Tréteaux Roof Framing Kernel and Claw Lines

The basic roof framing kernel can be used to develop the footprint of the hip rafter rotated into the roof surface. The roof framing kernel can also be used to verify the geometry of the claw lines on the fold down roof surface for the geometric development of the jack rafters, purlin rafters and purlin rafters skewed from the plate line.

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Here's the basic roof framing kernel.

Add the hip rafter backing triangle to the roof framing kernel.

The red square at the top of the hip rafter backing triangle represents the hip rafter material you're using and it is rotated to align with the top of the hip rafter backing triangle. 

Geometric drawing steps for the rotated hip rafter foot print.

1. Draw line FP #1 parallel with the hip rafter run line.
2. Draw lines TC #1 & TC #2 parallel with the sides of the rotated hip rafter block in the hip rafter backing triangle.
3. Where the TC # 1 line intersects at the bottom of the hip rafter backing triangle draw the line DP #1 to the eave line where it intersects with FP #1.
4. Draw DP #2 parallel to DP #1.
5. Draw the FP #2 line for the footprint line in plan view of the hip rafter. 
6. The hip rafter foot print line intersects the two DP lines and forms the , yellow, hip rafter prism footprint. 

To develop the claw lines for the jack rafters, purlin rafters and skewed purlin rafters develop the roof surface. Use the Tréteaux Angles Check Online Calculator to draw out the claw lines on the roof surface or develop the claw lines from the geometry circled in the following drawings. 

Drawing #1

Roof Eave Angle = 90.00000

SS = Main Rafter Slope Angle = 30.25644

S = Adjacent Rafter Slope Angle = 39.80557

DD = Main Plan Angle = 55.00798

D = Adjacent Plan Angle = 34.99202

DD = arctan (sin W ÷ (tan SS ÷ tan S + cos W)) = 55.00798

R1 = Hip Rafter Slope Angle = 25.54245

C5m Main Hip Rafter Backing Angle = 16.79517

C5a Adjacent Hip Rafter Backing Angle = 31.63186

Valley Sleeper Saw Blade Bevel Angle = 90° - (C5m + C5a) = 41.57297

P2m = Main Jack Rafter Side Cut Angle = 31.15906

90° - P2m = Main Roof Sheathing Angle = 58.84094

P2a = Adjacent Jack Rafter Side Cut Angle = 47.66035

90° - P2a = Adjacent Roof Sheathing Angle = 42.33965

Purlin Depth = 1

Hip Rafter Depth = 2

Hip Rafter Width = 2

Purlin Rafter Seat Line Dimension for Plan View = 1.56205

Purlin Rafter Seat Line Dimension for Roof Surface = 1.20000

Hip Rafter Footprint Seat Line Dimension = 3.96927

Hip Rafter Dimension for Claw Lines from A to B = 1.78640

Hip Rafter Dimension C for Upper Claw Lines from A to C = 1.12740

Hip Rafter Dimension D for Lower Claw Lines from B to D = 0.88700

Drawing # 2

Roof Eave Angle = 120.00000

SS = Main Rafter Slope Angle = 30.25644

S = Adjacent Rafter Slope Angle = 39.80557

DD = Main Plan Angle = 76.99608

D = Adjacent Plan Angle = 43.00392

R1 = Hip Rafter Slope Angle = 29.61276

C5m Main Hip Rafter Backing Angle = 6.51019

C5a Adjacent Hip Rafter Backing Angle = 27.91566

Valley Sleeper Saw Blade Bevel Angle = 90° - (C5m + C5a) = 55.57415

P2m = Main Jack Rafter Side Cut Angle = 11.28135

90° - P2m = Main Roof Sheathing Angle = 78.71865

P2a = Adjacent Jack Rafter Side Cut Angle = 39.47840

90° - P2a = Adjacent Roof Sheathing Angle = 50.52160

Purlin Depth = 1

Hip Rafter Depth = 3.5

Hip Rafter Width = 3.5

Purlin Rafter Seat Line Dimension for Plan View = 1.56205

Purlin Rafter Seat Line Dimension for Roof Surface = 1.20000

Hip Rafter Footprint Seat Line Dimension = 6.94622

Hip Rafter Dimension for Claw Lines from A to B = 1.39221

Hip Rafter Dimension C for Upper Claw Lines from A to C = 0.68538

Hip Rafter Dimension D for Lower Claw Lines from B to D = 1.45905

Drawing #3
Roof Eave Angle = 90.00000
SS = Main Rafter Slope Angle = 33.69007
S = Adjacent Rafter Slope Angle = 33.69007

