Ed,
Thank you for a speedy reply to my questions concerning Hawkindale Angles. From your response it appears that my questions were not explicit enough.

Firstly I appologise for making a math mistake, not a formulation mistake in my question. To try and understand the formulas as detailed on this web site I used two examples, both were for irregular pitches, regular plans not an irregular pitch and plan.

After reading your reply it appears my confussion lies around the basic formulas.

DD = Arctan[Tan(S)/Tan(SS)]
D = Arctan[Tan(SS)/Tan(S)]
SS = Arctan[Tan(D)/Tan(S)]
S = Arctan[Tan(DD)/Tan(SS)]

My first example was as follows:

SS Main Roof 12/12 or 45°
S Adjacent Roof 7/12 or 30.2564°

DD = Arctan[Tan(30.2564)/Tan(45)]= 30.2564°

D = Arctan[Tan(45)/Tan(30.2564)] = 59.74358°

SS= Arctan[Tan(59.74358)/Tan(30.2564)]= 71.20782°

S = Arctan[Tan(30.2564)/Tan(71.20782)]=11.22672°

R1 = Arctan[Tan(S)xSine(D)
= Arctan[Tan11.22672xSine59.74358]
= 62.54703°

Example 2:

SS Main Roof 10/12 or 39.80557°
S Adjacent Roof 8/12 or 33.69007°

DD= Arctan[Tan33.69007/Tan39.80557]= 38.65983°

D = Arctan[Tan39.80557/Tan33.69007]=51.34007°

SS= Arctan[Tan51.34007/Tan33.69007]=61.92739°

S = Arctan[Tan38.65983/Tan61.92739]=23.10646°

R1 = Arctan[Tan23.10646xSine51.34007]=18.42659°

Based on the above formulas and calculations the hip pitch angles for both roofs differ from those calculated from the more traditional methods.

A 12/12 - 7/12 roof would have a hip pitch of 26.74°

A 10/12 - 8/12 roof would have a hip pitch of 27.50045°

However if the pitch angle 'S' in the formula R1 was changed to the actual pitch angle of the adjacent roof them the R1 formula would produce the correct answer.

This raises the question what do 'SS' and 'S' really mean. If in the previous calculations the known roof pitches were used instead of the calculated pitches then different angles are created.

In a further attempt to try and understand the Hawkindale formulas I notice that some of the formulas do not account for the adjacent angles in hip and valleys of irregular roof and other formulas seem to have been omitted, for example the edge cut angle at the top of a hip or valley rafter and the top edge cut of a jack rafter or is the formula P2 to be used to obtain them.Looking at the drawing of the intersecting roof am I right in assuming that the formula P2 could also be Arctan[Cos SS/TanDD]