Ad Triangulum ( triangle within the circle)
Ad Triangulum (hexagonal base)
Ad Triangulum (√3 base)
Ad Triangulum (three point geometry)
Ad Triangulum (six point geometry)
Ad Triangulum (human consciousness)
Ad Triangulum (Heaven)
Star of David or Solomon's seal
Thunder Mark's

Ad Quadratum (square within the circle)
Ad Quadratum (octagonal based)
Ad Quadratum (√2 based)
Ad Quadratum (four point geometry)
Ad Quadratum (eight point geometry)
Ad Quadratum (earth geometry)
Ad Quadratum (physical world)

Philibert De l'Orme, LE TROISIEME LIVRE DE L’ARCHITECTURE
I will not dwell further on this point, to regain our lines, which are traits of a crossbow to offend, but traits and practices to teach geometry, architecture and the secret worthy of being known, and executed.

Six kinds of lines or geometric figures extracted from Euclid and Archimedes. The first kind will be used for all runs and vaulted cellars strange as they please, as we have said and shown at the beginning of the third book, the other will be used to find all kinds of arches and doors, and the third for all tubes, and the fourth for all kinds of spherical vaults and other developments, the fifth for all ways of stairs, and the sixth for all kinds of screws.

Tetrahedron Angles
D Angle = 30.00000
A Angle = 70.52878
C Angle = 54.73561
E Angle = 30.00000
B Angle = 54.73561

90-D Angle = 60.00000
90-A Angle = 19.47122
90-C Angle = 35.26439
90-E Angle = 60.00000
90-B Angle = 35.26439

Hip Pitch Angle = arctan( tan( Pitch Angle ) * sin( Plan Angle ))
Hip Pitch Angle = arctan( tan( 70.52878° ) * sin( 30° )) = 54.73561°

Hip Backing Angle = arctan( sin( Hip Pitch Angle) ÷ tan( Plan Angle ) )
Hip Backing Angle = arctan( sin( 54.73561°) ÷ tan( 30° ) ) = 54.73561°

Dihedral Angle = ( 90° - Hip Backing Angle ) * 2
Dihedral Angle = ( 90° - 54.73561 ) * 2 = 70.52878°

The dihedral angle is the angle measured between two planes.

Dihedral Angle of Tetrahedron

Hip Pitch Angle = arctan( tan( Pitch Angle ) * sin( Plan Angle ))
Hip Pitch Angle = arctan( tan( 39.81° ) * sin( 38.66° )) = 27.50°

Hip Backing Angle = arctan( sin( Hip Pitch Angle) ÷ tan( Plan Angle ) )
Major Pitch Hip Backing Angle = arctan( sin( 27.50°) ÷ tan( 38.66° ) ) = 29.99°
Minor Pitch Hip Backing Angle = arctan( sin( 27.50°) ÷ tan( 51.34° ) ) = 20.27°

Dihedral Angle = 180° - 29.99° -20.27° = 129.74°
The dihedral angle is the angle measured between two planes.

Dihedral Angle of Rectangular Pyramid Roof

Rectangular Pyramid Roof With Square Tail Fascia Geometric & Trigonometric Roof Framing Development
Deck Angle = 90.00000
Plan Angle = 51.34019
Roof Pitch = 8:12
Roof Pitch Angle = 33.69007
R1 Hip Rafter Angle = 27.50055
C5m Hip Rafter Backing Angle = 20.27452
C5m 90° - Hip Rafter Backing Angle = 69.72548
R4m Hip Rafter Side Cut Angle = 35.35971
P2m Jack Rafter Side Cut Angle = 33.64933
90° - P2m Roof Sheathing Angle = 56.35067
P1m Purlin Miter Angle = 23.92974
C1m Purlin Saw Bevel Angle = 31.31734
SFMm Fascia Miter Angle = 34.73648
SFBm Fascia Saw Bevel Angle = 28.67913
R2m Hip Rafter Square Tail Miter Angle = 12.98849
C1m Hip Rafter Square Tail Saw Bevel Angle = 31.31734