DD = Main Plan Angle = 45.00000
D = Adjacent Plan Angle = 45.00000

R1 = Hip Rafter Slope Angle = 25.23940
C5m Main Hip Rafter Backing Angle = 23.09347
C5a Adjacent Hip Rafter Backing Angle = 23.09347
Valley Sleeper Saw Blade Bevel Angle = 90° - (C5m + C5a) = 43.81306

P2m = Main Jack Rafter Side Cut Angle = 39.76216
90° - P2m = Main Roof Sheathing Angle = 50.23784
P2a = Adjacent Jack Rafter Side Cut Angle = 39.76216
90° - P2a = Adjacent Roof Sheathing Angle = 50.23784

Purlin Depth = 1
Hip Rafter Depth = 2
Hip Rafter Width = 2

Purlin Rafter Seat Line Dimension for Plan View = 1.80278
Purlin Rafter Seat Line Dimension for Roof Surface = 1.50000
Hip Rafter Footprint Seat Line Dimension = 3.60555

Hip Rafter Dimension for Claw Lines from A to B = 1.81221
Hip Rafter Dimension C for Upper Claw Lines from A to C = 1.04231
Hip Rafter Dimension D for Lower Claw Lines from B to D = 0.95940

Drawing for location of Miter & Claw lines on a roof with a Plumb Hip Rafter.

Roof Eave Angle = 90.00000
SS = Main Rafter Slope Angle = 30.25644
S = Adjacent Rafter Slope Angle = 39.80557

DD = Main Plan Angle = 55.00798
D = Adjacent Plan Angle = 34.99202

R1 = Hip Rafter Slope Angle = 25.54245
C5m Main Hip Rafter Backing Angle = 16.79517
C5a Adjacent Hip Rafter Backing Angle = 31.63186
Valley Sleeper Saw Blade Bevel Angle = 90° - (C5m + C5a) = 41.57297

P2m = Main Jack Rafter Side Cut Angle = 31.15906
90° - P2m = Main Roof Sheathing Angle = 58.84094
P2a = Adjacent Jack Rafter Side Cut Angle = 47.66035
90° - P2a = Adjacent Roof Sheathing Angle = 42.33965

Purlin Depth = 2
Hip Rafter Depth = 2
Hip Rafter Width = 2

Hip Rafter Claw & Miter Line Dimensions for Plumb Hip Rafters

Main SS Hip Rafter Offset Perpendicular to Hip Rafter Run Line = 1.34228
Man SS Hip Rafter Offset Along Eave Line = 1.63846
Adjacent S Hip Rafter Offset Perpendicular to Hip Rafter Run Line = 0.65772
Adjacent S Hip Rafter Offset Along Eave Line = 1.14692
Hip Rafter Backing Depth on Side of Hip Rafter = 0.40514

Hip Rafter Claw & Miter Line Dimensions for Plumb Hip Rafter on Main Side SS-DD
Hip Rafter Line Dimensions for Line 1 = 0.00000
Hip Rafter Miter Line Dimensions for Line 2 = 1.40209
Hip Rafter Miter Line Dimensions for Line 3 = 2.00574
Hip Rafter Claw Line Dimensions for Line 4 = 0.29525
Hip Rafter Claw Line Dimensions for Line 5 = 6.92159

