3D Solids modeling the
Tangent Handrailing Planes and Angles


Lower Tangent Plane Slope Angle = 50
Upper Tangent Plane Slope Angle = 40
Angle negotiated by Handrailing = 60
(Angle between Tangents in Plan View = 120)


Tetrahedron modeling the Lower (50) Tangent Plane Angles
... not seen on the developed drawing to the scale of the model, a downsized version
of this tetrahedron has a vital role to play and will make an appearance later.


Development of the Tetrahedron
Lower Tangent Plane Slope Angle = r5P = 50
Lower Tangent Plane Plan Angle = dd = 76.73958
Angle produced on Oblique Plane by trace
of Lower Tangent Plane = r4P = 8.47866
Dihedral Angle measured between the
Level (Plan View) Plane and the Oblique Plane = R1 = 50.76055
Dihedral Angle measured between the
Lower Tangent Plane and the Oblique Plane = 90 a5P = 79.76680

Development ... Tetrahedron modeling the Lower (50) Tangent Plane Angles

Elevation ... View of 50 Tangent Plane
Side View ... Tetrahedron modeling the Lower (50) Tangent Plane Angles

View of Oblique Plane ... r4P = 8.47866
View of Oblique Plane ... Tetrahedron modeling the Lower (50) Tangent Plane Angles

Tetrahedron modeling the Upper (40) Tangent Plane Angles
... the location of this group of triangles on the drawing is self-evident.

Development of the Tetrahedron
Upper Tangent Plane Slope Angle = R5P = 40
Upper Tangent Plane Plan Angle = DD = 43.26042
Angle produced on Oblique Plane by trace
of Upper Tangent Plane = R4P = 33.90856
Dihedral Angle measured between the
Level (Plan View) Plane and the Oblique Plane = R1 = 50.76055
Dihedral Angle measured between the
Upper Tangent Plane and the Oblique Plane = 90 A5P = 55.66480

Development ... Tetrahedron modeling the Upper (40) Tangent Plane Angles

Elevation ... View of 40 Tangent Plane
Side View ... Tetrahedron modeling the Upper (40) Tangent Plane Angles

View of Oblique and Section Planes
Vertex of the Oblique Plane Triangle ... Angle R4P = 33.90856
Angle R1 = 50.76055 on the Section Plane is the dihedral angle
measured between the Plan View (Level) Plane and the Oblique Plane.

View of Oblique and Section Planes ... Tetrahedron modeling the Upper (40) Tangent Plane Angles

Views of the Tetrahedra modeling the Tangent Plane Angles
juxtaposed along their respective Section Planes

... these tetrahedra appear on the development on a smaller scale (see below).

Top View of Tetrahedra modeling the Tangent Planes juxtaposed along their Section Planes
Oblique View of Tetrahedra modeling the Tangent Planes juxtaposed along their Section Planes

Pentahedron modeling the Tangent Plane Angles
... the tetrahedron of the upper (40) tangent plane angles with scaled
down juxtaposed tetrahedra extracted from the lower right hand corner.


Development of all Planes and Angles
Development of all Planes and Angles

Development of Pentahedron extracted from the drawing above
Development ... Pentahedron modeling the Tangent Plane Angles

View of Oblique Plane ... line K P is tangent to the ellipse at point T2.
Angle K P T1 is the "Angle between Tangents"; the angle
produced on the oblique plane by the traces of the tangent planes
= 180 (r4P + R4P) = 180 (8.47866 + 33.90856) = 137.61278.

View of Oblique Plane ... Pentahedron modeling the Tangent Plane Angles

View from Corner at Point of Tangency (T1)
Note that the dihedral angle measured between the lower
tangent plane and the oblique plane is 90 + a5P = 100.23320.

View from Corner at Point of Tangency (T1) ... Pentahedron modeling the Tangent Plane Angles

Views of the Tangent Planes intersecting along plumb line V P
View of Tangent Planes ... Pentahedron modeling the Tangent Plane Angles
View of Tangent Planes ... Pentahedron modeling the Tangent Plane Angles

Juxtaposed Tetrahedra modeling the Tangent Plane Angles
... the scaled down pair of tetrahedra removed from the lower right hand corner
of the upper tangent plane tetrahedron to produce the pentahedron above.
The tetrahedra are shown sitting on the construction of the twist angles
(the dihedral angles measured between the oblique plane and the tangent planes).


Development of the Juxtaposed Tetrahedra
Development ... Juxtaposed Tetrahedra modeling the Tangent Plane Angles

View of the Oblique Plane
View of the Oblique Plane ... Juxtaposed Tetrahedra modeling the Tangent Plane Angles

Views of the Tangent Planes
View of the Lower (50) Tangent Plane ... Juxtaposed Tetrahedra modeling the Tangent Plane Angles
View of the Upper (40) Tangent Plane ... Juxtaposed Tetrahedra modeling the Tangent Plane Angles

Prismatic Solid (Post-Type Model)
... beginning with the pentahedron, raise a plumb plane through
points TP and CC, and a plumb plane through points T1 and CC.


Development of all Planes and Angles
Development of all Planes and Angles

Development of Prismatic Solid extracted from the drawing above
Development ... Prismatic Solid modeling the Tangent Handrailing Angles

View of Oblique Plane
View of Oblique Plane ... Prismatic Solid modeling the Tangent Handrailing Angles

View of Corner at Point of Tangency (T1)
As in the pentahedron, the dihedral angle measured between the
lower tangent plane and the oblique plane is 90 + a5P = 100.23320.

View from Corner at Point of Tangency (T1) ... Prismatic Solid modeling the Tangent Handrailing Angles

View of plumb line through Points CC and EC
View from plumb line through Points CC and EC ... Prismatic Solid modeling the Tangent Handrailing Angles

View of the Tangent Planes and plumb line through Points V and P
View of the Tangent Planes and plumb line through Points V and P ... Prismatic Solid modeling the Tangent Handrailing Angles

View of plumb line through Points TP and T2
(T2 is the second Point of Tangency on Oblique Plane)

View of plumb line through Points TP and T2 ... Prismatic Solid modeling the Tangent Handrailing Angles

Juxtaposed Prismatic Solid and Tetrahedra modeling the Tangent Plane Angles
... the dihedral angles between the oblique plane and tangent planes may be measured on any of the solids.
These dihedral angles are the twist angles seen on the handrailing layout on the end grain of the block.

All of the solids above are found in the developed drawing and describe the same group of planes.


View of the Oblique Plane
... it's always the same surface, angled at 50.76055 with respect to level.

Juxtaposed Prismatic Solid and Tetrahedra modeling the Tangent Plane Angles

Joe Bartok