Developments of Hip and Valley Roof Angles based on the Roof Lines

Overview of Tetrahedra and Triangles defining a Hip or Valley Roof

An ALTERNATE MODEL for HIP / VALLEY RAFTERS:

Isometric Projection: Hip orientation R1 is the dihedral angle between the deck and the bottom face of the hip/valley rafter.

DD

Plan

D

90 – DD

R4Bm

R4Pm

R4Pa

R4Ba

R1

P6m

SS

R5Pm

R5Bm

Dihedral angle = 90 – A5Ba

tan R5B = (cos2 DD tanR1) / cos DD = tanR1 cos DDtan R5P = tanR1 / (1 / sin DD) = tanR1 sin DD Same R5 angle formulas as the previous analysis. Also: P6 = (90 – R5P) – (90 – SS) = SS – R5P

Dihedral angle = 90 – A5Bm

Dihedral angle = 90 – A5Pm

Dihedral angle = 90 – A5Pa

DD

D

SS

cos DD

cos2 DD

cos2 DD tan R1cos2 DD

tan R1

1 / sin DD

Hip run = 1

Valley orientation: The deck is a level plane, with the plane of the bottom shoulder of the valley dropping away from the deck.

An ALTERNATE MODEL for HIP / VALLEY RAFTERS:

Kernels of R4, R5 and A5 Angles

90 - DD

R5P

DD

R4Bm

R4Pm

R4Pa

R4Ba

R1

R4P

SS

R5Pm

R5Bm

Dihedral angle = 90 – A5B

R1

1

Kernels re-scaled to “Hip run” = 1: Compare the drawings below to the models extracted directly from the Valley rafter in the previous section, as well as the kernels extracted from the stick.

R5B

R1

R4B

1

Dihedral angle = 90 – A5P

GENERAL HIP / VALLEY MODEL:

VALLEY meets RIDGE or HEADER VALLEY meets MAIN COMMON RAFTER

VALLEY meets MAIN PURLIN

The triangle showing angles labeled Q2and P1 defines the plane of the Main Purlin perpendicular to the Roof plane. Other angular values to expect at the intercept of the Purlin plane and the bottom face of the Hip/Valley: R3, P3, C2, C1, R2

R4B

Q2

Dihedral angle between side face and bottom face of Valley rafter = 90 degrees. Dihedral angle between Roof plane and bottom face of Valley rafter = C5.

DD

Angles to expect at intercept of Valley Peak meets Ridge or Header: DD, R4B, R5B, A5B

MAIN SIDE VIEW

For the sake of clarity, only some of the major angles on the model faces have been labeled. Exploded views will show the remaining angles in more detail.

The planes that form the boundaries of the model (planes of R4 and R5 angles) are the same planes created by cutting a plane on the top of a post to conform to the bottom face of a Hip/Valley rafter. Expect R1, A5, R4, and R5 angles at Valley meets Post.

Working point for purlin related angles.

Anticipated angles where Valley Foot meets Main Common Rafter or Eave: 90 – DD, R4P, R5P, A5P, P6

R5B

R1 P1

C5

SS

GENERAL HIP / VALLEY MODEL:

Kernels extracted from the general model

R4B

Q2

DD

R5B

R1 P1

C5

SS

Mutually perpendicular lines

R4P 90-R5P

90-P6

90-R1

SS

90-R2

P1

Q2

C5 C5 90-P3

90-R3

Dihedral angle = 90-C1

Dihedral angle = 90-C2

* Dihedral angle = 90-P2

* Dihedral angle = P2

Valley peak meets Main Purlin

Valley foot meets Main Common Rafter

* Note: See alternate diagram of kernels extracted from model.

Note the lines that represent Common Rafter or Purlin Depth : Valley Depth. The ratio cosC5 = cosSS / cosR1 = cosP1 / cosR2plays a role in determining the location of angle C5.

Dihedral angle = 90-A5P

Dihedral angle = 90-DD

GENERAL HIP / VALLEY MODEL:

Kernels rotated and re-scaled

Valley peak meets Main Purlin (opposite hand)

Valley foot meets Main Common Rafter

1

Dihedral angle = 90-C1

90-R5P

Q2 1

90-R3

C5 90-R2 C5

90-R1

R4P

Dihedral angle = 90-DD

Dihedral angle = 90-A5P

Dihedral angle = 90-P2

Dihedral angle = 90-C2

Dihedral angle = P2

NOTES: Three mutually perpendicular lines form one of the vertices (see “Purlin kernel extracted from Hip kernel”); the kernels may positioned with any face containing a right angle as the “deck”. The angle arrangements shown above match those of the kernels extracted from the stick. The kernels may be split along their respective dihedral angles: Valley peak meets Main Purlin along 90-C2, 90-C1 or P2, and Valley foot meets Main Common along 90-A5P, 90-DD or 90-P2. Each split produces an arrangement of angles as per a standard Hip kernel. The alternate exploded view depicts the kernels split along dihedral angles P2 and 90-P2. Angles that upon casual inspection seem to have no direct connection to one another are now related through their respective kernels, which may be used for dimensioning as well as simply producing formulas. In addition, since the groups of angles are now defined, and a given angle occurs in more than one kernel, angular values may be determined empirically by developing the kernels using compass and straightedge only. The sample equations given on the following page cover only a very few possible formulas.