Jack Rafter Miter Angle On Stick = 59.74356
Jack Rafter Bevel Angle On Stick= 31.15906
Jack Rafter Claw Angle on Stick = 8.87615
Jack Purlin Rafter Miter Angle On Side Face Perpendicular To Roof Surface On Stick= 70.57176
Jack Purlin Rafter Bevel Angle On Face Set In Roof Surface On Stick = 58.84094
Jack Purlin Rafter Claw Angle on Stick= 14.48201

Tréteaux Fold Down Roof Surface

We're getting closer to defining the claw lines on the Tréteaux  Fold Down Roof Surface correctly. Once the claw lines on the fold down roof surface are drawn correctly you can place the lines for the timber in plan view at any skewed angle from the eave line and still develop the timber angle cuts on the the fold down roof surface.

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First, here's an example of a purlin rafter on the DP side of the hip rafter rotated into the roof surface and a jack rafter on the rotated side of the hip rafter.

Here's a drawing using the same folding roof surfaces with the purlin rafters skewed from the plate line.

Here's the drawing of how the claw lines on the folding roof surface are developed from the seat lines in plan view and the seat lines on the folding roof surface. There should always be 4 lines that define the claw-miter-bevel angles of the rafters on the roof surface. These four lines represent the center line of the hip rafter, the edge of the hip rafter, the bottom of the hip rafter plane that intersects with the roof surface and the bottom of the jack rafter where it intersects the bottom of the hip rafter.

These pictures are from Louis Mazerolle's book, Traite Theorique Et Pratique De Charpente, planche 29. Now that I can develop the folding roof surface with the claw lines this Tréteaux-- Trestle -- Sawhorse doesn't look all that hard anymore. This Sawhorse has two legs rotated into the roof surface, that Louis Mazerolle refers to as Liens Mansards, and two legs of the Sawhorse that are Plumb.

Here's my first drawing of  the Tréteaux Planche 29 with the two legs rotated into the roof surface and the two legs, hip rafters, that are plumb to the ground plane.

Here's an updated version of my drawing. Using the Tréteaux Angles Check I can see that the green footprint prism was not drawn correctly.

Tréteaux Angles Trigonometry Formulas

Tréteaux Angles Trigonometry Formulas

Link to Online Script to check the Tréteaux Angles

Input Variables

Works for any Roof Eave Angle with equal or unequal rafter slope angles.

Roof Eave Angle
SS = Main Rafter Slope Angle
S = Adjacent Rafter Slope Angle

DD = Main Plan Angle
D = Adjacent Plan Angle

R1 = Hip Rafter Slope Angle
C5m Main Hip Rafter Backing Angle
C5a Adjacent Hip Rafter Backing Angle
Valley Sleeper Saw Blade Bevel Angle = 90° - (C5m + C5a)

P2m = Main Jack Rafter Side Cut Angle
90° - P2m = Main Roof Sheathing Angle
P2a = Adjacent Jack Rafter Side Cut Angle
90° - P2a = Adjacent Roof Sheathing Angle

Tréteaux Angles

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R8-DP = Tilted Hip Rafter Slope Angle on DP Side = arctan( tan ( Hip Rafter Backing Angle) ÷ tan (Jack Rafter Side Cut Angle ))
R8-DP = arctan( tan ( C5m )  ÷  tan ( P2m ))

R9-DP = Horizontal Plane Rotation for Tilted Hip Rafter on DP Line
R9-DP = arctan( sin ( R1 ) × tan ( C5m ))

D-DP = Horizontal Plane Rotation for Tilted Hip Rafter on DP Line from Eave Line
D-DP = D - R9

R10-DP = Vertical Plane Rotation for Tilted Hip Rafter on DP Line
R10-DP = arctan((cos(R1) × tan(R9)) / sin(R1));

PSB-DP = Prism Saw Blade Bevel Angle Along side of Hip Rafter Rotated into the Roof Surface Plane
PSB-DP = arctan( sin(90° - P2m) ÷ tan(R8))

SR4Bm-DP = Prism Footprint Angle Along side of Hip Rafter Rotated into the Roof Surface Plane
SR4Bm-DP = arccos( cos(90° - P2m) × cos(R8))

