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.
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 semimajor axis = Radius ÷ cos(pitch of plank) semimajor 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 TangentHandrailingSectionPlanesTetrahedrons.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 HandRailing, by Robert Riddell 1871 31: The Ordinate, its Power and value in the Construction of Wreaths From New Elements of HandRailing, by Robert Riddell 1871
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