VALLEY ANGLE FORMULAS:

Valley Peak meets Main Purlin

KNOWN EQUATION

Valley Foot meets Main Common Rafter

tanR2 = tanC5 tanR3 tanR3 = tanR2 / tanC5

(From “Extracting the Purlin kernel from the Hip kernel”) tanP1 = tanSS tanP2

tanR1 = tanC5 tan(90-R4P) tanR4P = tanC5 / tanR1

Divide the kernel along dihedral angle90–C2, producing two standard Hip kernels. Consider the kernel on the left hand side:

Divide the kernel along dihedral angle 90–A5P, producing two standard Hip kernels. Consider the kernel on the left hand side:

cosC1 = sinC2 / sinR2 sinC2 = sinR2 cos C1

cosC1 = sinC5 / sinP1

cosDD = sinA5P / sinR1 sinA5P = sinR1 cos DD

cos(90- Q2) = cosR3 / cosC1sinQ2 = cosR3 / cosC1

cosR2 = cosP2 / cosC1

cosR5P = cos(90-R4P) / cosDD cosR5P =sin R4P / cosDD

tan(90-R2) = tan(90-C2)sin(90-R3)

tanC2 = tanR2 cosR3

(From Standard Hip kernel) tanR1 = tanSS sinDD

tan(90-R1) = tan(90-A5P) sinR4P

tanA5P = tanR1 sinR4P

Consider the standard Hip kernels created on the right hand side:

tan(90-C5) = tan(90-C2) sinR3 tanC2 = tanC5 sinR3

tanR1 = tanSS sinDD

tan(90-C5) = tan(90-A5P) sin(90-R4P)tanA5P = tanC5 cosR4P

tan(90-P2) = sin(90-C5) / tanR3 tanR3 = tanP2 cosC5

tanC5 = sinR1 / tanDD

tanP2 = sin(90-C5) / tan(90-R4P)

tanR4P = tanP2 / cosC5

The process may be continued, using any known formula as a

template (for example, tanP2 = cosSS / tanDD), and substituting cognate angles from the unsolved kernel. Remember to

compensate for trig functions of complementary angles. The next diagram depicts an alternate method of extracting Hip kernels

from the general model. Rotate any appropriate face to the deck, and apply the methods outlined above to obtain solutions.

GENERAL HIP / VALLEY MODEL:

Standard kernels extracted from General Model (kernels re-scaled for clarity)

R4B

Q2

DD

R5B

R1 P1

C5

SS

Dihedral angle = 90-C2

Dihedral angle = 90-DD Dihedral angle = 90-C1

Dihedral angle = 90-A5P

R4P

C5 C5

C5

C5

P2

P2

90-P2

90-P2

P3

90-R3

P6

90-P1

90-R2 90-R190-SS

Note that the two kernels above may be combined along the plane of C5 to create a new kernel.

The two kernels may be combined along the plane of C5 to create the kernel examined in “Extracting the Purlin kernel from the Hip kernel”.

EXTRACTING KERNELS from “the STICK”:

DD

90-C2

Mai

n C

omm

on R

afte

r Main Purlin

D

90-DD

Valley Peak

90-R3

90-R2

Dihedral Angle = 90-C1

90-R3

Q2 90-R2

For the purpose of clarity, backing angles and other cuts are not shown.

Valley Foot meets Adjacent Common Rafter: same kernel as Valley Foot meets Main Common Rafter, substitute adjacent side values. Also anticipate these angular values at Valley meets Post or Eaves.

90-R5P 90-R1

R4P

90-A5P

Dihedral Angle = 90-DD

R4P 90-R1

Valley Foot

Valley Peak meets Main Purlin DD projected to bottom shoulder of Valley is 90-R3

Valley Foot meets Main Common Rafter 90-DD projected to bottom face of Valley is R4Pm (90-D projected is R4Pa at Adjacent Common Rafter)

Plan

Valley Peak

EXTRACTING KERNELS from “the STICK”:

DD

90-A5Bm

Header

D

Valley Peak

R4Bm

90-R1

Dihedral Angle = DD

90-R1

90-R5Bm

R4Bm

90-R5Ba 90-R1

R4Ba

90-A5Ba

Dihedral Angle = D

R4Ba

90-R1

Valley Peak

Valley Peak meets Header DD projected to bottom plane of Valley is R4Bm

Valley Peak meets Valley Peak D projected to bottom face of Valley is R4Ba

The same angles occur at Valley meets Post.

