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Strong Built Establishments Builders

SBE Builders is a commercial and residential framing contractor located in Discovery Bay California. Dedicated to the art of Carpentery & Roof Framing. We are the professionals for the future of the craft of carpentry.

Tangent Hand Railing Trigonometric Development
or
Elucidation of the stuff is self evident
Tangent Handrail
PDF file of Minor Axis Offset

This script returns the Upper & Lower Twist Bevel Angles, Dihedral Angle at Upper Tangent Angle,Angle of the Tangents,Ellipse Semi Major Axis for the tangent handrail geometric drawing. It allows the user to check their geometric drawing of the tangent handrail system. The Rail Block Radius is half the width of the handrailing. It's use to calculate the semi-major axis for the inside and out side of the face mold template. It is used to calculate the Face Mold Width at the Spring Lines. It is used to calculate the Width of Rail Blocks at Twist Bevels. The script also returns the Prism Angles and Prism Heights for any type of prism. Square(quarter circle), obtuse and acute plan angles. The Prism Angles are used to calculate the Face Mold Width at the Spring Lines. The Prism Height at the Spring Lines is calculated from the pitch of plank.

 

The section plane minor axis offset length can be used to locate the center of the ellipse if your drawing out section planes, like W & A Mowat. The pitch of the plank can be used to check your geometric drawing of the tangent handrail system, if your drawing is based on Robert Riddell's embodying the perfect elucidation of the tangent system.

 

Plan View Angle of the Tangents Upper Tangent Pitch Angle Lower Tangent Pitch Angle Radius of the Tangents Rail Block Radius
Download Development of Tetrahedron Geometry DXF
Development of Tetrahedron Geometry .dxf -- auto generated DXF CAD file of Development of Tetrahedron geometry drawing each time script is run


Tangent Handrailing Section Planes Tetrahedrons
50 pages of drawings on the study of Tangent Handrailing
 

Tangent Handrailing Fold out Poster Board Drawings
for 15 different plan tangents

Elucidation of the Tangent Handrailing Angles

Plan View Radius = 8.04494
Upper Tangent Angle = 50.56066° 
Lower Tangent Angle = 39.12785°
Central Angle = 105°
Plan View Angle = 75°

cord = 2 * Radius * sin(central angle ÷ 2) = 12.76496
tangent length = (Radius* sin(central angle ÷ 2)) ÷ cos(central angle ÷ 2) = 10.48436
lower tangent rise = tangent length * tan(lower  tangent angle)
lower tangent rise = 10.48436 * tan(39.12785)  = 8.52888
tangent base offset = lower tangent rise ÷ tan(upper  tangent angle)
tangent base offset= 8.52888 ÷ tan(50.56066)  =   7.015515
upper tangent base =tangent length + tangent base offset upper tangent base = 10.48436 + 7.015515 = 17.49988
upper tangent height = upper tangent base * tan(upper  tangent angle)
upper tangent height= 17.49988 * tan(50.56066)  = 21.2749033461
    

ordinate length = square root ((cord˛ +upper tangent base˛ ) - (2 * cord * upper tangent base * cos(central angle ÷ 2))
ordinate length = square root ((469.1900038160 ) - (2 * 12.76496 * 17.49988 * cos(52.5))
ordinate length = square root ((469.1900038160 ) - (271.9766701837) = 14.0432664873

angle D = arccos((upper tangent base˛ + ordinate length˛ - cord˛) ÷ (2 * upper tangent base * ordinate length) )
angle D = arccos((17.49988˛ + 14.0432664873˛ - 12.76496˛) ÷ (2 * 17.49988 * 14.0432664873) )
angle D = arccos((340.515206998) ÷ (491.511223015))  = 46.148445275

angle DD = 180° - central angle - angle D
angle DD = 180°- 105- 46.148445275 = 28.851554725

seat = (Radius* cos(angle DD)) + (Radius* cos(angle D))
seat= (8.04494 * cos(28.851554725)) + (8.04494 * cos(46.148445275))  = 12.6195363496

pitch of plank = arctan(upper  tangent height ÷ seat)
pitch of plank = arctan(21.2749033461 ÷ 12.6195363496)  = 59.3250488281

semi-major axis  = Radius ÷ cos(pitch of plank)
semi-major axis  = 8.04494 ÷ cos(59.3250488281)  = 15.7692298976

slant_angle = upper_tangent_angle
SS = arctan(tan(slant_angle) ÷ sin(Angle_DD))
S = atan(tan(slant_angle) ÷ sin(Angle_D))

Plan View Minor Axis Offset Length = radius * cos(Angle_D)
Section Plane Minor Axis Offset Length = Plan View Minor Axis Offset Length ÷ cos(SS)
major_axis = radius ÷ cos(SS)
utb = 90° - arctan(sin(upper_tangent_angle) ÷ tan(Angle_DD))
ltb = 90° - arctan(sin(lower_tangent_angle) ÷ tan(Angle_D))

R1 = arctan( tan(upper_tangent_angle) ÷ sin(Angle_DD))
R4Pm = arctan( cos(R1) ÷ tan(Angle_DD))
R4Pa = arctan( cos(R1) ÷ tan(Angle_D))
Angle of the Tangents = 180 - R4Pm - R4Pa




