the times of rise and set when the path of light through the atmosphere is longest.
So, if you select the time of sunrise in TPE, the az/alt of the sun will likely show a slightly negative value for the altitude.
This raises one other important point: the altitudes shown are apparent altitudes, i.e. altitude to the centre of the disk adjusted for atmospheric refraction. As photographers, this is what you care about - it's the altitude at which you would observe the sun or moon (as opposed to "true altitude", where it would appear were the earth stripped of its atmosphere - we don't want that!).
If you'd like to understand more about how atmospheric refraction is calculated, Wikipedia has a good article. The key thing to know is that it varies with temperature, humidity, and elevation above sea level, and is most significant around the times of rise and set.
Note: there is no concept of apparent azimuth here. The azimuth of the sun and moon is unaffected by atmospheric refraction.
Vertical alignment shots
If you are planning a shot where the sun or moon will appear sat atop a mountain/building, or hanging underneath the span of a bridge, for example, you will want to align to either the lower or upper limb of the body instead of the centre.
The rule of thumb is to add or subtract 0.25° from the altitude: add to align to the lower limb and subtract to align to the upper limb.
In TPE for iOS, in the Visual Search screen, you can select this adjustment programmatically, as shown below. The adjustment is calculated for each date - the moon's semidiameter various by around 14% from smallest to largest over the lunisolar cycle.
If you have any further questions on this topic, please leave a comment or create a support ticket using the links on this page.