SR4Ba-DP = Prism Footprint Angle Along side of Hip Rafter on Main Plan Angle
SR4Ba-DP = arccos( cos(90° + P2m) × cos(R8))

R11m-DP = Hip Rafter Miter Angle on Side Of Hip Rafter at Peak
R11m-DP = arctan( (sin (90° - P2m ) ÷ tan ( C8 )) ÷ cos (90° - P2m ))

P7m-DP = Purlin Rafter Bevel Angle on Side Face Parallel To Roof Surface
P7m-DP = 90° + P2m
P7m-DP = 90° + 19.01633 = 109.01633
P8m-DP = Purlin Rafter Miter Angle On Top Edge Face Perpendicular to Roof Surface = 90.00°
P9m-DP = Purlin Rafter Saw Blade Bevel Angle Along Miter Angle on Side Face = 0.00°

P10m-DP = Jack Rafter Miter Angle on Side Face Perpendicular to Roof Surface at Hip Rafter Rotated Into Roof Surface
P10m-DP = 90°
P11m-DP = Jack Rafter Bevel Angle On Top Edge Face Set in Roof Surface = P2m  
P12m-DP = Jack Rafter Saw Blade Bevel Angle Along Miter Angle on Side Face = P2m 

P13a-DP = Purlin Rafter Miter Angle on DP Side Of Hip Rafter on Face Perpendicular To Roof Surface, Lower Claw Angle
P13a-DP = arctan( sin (90° - P2a ) ÷ tan ( Valley Sleeper Saw Blade Bevel Angle )) 

P14a-DP = Purlin Rafter Miter Angle on DP Side Of Hip Rafter on Face Perpendicular To Roof Surface, Upper Claw Angle
P14a-DP = arctan( cos ( P2a ) ÷ tan ( 90° -  Valley Sleeper Saw Blade Bevel Angle )) 

P15a-DP = Purlin Rafter Bevel Angle on DP Side Of Hip Rafter on Face Set in Roof Surface
P15a-DP = 90° - P2a

P16a-DP = Jack Rafter Miter Angle on DP Side Of Hip Rafter on Face Perpendicular To Roof Surface, Lower Claw Angle
P16a-DP = arctan( sin ( P2a ) × tan (90° - Valley Sleeper Saw Blade Bevel Angle )) 

P17a-DP = Jack Rafter Miter Angle on DP Side Of Hip Rafter on Face Perpendicular To Roof Surface, Upper Claw Angle
P17a-DP = arctan( sin ( P2a ) × tan ( Valley Sleeper Saw Blade Bevel Angle )) 

P18a-DP = Jack Rafter Bevel Angle on DP Side Of Hip Rafter on Face Set in Roof Surface
P18a-DP = P2a

Control files for Roof Eave Angle = 90° , 7:12 & 10:12 roof pitch

Control files for Roof Eave Angle = 120° , 7:12 & 10:12 roof pitch

Parallelogram Prism Joe Bartok

I asked Joe Bartok to help me with the saw blade bevel angle on the Tréteaux Dit Cadet task model. I had some inconsistencies in my SketchUp drawings and it resulted in inconsistencies in the results of Joe's formulas. So I made drew up some new control files and sure enough Joe's formulas for the saw blade bevel angle at the foot of the trestle-sawhorse-hip rafter rotated into the roof surface plane for the Parallelogram Prism were correct.

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Research notes:
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Parallelepiped

In three dimensional geometry a parallelepiped is a prism whose faces are all parallelograms.

In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms.

Trihedral Angle

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First I drew out an equal pitched 8:12 roof.

