Having seen the the wall sundial on Eyam church, I was inspired to find out how the layout of the dial was arrived at.
In the local village bookshop I came across a volume first published in 1936 by Lancelot Hogben called "Mathematics for the Millions". Amongst the many interesting topics is a chapter explaining spherical triangles, and as an example shows how to calculate the shadow angles of a horizontal sundial.
That was some years ago, and having since learnt more on the subject, I've produced the sundials shown below on my house at Cromford.

In order to make a declining wall dial as large as possible, the only practical approach was to attach the dial furniture directly to the brick wall. The gnomon bracket is fabricated from steel, the style being part of an old wrought iron fence.
The steel numerals are hooked onto the wall, so can be moved to correspond with summer and winter changes of the clock.

The picture shows the dial set for British Summer Time, longitude correction was made in the dial calculations to give more accurate time keeping.

The aim of this dial was to create a sundial that tells the time on the hour to within an accuracy of around 2 minutes.

The gnomon is the rod that stands squarely on the dial face near the top and casts a shadow down the dial. The time is indicated by the very end of the shadow only.
In the photograph above, the time shown is 3pm on June 21st.

Due to the equation of time, the hour lines are shaped like a figure of eight . This is known as an analemma.

Also the end of the shadow tracks the month lines (lines of declination) along the width of the dial on approx' the 21st of each month.
The curved line at the top of the dial is followed on the 21st of December, and the curved line at the bottom on the 21st of June.
The straight line is followed at the equinoxes, 21st of March and 21st of September.

The dial measures 25cm x 21cm, and is engraved into slate using a home made computer controlled engraving machine. The 365 points for each hours analemma were calculated via a computer program, and the results were translated to control the machine stepper motors. The engraving was then finished by hand.

Laser cut from stainless steel, this sundial is photographed before being fitted to the wall.

It is 28cm in diameter. After using the Calculator for vertical dials,
the sundial was designed on a computer using CAD.
The resultant CAD file was then sent off to the laser cutters to produce the parts from 3mm sheet stainless steel.
All the dial markings are laser cut perforations, and the gnomon fits into slots to ensure accurate positioning.

Equation of Time

The equation of time is the disparity between the arrival of the sun at its highest point and the clock definition of noon, this difference may be as wide as 16 minutes, and renders normal sundials inaccurate.

For more details see the Derbyshire Sundials Equation of Time page.

Analemma

An analemma is the resultant shadow track if a board is stood vertically facing south, and a rod is standing squarely from the face of the board near the top.

A shadow will be cast on the board from the rod (gnomon). If the end of the shadow is marked on the board at noon by the clock time for all of the days of the year, the result would be a figure of eight shape called an analemma.
The shape would be generated by a combination of the equation of time and the variation in the altitude of the sun through the year.

In order to tell the time on the hour, you need to know which side of the figure of eight is relevant for that time of year (as dictated by the equation of time).
For example, on March 21st, the hour is read on the left side, and on September 21st it is read on the right side.

In fact an analemma can be made on any hour line as shown on the slate dial further up this page.
One method of doing this using the Vertical dial Calculator is plotting point 'F' for various times of the year for each hour, though it would be a tedious job.