Tangent Handrailing Trigonometric and Geometric Development

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.

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 ((cord² +upper tangent base² ) - (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 base² + ordinate length² - cord²) ÷ (2 * upper tangent base * ordinate length) )
angle D = arccos((17.49988² + 14.0432664873² - 12.76496²) ÷ (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