Angular Dimension (27.13381°)
Hip Rafter Rotated Slope Angle = arctan( tan ( Hip Rafter Backing Angle) ÷ tan (Jack Rafter Side Cut Angle ))
Hip Rafter Rotated Slope Angle = arctan( tan ( 23.09347) ÷ tan (39.76216 )) = 27.13381°
Prism Miter Angle =  50.23784°
Prism Saw Blade Bevel Angle = arctan(sin 50.23784° ÷ tan 27.13381°) = 56.30993°
Profile Rafter Slope Angle = 33.69007°
90.00° - 33.69007° = 56.30993°

Angle on Sloped Face of Prism = arccos(cos 50.23784° * cos 27.13381°) = 55.30485°

Then I re-drew the Tréteaux Dit Cadet task model, but without all of the actual sawhorse lines.

Angular Dimension (75.51397°)

Hip Rafter Rotated Slope Angle = arctan( tan ( Hip Rafter Backing Angle) ÷ tan (Jack Rafter Side Cut Angle ))

Hip Rafter Rotated Slope Angle = arctan( tan ( 45.26145°) ÷ tan (14.61316°)) = 75.51399°

Prism Miter Angle =  14.61316°

Prism Saw Blade Bevel Angle = arctan(sin 75.38684° ÷ tan 75.51399°) = 14.03622°

90.00° - 14.03622° = 75.96378°

Angle on Sloped Face of Prism = arccos(cos 75.38684° * cos 75.51399°) = 86.38170°

After I developed the formula to calculate the Hip Rafter Rotated Slope Angle everything fell into place.

Hip Rafter Rotated Slope Angle = arctan( tan ( Hip Rafter Backing Angle) ÷ tan (Jack Rafter Side Cut Angle ))

Joe Bartok's formula for Compound Angle Formulas

tan Saw Blade Bevel = sin Angle of Saw Travel ÷ tan Angle on the Adjoining Face

Angle of Saw Travel = 75.38684° = Roof Sheathing Angle

Angle on the Adjoining Face = 75.51399° = Hip Rafter Rotated Slope Angle

Prism Saw Blade Bevel Angle = arctan(sin 75.38684° ÷ tan 75.51399°) = 14.03622°

Profile Rafter Slope Angle = 75.96376°

90.00° - 14.03622° = 75.96378°

It turns out that the Prism Saw Blade Bevel Angle = 90° - Profile Rafter Slope Angle.

The next angle we need to check is the Hip Rafter Tilt from Plumb, Angular Dimension (19.17417°).

Math Notes:

Profile Slope Angles

SS = side of hip rafter rotated into roof surface

S = DP side of hip rafter …. DP = Devers De Pas -à tilt/ skewed from ground plane

SS = 75.96376 -- plan angle DD = 42.92997° -- backing angle C5m = 45.26145°

S = 74.95720 -- plan angle D = 47.07003° -- backing angle C5a = 41.12961°
Hip Slope Angle R1 = 69.84428°

P2m = arctan (cos SS ÷ tan DD)
P2m = arctan (cos 75.96376° ÷ tan 42.92997°) = 14.61316°
90 - P2m = arctan (cos 75.96376° ÷ tan 42.92997°) = 75.386839°

H1 = Tilted Hip Rafter Slope Angle = arctan( tan ( Hip Rafter Backing Angle) ÷ tan (Jack Rafter Side Cut Angle ))
H1 = arctan( tan ( C5m )  ÷  tan ( P2m ))

H1 = arctan( tan (  45.26145° )  ÷  tan ( 14.61316° )) = 75.51399°

ΣR4B = Angle on Sloped Face of Prism = arccos( cos P2m  × cos H1 )

ΣR4B  = Angle on Sloped Face of Prism = arccos( cos 75.38684°   × cos 75.51399° )  = 86.38170°

H2 = Horizontal Plane Rotation for Tilted Hip Rafter -- DD-dp

H2 = D   - ( 90° - ΣR4B )

H2 =  47.07003°    - ( 90° - 86.38170° )  = 43.45173°
or

H2 = arctan( sin R1  × tan C5m )
H2 = arctan( sin 69.84428°  × tan  45.26145°) = 43.45178°

HDP = Horizontal Plane Rotation for Tilted Hip Rafter for Footprint From Eave Line
HDP = D - H2
HDP = 47.07003°  - 43.45173° = 3.6183°