Plan Valley Peak

Images of 3D Model ... Right Triangles composing the Tetrahedra modeling the Hip Rafter Side Cut Angles at the Peak of the Hip Roof

Main Slope = 10/12 Adjoining Slope = 7/12 Corner Angle measured between Eaves in Plan View = 90°

Development of Tetrahedron modeling the 10/12 Side Angles (Trigonometric Scale)

Development of Tetrahedron modeling the 7/12 Side Angles (Trigonometric Scale)

Plan View Lines and Angles

Views of the Tetrahedra modeling the Hip Side Cut Angles juxtaposed along the Triangle of the Hip Slope

Hip Side Cut Angles at Peak of Hip Rafter Juxtaposed Tetrahedra viewed from

Triangles representing upper shoulder of unbacked Hip Rafter

The triangles follow the plane of the upper shoulder of an unbacked Hip Rafter. This planepasses through a line perpendicular to the Hip Run, and the slope of the plane, measured fromlevel, is equal to 25.54245°, the Hip Slope Angle. In other words, the Dihedral Angle betweenthe plane containing the Hip Side Cut Angles and the plane following level is the Hip SlopeAngle.

Juxtaposed Hip Side Cut Tetrahedra viewed from 10/12 Side

Juxtaposed Hip Side Cut Tetrahedra viewed from 7/12 Side

Juxtaposed Hip Side Cut Tetrahedra viewed from the line of the Rise

Views of the Tetrahedra modeling the Hip Side Cut Angles separated along the Triangle of the Hip Slope

Oblique View of Hip Side Cut Tetrahedra

View of Hip Side Cut Tetrahedra from the Triangles of the Hip Slope

Cutting the Backing Angle and comparing 3D Geometry Notes ... Images of Right Triangles composing the Tetrahedra extracted from the Hip Roof

A Study of the Jack Rafter Side Cut Angles

Developments of Tetrahedra extracted from the Hip Roof

Views of the Tetrahedra extracted from the Hip Roof juxtaposed along the Triangle of the Hip Slope

Bird's Eye view of juxtaposed Tetrahedra extracted from the Hip Roof Jack Rafter Side Cut Triangles about centerline of Backed Hip Rafter

Oblique view of juxtaposed Tetrahedra extracted from the Hip Roof Jack Rafter Side Cut Triangles about centerline of Backed Hip Rafter

Oblique view of juxtaposed Tetrahedra extracted from the Hip Roof showing the hypotenuses of the Hip Side Cut Triangles (unbacked upper shoulder of Hip Rafter)

Comparison of juxtaposed Hip Rafter Side Cut Angle Tetrahedra (left) and juxtaposed Tetrahedra modeling the Jack Rafter Side Cut Angles (right)

Juxtaposed Tetrahedra extracted from the Hip Roof View from side of the 10/12 Common Slope Triangle

Juxtaposed Tetrahedra extracted from the Hip Roof View from side of the 7/12 Common Slope Triangle

Juxtaposed Tetrahedra extracted from the Hip Roof View from the line of the Rise

Views of the Tetrahedra extracted from the Hip Roof

separated along the Triangle of the Hip Slope

Oblique view of 10/12 Roof Surface ... Jack Rafter Side Cut Angle

Oblique view of 7/12 Roof Surface ... Jack Rafter Side Cut Angle

Tetrahedra extracted from the Hip Roof viewed from the Triangles of the Hip Slope

Images of Post Type 3D Model of the Hip Rafter Side Cut Angles

Main Slope = 10/12 Adjoining Slope = 7/12 Corner Angle measured between Eaves in Plan View = 90°

Development of Post Type Model of the Hip Rafter Side Cut Angles (Trigonometric Scale)

Comparison of all the Model Types ... Bird's Eye View Models to left and center ... Side Cut Angles on upper shoulder of unbacked Hip Rafter

The Tetrahedra extracted from the Hip Roof (right) describe a backed Hip Rafter

Comparison of Hip Side Cut Angle Model Types ... Bird's Eye View Plumb planes passing through the lines on the surface of the

Tetrahedral Model (left) will produce the Post Type Model (right)

Wireframe Sketch illustrating the plumb planes passing through the Eave Lines and perpendiculars to the Eave Lines superimposed on the Tetrahedra extracted from the Hip Roof

Tetrahedral and Post Type Hip Side Cut Angle Models View from 10/12 Side of Roof

Tetrahedral and Post Type Hip Side Cut Angle Models

View from 7/12 Side of Roof

Employing the Post Type 3D Model of the Hip Rafter Side Cut Angles

to describe the Valley Rafter Side Cut Angles

Hip Model upside down ... Valley Plan Angles View of the level plane of the Plan Angles