Understanding Tangent Handrailing

The file-- Tangent-Handrailing-Section-Planes-Tetrahedrons.pdf now contains:

Plates
1: Quarter Circle Plan with Equally Pitched Tangents.
2: Quarter Circle Plan with Short Lower Pitched Tangent.
3: Quarter Circle Plan with the Upper Tangent being Pitched with Level lower Tangent.
4: Quarter Circle Plan with the Upper Tangent  Level with Pitched lower Tangent.
5: Quarter Circle Plan with Short Upper Pitched Tangent.
6: - 15: Yet To do.
16: Quarter Circle Plan with Short Lower Pitched Tangent, correct way to find twist bevel angles.
17: Obtuse Plan with Short Lower Pitched Tangent, correct way to find twist bevel angles.
18: Acute Plan with Short Lower Pitched Tangent, correct way to find twist bevel angles.
19.1: Tangent Handrailing Folding Template. This is the correct way to make a folding template for equal or unequally pitched tangents.
19: Tangent Handrailing Folding Template.  Tetrahedron folding template for equal or unequally pitched tangents.
20: Tangent Handrailing trigonometry code to calculate the plan angles, major axis of the ellipse, upper and lower twist bevel angles and angle of the tangents.
21: Tangent Handrailing tetrahedron relationships to the dihedral angle and twist bevel angles.
22: Rotated Section Planes for Quarter Circle Plan with Equally Pitched Tangents.
23: Rotated Section Planes for Quarter Circle Plan with Unequal Pitched Tangents.
24: Rotated Section Planes for Obtuse Plan with Equally Pitched Tangents.
25: Rotated Section Planes for Obtuse Plan with Unequal Pitched Tangents.
26: Rotated Section Planes for Acute Plan with Equally Pitched Tangents.
27: Rotated Section Planes for Acute Plan with Unequal Pitched Tangents.
28: Step by step instructions for drawing Equally Pitched Tangents.
29: Tangent to Ellipse Geometric Drawing Instructions
30: The Tangent, Bevel and Elliptic Curve
From New Elements of Hand-Railing, by Robert Riddell 1871
31: The Ordinate, its Power and value in the Construction of Wreaths
From New Elements of Hand-Railing, by Robert Riddell 1871


Tangent handrailing wreath developed from both books. A Treatise on StairBuilding & Handrailing is theoretically geometrical correct, but you need A Simplified Guide to Custom StairBuilding and Tangent Handrailing to understand W & A Mowat.
Tangent handrailing wreath
Rail Block Laid out for Oblique Cut Method
Rail Block
Oblique Cut Method
Rail Block Oblique Cut Method
Squaring the Wreath
Rail Block

Development of Tetrahedron Geometry
based on Joe Bartok's Tetrahedron Developmental Geometry
Development of Tetrahedron Geometry
Development of Tetrahedron and Dihedral Angle for Polygons
Development of Tetrahedron for Compound Angles Geometry
Compound Miter and Bevel Calculator, To cut compound angles on the edge
Development of Tetrahedron for Purlin Geometry
Development of Tetrahedron using Hip Rafter Angle
Development of Tetrahedron using Jack Rafter Side Cut Angle
Development of Tetrahedron using Sheathing Angle
Development of Tetrahedron using Hip Rafter Backing Angle
Canadian & American Geometric Roof Framing Development Tetrahedron Calculator & DXF File
Development of Tetrahedron for Square Tail Fascia Geometry
Development of Tetrahedron for Hawkindale Formulas Geometry
Development of Tetrahedron for Crown Molding Geometry Using Spring Angle and Corner Wall Angle
Development of Tetrahedron for Crown Molding Geometry Using Slope Angle and Corner Plane Angle
Development of Tetrahedron for Crown Rake Molding Geometry

 X + 1 

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SBE Builders
SBE Builders
5305 Laguna Ct.
Discovery Bay, California 94505
(925) 634-6022 • Fax: (925) 634-6022
Commercial & Residential Framing Contractor
Build it Green Framing Contractor
American Carpenter and Builder
California Licensed General Building Contractor
CA LIC.#  546126
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Roof Framing Geometry

  • Roof framing geometry is empirical-type knowledge.
  • Information gained by means of observation, experience, or experiment.
  • From ground Line 0: Z1, Z2, Z3 etc...
  • To Roof Axis 0: A1, A2, A3 etc...
  • To the bring back line 0 for scribing: B1, B2, B3 etc...

Roof Framing Geometry Blog

Rafter Tools for Android
With Quad Tetrahedra Analytic Algorithm Technology.
Helping apprentice carpenters on their journey to becoming master carpenters with Certification in roof framing.

Rafter Tools+ for iPhone

Rafter Tools+ is a rafter calculator for complex roof framing angles that other roof framing calculators don't offer.

RafterTools+

Crown Molding Tools
crownmoldingtools.com

Crown Molding Tools  

Carpenters Of Steel

Canadian & American Geometric Roof Framing Development
using a Steel Framing Square

Geometric Roof Framing Development for Irregular Hip Roof Calculations
using a Steel Framing Square

Geometric Development of Tetrahedron
using a Steel Framing Square

Hip Rafter Backing Bevel Angles from 1565 to 2013

Applied Geometry and Euclidean GeometryA lesson in Applied Geometry and Euclidean Geometry

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