H3 = Vertical Plane Rotation for Tilted Hip Rafter

H3 = arctan( cos R1  × tan H2 ) ÷ sin R1

H3 = arctan( cos 69.84428°   ×  tan  43.45173° ) ÷ sin 69.84428°  = 19.17416°


H4 = Prism Saw Blade Bevel Angle Along side of Hip Rafter Rotated into the Roof Surface Plane

H4 = arctan( sin 90° – P2m  ÷  tan H1 )

H4 = arctan( sin 75.38684°  ÷  tan 75.51399° )  = 14.03622°

Joe's Sketch of  an isometric projection of the Adjusted Dihedral Angle.

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H5 = Hip Rafter Miter Angle on Side Of Hip Rafter at Peak
H5 = arctan(( sin 90-P2m  ÷ tan C8) ÷ cos 90-P2m)

Results for an equal pitched roof
8:12 , 90° Eave Angle
C8 = 23.09341 + 23.09341 = 46.18682

arctan(( sin 50.23784  ÷ tan 46.18694) ÷ cos 50.23784)=49.06626°
arctan(cos 50.23784 ÷ ( sin 50.23784  ÷ tan 46.18694)) = 40.93374
arctan(( sin 90-P2m  ÷ tan C8) ÷ cos 90-P2m)=49.06626°

SS = 75.96376 -- plan angle DD = 42.92997° -- backing angle C5m = 45.26145°
S = 74.95720 -- plan angle D = 47.07003° -- backing angle C5a = 41.12961°
Hip Slope Angle R1 = 69.84428°

C8 = C5m + C5a = 45.26145° + 41.12961° = 86.39085°
P2m = arctan (cos SS ÷ tan DD)
P2m = arctan (cos 75.96376° ÷ tan 42.92997°) = 14.61316°
90 - P2m = arctan (cos 75.96376° ÷ tan 42.92997°) = 75.386839°

H5 = Hip Rafter Miter Angle on Side Of Hip Rafter at Peak
H5 = arctan(( sin 90-P2m  ÷ tan C8) ÷ cos 90-P2m)
H5 = arctan(( sin 75.386839°  ÷ tan 86.39097°) ÷ cos 75.386839°) = 13.59932°

tan Saw Blade Bevel = sin Angle of Saw Travel ÷ tan Angle on the Adjoining Face
H6 = Saw Blade Bevel Angle along Miter Line H5
H6 = arctan(sin 13.59932° ÷ tan 75.38684°) = 3.508105°

Checking the traditional layout geometry against the trigonometry formulas. The traditional layout geometry is correct. This is one of the main reasons to study the trigonometric formulas. It's easy to draw out the traditional layout geometry incorrectly for rafters. Using trigonometry we can check the traditional layout geometric techniques in books, like Louis Mazerolle's book, Traite Theorique Et Pratique De Charpente, planche 27-28 and know we understand the traditional layout geometry technique and we're drawing it out correctly.  There's nothing worst then  to draw out and cut the rafters only to find out that the traditional layout geometry was never correct, or you didn't interpret the traditional layout geometry in the books correctly.

Purlin Rafters at Hip Rafter Rotated Into Roof Surface

H7 = Miter Angle on Side Face Parallel To Roof Surface = 90° + P2m = 104.61316
H8 = Bevel Angle On Top Edge Face Perpendicular to Roof Surface = 90.00°
H9 = Saw Blade Bevel Angle Along Miter Angle on Side Face = 0.00°

Jack Rafters at Hip Rafter Rotated Into Roof Surface

H10 = Miter Angle on Side Face = 90°
H11 = Bevel Angle On Top Edge Set in Roof Surface = P2m
H12 = Saw Blade Bevel Angle Along Miter Angle on Side Face = P2m

For the jack purlin rafter on the DP side.

