Figure 35: Vertical direct south dial
The dial is placed on walls which face directly north. It is not popularly used because the Sun is far too south in the sky to strike a north wall between 22nd September and 21st March. This is illustrated in the animation below.
Only the early morning and late evening hour lines are drawn on the dial plate, because the Sun will not shine on the dial face at other hours. The earliest hour line and the latest hour line on the dial plate are dependent on the sunrise and sunset times of the latitude at which the sundial is designed to be used. The hour lines run clockwise.
We will show in the below animation how the shadow of the gnomon is cast for a year.
Click here for the animation.
The dial is placed on walls which face directly south. It is the most commonly used vertical direct dial because it can measure the greatest duration of time each day.
Unlike the vertical direct north dial, the hour lines run anti-clockwise. The hour lines on any vertical direct south dial are precisely the same as those on a horizontal dial at the colatitude. (This will be explained under the mathematics of vertical direct south dial.) The 6-o'clock hour line is a horizontal line at the top, and the 12-o'clock line is vertical. The Sun can never shine on these dials earlier than 6a.m or later than 6p.m., so we need not include any hour lines other than those shown in the following animation.
Similar to a horizontal sundial, to find the hour lines of a vertical direct south dial, we use Figure 36.
In Figure 36, Z is the zenith, Z' the nadir. (The point on the celestial sphere that lies directly beneath an observer. It is diametrically opposite the observer's zenith.) P is the north celestial pole and P' the south celestial pole. By using one or more of the relations of spherical trigonometry, we can deduce that
cos Z'P' cos TZ'P' = sin Z'P' cot Z'T - sin P'Z'T cot Z'P'T
Since TZ'P' = 90º, Z'T = 
, therefore
0 = sin (90 - Ø) cot - cot (HA) --(equation 1)
tan = tan (HA) cos Ø
Equation 1 is actually the formula for the horizontal dial but with sin (90 - Ø) replacing sin Ø.; In this case, equation 1 is actually the formula to find the hour lines for a horizontal dial at the colatitude.
We will show in the below animation how the shadow of the gnomon is cast in a year.
Click here for the animation.
The vertical direct east dial is placed on walls which face directly east. They can be used on any latitude.
Only the morning hours are drawn on the dial plate because the Sun only shines on the dial face in the morning. The hour lines are parallel with one another, so there is no dial center. Since the dial plate lies in the meridian, the gnomon must be parallel to the dial plate in order for it to be parallel to the Earth's axis.
The gnomon stands vertically on the hour line of 6a.m. The height of the style above the dial plate is always equal to the distance between the hour lines of 6a.m and 9a.m as shown by x in Figure 37. This is because at 9a.m., the Sun's ray is 45° away from the Sun's ray at 6a.m. Hence, the height of the gnomon and the Sun's ray at 9a.m. forms an isosceles triangle with the wall. (See Figure 38.)
We will show in the below animation how the shadow of the gnomon is cast in a year.
Click here for the animation.
The vertical direct west dial is placed on walls which face directly west. They can be used on any latitude.
Only the afternoon hours are drawn on the dial plate because the Sun only shines on the dial face in the afternoon. Again, since the hour lines are parallel with one another, there is no dial center. As in the case of the vertical direct east dial, the dial plate lies in the meridian, hence the gnomon must be parallel to the dial plate in order for it to be parallel to the Earth's axis.
The gnomon stands vertically on the hour line of 6p.m. The height of the style above the dial plate is always equal to the distance between the hour lines of 6p.m. and 3p.m. This is because at 3p.m., the Sun's ray is 45° away from the Sun's ray at 6p.m. Hence, the height of the gnomon and the Sun's ray at 3p.m. forms an isosceles triangle with the wall. (See Figure 40.)
We will show in the below animation how the shadow of the gnomon is cast in a year.
Click here for the animation.