Deck Angle = 90.00000
Plan Angle = 38.65981
Roof Pitch = 10:12
Roof Pitch Angle = 39.80557
R1 Hip Rafter Angle = 27.50055
C5a Hip Rafter Backing Angle = 29.99339
R4a Hip Rafter Side Cut Angle = 47.95238
P2a Jack Rafter Side Cut Angle = 43.83911
90° - P2a Roof Sheathing Angle = 46.16089
P1a Purlin Miter Angle = 38.66786
C1a Purlin Saw Bevel Angle = 36.86131
SFMa Fascia Miter Angle = 27.11914
SFBa Fascia Saw Bevel Angle = 40.52065
R2a Hip Rafter Square Tail Miter Angle = 25.64298
C1a Hip Rafter Square Tail Saw Bevel Angle = 36.86131

Rectangular Pyramid Roof With Square Tail Fascia Geometric & Trigonometric Roof Framing Development

Treatise on Stair Building & Handrailing
by W & A Mowat
page 178, fig.214
Twist Bevel Angle Geometric Development based on Tetrahedron of Triangular Pyramid
Dihedral Angle = mPn
Twist Bevel Angle = bVc
Bevel Angle on joint line tangent = aV1c
D A C E B Tetrahedron Angles
D S R1 P2 C5 Hawkindale Angles

Twist Bevel Angle Geometric Development based on Tetrahedron of Triangular Pyramid

Barrel Vault Intersects Main Roof Vertical & Horizontal Trace Development.

If a line be perpendicular to an oblique plane, the projections of the line are
respectively perpendicular to the traces of the plane; that is, the horizontal
projection to the horizontal trace, and the vertical projection to the vertical trace.

Barrel Vault Intersects Main Roof Vertical & Horizontal Trace Development

Hip Rafter Pitch Angle = arctan( tan( Pitch Angle ) * sin( Plan Angle )) = 31.63°
Hip Rafter Backing Angle = arctan( sin( Hip Rafter Pitch Angle) ÷ tan( Plan Angle ) ) = 12.26°
Hip Rafter Side Cut Angle = arctan( cos( Hip Rafter Pitch Angle ) ÷ tan( Plan Angle )) = 19.43°

Jack Rafter Side Cut = arctan( cos( Pitch Angle ) ÷ tan( Plan Angle )) = 19.01°
Roof Sheathing Angle = arccos( cos( Plan Angle ) * cos( Hip Rafter Pitch Angle )) = 70.98°

Roof Sheathing Cut Measurement = Plywood Width ÷ tan( Roof Sheathing Angle) = 16.54"
Jack Rafter Difference = tan( Roof Sheathing Angle ) * Jack Rafter Spacing = 69.62"

Frieze Block Saw Miter Angle = arctan( sin ( Pitch Angle ) ÷ tan( Plan Angle )) = 12.94°
Frieze Block Saw Blade Bevel Angle = arctan( sin( Frieze Block Saw Miter Angle ) ÷ tan( Pitch Angle )) 18.57°

Common Rafter Rise = Run * tan( Pitch Angle )
Common Rafter Length = Run ÷ cos( Pitch Angle )
Common JackRafter Run = Jack Rafter Spacing * tan( Plan Angle )
Common JackRafter Length = Jack Rafter Runn ÷ cos( Pitch Angle )

Hip Rafter Run = Run ÷ cos( Plan Angle )
Hip Rafter Length = Hip Rafter Run ÷ cos( Hip Rafter Pitch Angle )

Polygon 2 Cord Gables ( Prow Rafters ) can be calculated with the same angles used to calculate the polygon rafters. The working angle of the polygon is used to calculate the length of the 2 Cord Gable. The length of the 2 Cord Gable is calculated using the following formulas.

Example:
Polygon = Octagon
pitch = 8:12
pitch angle = 33.69°

side wall length = 59.64696
hip rafter run = side wall length
common rafter run = hip rafter run × cos ( working angle )
common rafter run = 59.64696 ÷ cos ( 22.5° ) = 52.33496
roof rise = common rafter run × tan ( pitch angle )
roof rise = 52.33496 × tan ( 33.69° ) = 34.88988hip pitch angle = arctan ( tan ( pitch angle ) × sin ( plan angle ) )
hip pitch angle = arctan ( tan ( 33.69° ) × sin ( 67.5° ) ) = 31.63°

common rafter length = common rafter run ÷ cos ( pitch angle )
common rafter length = 52.33496 ÷ cos ( 33.69° ) = 62.89874

hip rafter length = hip rafter run ÷ cos ( hip pitch angle )
hip rafter length = 59.64696 ÷ cos ( 31.63° ) = 70.05

polygon gable rafter length = hip rafter length

Mark the gable rafter at the same pitch angle as the hip pitch angle on the gable rafter material. Then cut the head cut angle with your saw set at the polygon miter angle. The Octagon polygon miter angle is 22.5°. The saw bevel angle is the same as the polygon miter angle.