H13 = Jack Purlin Miter Angle on Face Perpendicular to Roof Surface, Lower Claw

H13 = arctan( sin ( P2 ) ÷ tan ( Valley Sleeper Saw Blade Bevel Angle )) 

H13 = arctan( sin ( 76.42666 ) ÷ tan ( 3.60903))= 86.2876°

H14 = Jack Purlin Miter Angle on set in Roof Surface

H14 =  90 - P2 

H15 = Purlin Lip/Claw Angle on DP Side, Upper Claw

H15 = arctan( cos ( P2 ) ÷ tan (90 – Valley Sleeper Saw Blade Bevel Angle )) 

H15 = arctan( cos ( 13.57334 ) ÷ tan ( 90-3.60903))= 3.50849°

Angular Dimension (3.50849°) 

SS = 75.96376

S = 74.9572

P2a = 13.57334

Valley Sleeper Saw Blade Bevel Angle = arctan(tan(90-S) *sin(P2))

Valley Sleeper Saw Blade Bevel Angle = arctan(tan(90-74.9572) *sin(13.57334)) =3.60903

Jack Purlin Claw Angle  = arctan( cos ( P2 ) ÷ tan (90 – Valley Sleeper Saw Blade Bevel Angle )) 

Jack Purlin Claw Angle  = arctan( cos ( 13.57334 ) ÷ tan ( 90-3.60903))= 3.50849

For an equal pitched 8:12 roof.

For the jack rafter on the DP side.

H16 = Jack Rafter Miter Angle on Face Perpendicular to Roof Surface, Upper Claw Angle

H16 = Upper Jack Rafter Claw Angle = arctan( sin ( P2 ) × tan (Valley Sleeper Saw Blade Bevel Angle ) )

H16 = Upper Jack Rafter Claw Angle = arctan(sin 39.76216 ×  tan (43.81306)) = 31.53479°

H17 = Jack Rafter Miter Angle on set in Roof Surface
H17 = 90 - P2

H18 = Jack Rafter Lip/Claw Angle on DP Side, Lower Claw Angle

H18 = SS or S

Hip Rafter Housing Angles

H19 = Hip Rafter Housing Angle on Hip Rafter Rotated into the Roof Surface

H20 = Hip Rafter Housing Angle on  DP side of Hip Rafter on top Face

H21 = Hip Rafter Housing Angle on  DP side of Hip Rafter on Bottom Face

H22 = Valley Saw Blade Bevel Angle

Valley Sleeper Bevel Angle = 90 - arctan( tan ( SS ) ÷ sin ( P2m ))

or

Valley Sleeper Bevel Angle = 90 - (C5m + C5a)

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Time to make some more test cuts.

Here I cut a 4x4 using
Miter Angle = 90° - 75.38684° = 14.6°
Saw Blade Bevel Angle = 14.03622° = 90° - 14° =  76°.

The test cut post fit next to the  Tréteaux Dit Cadet task model post correctly.

I printed out the Parallelogram Prism at a 1" to 1" scale and placed the test post on the paper with the Parallelogram Prism. It fit correctly.

Another view of the post on the Parallelogram Prism.

Saw Blade Bevel Angle set to 76° = 90° - 14°

After testing the Parallelogram Prism post cut I decided to make another sawhorse. However, I didn't have any more 4x4 material. So I used 2x4's. It can be a job site Tréteaux -- Trestle sawhorse.

I cut 4 2x4's with the same miter and saw blade bevel. I layed them out using my Sawhorse template, to make sure they were exactly the same length.

After I cut the legs, I measured 1" from the outside of the leg and used my skillsaw ,set at 0°, to make the plumb cut on the end of the sawhorse leg. When you first cut the sawhorse legs use the same miter and bevel angle at each end of the leg with both cuts going the same direction. If the leg were placed in the correct vertical position in the sawhorse model the top of the saw horses prism cut would represent a level plane. Cutting across the top of the  level plane produces the plumb cut on the sawhorse leg.

Here I laid out the top 2x4 of the sawhorse using a 75° angle.

Test sawhorse assembled.

Checking the 4x4 template block against the assembled sawhorse leg.

Added some plywood gussets at each end of the sawhorse.

Comparison of the two Tréteaux -- Trestle sawhorses.