The "eaves" now represent the intersecting ridge lines at the Valley roof peak

Hip Model upside down ... Valley Side Cut Angles View of the unbacked shoulder of the Valley Rafter

The "eaves" describe the intersecting ridge lines at the Valley roof peak and the "Hip Length" represents the Valley sloping down from the level plane of the Plan Angles

A Review of the Juxtaposed Tetrahedral Models ... their Relationship to the Post Type Model

Application to describe both the Hip and/or Valley Side Cut Angles

Lines and Angles seen in Plan View

Wireframe Sketch of Plumb Section Planes superimposed on the Tetrahedra extracted from the Hip Roof

Bird's Eye View of the Juxtaposed Tetrahedra illustrating the lines and Side Cut Angles formed by the Plumb Section Planes

intersecting the plane of the upper shoulder of an unbacked Hip or Valley Rafter

Unequal Roof Slopes, Non-Rectangular Corner Angle Images of Post Type 3D Model of the Irregular Hip Rafter Side Cut Angles

Main Slope = 8.5/12 Adjoining Slope = 10/12 Corner Angle measured between Eaves in Plan View = 120°

Development of Post Type Model of Irregular Hip Rafter Side Cut Angles (Trigonometric Scale)

Post Type Model of Irregular Hip Rafter Side Cut Angles View of the upper shoulder of the Hip Rafter

Post Type Model of Irregular Hip Rafter Side Cut Angles View from 8.5/12 Side of Roof

Post Type Model of Irregular Hip Rafter Side Cut Angles View from 10/12 Side of Roof

Employing the Post Type 3D Model

of the Irregular Hip Rafter Side Cut Angles to describe the Irregular Valley Rafter Side Cut Angles

Irregular Hip Model upside down ... Irregular Valley Plan Angles View of the level plane of the Plan Angles

The "eaves" now represent the intersecting ridge lines at the Valley roof peak

Irregular Hip Model upside down ... Irregular Valley Side Cut Angles View of the unbacked shoulder of the Valley Rafter

The "eaves" describe the intersecting ridge lines at the Valley roof peak and the "Hip Length" represents the Valley sloping down from the level plane of the Plan Angles

Overview of the related Hip Side Cut Angle and Jack Rafter Side Cut Angle Models

Tetrahedra extracted from the Hip Roof illustrating the Jack Rafter Side Cut Angles

Tetrahedron relating the Hip Side Cut Angle at the Hip Peak (R4P) the Backing Angle (C5), and the Jack Rafter Side Cut Angle (P2)

tan R4P = tan P2 ÷ cos C5

Main Slope = 10/12 Adjoining Slope = 7/12 Corner Angle measured between Eaves in Plan View = 90°

Comparison of nested Tetrahedra (right) extracted from the Post Type Model (left)

Development of the Tetrahedron extracted from the Post Type Model of the Hip Rafter Side Cut Angles

(Trigonometric Scale)

Nested Tetrahedra ... Triangles of Common Slope and P6

Nested Tetrahedra ... Triangle of Backing Angle

Nested Tetrahedra ... Triangle of Hip Side Cut Angle at the Hip Peak

Juxtaposed Tetrahedra relating the Hip Side Cut Angle at the Hip Peak (R4P)

the Backing Angle (C5), and the Hip Slope Angle (R1)

tan R4P = tan C5 ÷ tan R1

Main Slope = 10/12 Adjoining Slope = 7/12 Corner Angle measured between Eaves in Plan View = 90°

Juxtaposed Tetrahedra extracted from the Post Type Model of the Hip Rafter Side Cut Angles

Development of the juxtaposed Tetrahedra extracted from the Post Type Model of the Hip Rafter Side Cut Angles

(Angular Scale)

Juxtaposed Tetrahedra ... Triangles of Common Slope and P6

Juxtaposed Tetrahedra ... Triangles of Backing Angle

Juxtaposed Tetrahedra ... Triangle of Hip Side Cut Angle at the Hip Peak

Juxtaposed Tetrahedra and the Triangles of Hip Slope ... Plumb Section Plane taken through the line of the Hip Run

Application of Juxtaposed Tetrahedra ... Square Beam intersected by Hip Rafter

Main Slope = 12/12 Adjoining Slope = 12/12 Corner Angle measured between Eaves in Plan View = 140°

Model of Nonagonal Footprint Pyramid

Developments of the Juxtaposed Tetrahedra

Development of the Beam

Model of Beam ... Oblique View

Model of Beam ... View from Hip Rafter Foot

Hip Rafter intersects Beam ... View of Upper Surfaces (Roof Planes)

Hip Rafter intersects Beam ... View of Side Faces

Tetrahedron representing Roof Surface Angles juxtaposed with Model of Hip Rafter intersects Beam

Tetrahedra modeling all angles juxtaposed with Model of Hip Rafter intersects Beam