The polygon 2 cord gable rafter (prow rafter) is calculated at the same pitch angle as a polygon hip rafter.

Polygon 2 cord gable rafter (prow rafter) Geometric & Trigonometric Roof Framing Development
Polygon 2 Cord Gable Crown Molding Angles and Prow Rafter Development Calculators

Ellipse Formulas for Eyebrow Roof Design

Intersecting Slope Angle = arctan((Roof Pitch - Dormer Pitch ) ÷ 12)
Semi Minor Axis = Radius
Roof Surface Semi Major Axis = ( Radius ÷ tan( Intersecting Slope Angle ) ) ÷ cos( Roof Pitch Angle )
Plan View Semi Major Axis = Radius ÷ tan( Intersecting Slope Angle )
Elevation View Semi Major Axis = ( Radius ÷ tan( Intersecting Slope Angle ) ) * tan( Roof Pitch Angle )

Ellipse Formulas for Eyebrow Roof Design

Take the case that you have drawn the line QR, and hereunto an equilateral triangle, that is to say as great on one side than the other, as you see RST, T is the point where you pull another curved line marked Z, is tightened without moving the compass, and requires that the distance ST is similar to that of TZ. This makes you draw a straight line from point S to T, till it intersects the line Z, and this place, as you see the point marked X, you draw another line even to the point of R, which will be precisely perpendicular to the line QR, as you can judge by the figure followed.

Of these four main parts of the world four winds blowing directly appointed principal or cardinal knowledge is the point of East Subsolanus, marked the ensuing Figure A, where the quality and nature is hot and dry in the West Favonius sale, noted by C, its quality being cold and moist, Midi, Auster, signed B, whose nature and quality wet and hot and Septentrion Boreas, marked D, where quality is cold and dry.
So much for four parts and corners of the world with their own domestic wind.
It should therefore be noted that the ancients still divided equally into four one each space between the aforesaid principal winds, and gave one each to own a wind superabundant. Whereby between Subsolanus and Auster, that is to say between East and South, or if you want between A and B also, they have located the wind called Eurus, marked by E, between South and West, Africus noted by F . , Between West and north, signed by G. Caurus, and between East and Septentrion Aquilo, marked by H.

Vitruvius Roman Theater based Triangle, Ad Triangulum Geometry

Be given an equilateral triangle such width as you like, like ABC, the more it grows, so will the insurance and kindness. Where do I did not wish to help more than that which you see below figuratively, by as much as I used to be easier in my coffers, and do not usually point thereof, a Astrolable be, and ephemeris, with a few other books, and cases filled compass, and what it takes to portraire. Within this triangle imagine a circle, as you can see marked EFGH (almost as if it were a dial showing the hours) and divide into so many parts that will, like twenty-four, thirty-two, forty eight, the most that there is the best. I divided the latter into thirty-two, and is set amidst a magnetic needle, as well as marine dials and compasses, or small whom we help to find the hours to the Sun, but notice that said needle must be very good and very moving. When you want to help the triangle, you look through one side as you please, for the one marked in Figure D. This makes you discard your city view, castle or place from which you want to take the form and figure, and make a sketch first on paper scored coarsely and you can understand the decision. Can you make the trip at all. If you want that it should fit in memory or writing a bend and each face of the walls to measure the length as you will see below. Having done this, you can start at one end of the castle, town or place, keeping your triangle against the first section of wall with a ruler to have greater decision, against which must be your triangle as you see marked K.

Philibert De l'Orme, LES LIVRES D'ARCHITECTURE, Plumb Bob from Equilateral Triangle Geometry

I suggest you put three points to your will, and that from one point to another you draw the lines, you divide that by the middle, and then made a perpendicular thereon, and you see the two lines A and B and where they meet and intersect is the center and you see the place where is C, where you have to put one of the points of the compass, and mark another line precisely, which will go on three points, as you can see in figure marked C in the center.

You can also proceed in this case with the compass by the way you see kept in the following figure given, which is way more secure. So that those who are quick to wield said compass, do not square, as well as if it is just and right, the line can not be done precisely. So looking to find the lengthened with the three points is very useful and necessary, because you can not only do not lift a panel to a building on a round shape, it must always find you are looking lengthened, which can not be done quickly if not by those three points lost, they are in the panel as those marked D and I said, and are looking more and different.