8.7.2.3 IfcAlignmentVerticalSegmentTypeEnum
8.7.2.3.1 Semantic definition
The IfcAlignmentVerticalSegmentTypeEnum indicates the type of a segment of a vertical alignment segment (IfcAlignmentVerticalSegment).
Vertical curvature  Segmenttype  Enumeration Values 

No vertical curvature  constant gradient  CONSTANTGRADIENT 
Derivative of gradient with respect to horizontal projection of alignment is constant  Vertical curve, parabola  PARABOLICARC 
Derivative of vertical angle with respect to 3D arc length along the alignment is constant  Vertical curve, circular  CIRCULARARC 
Variation of vertical curvature is constant  Vertical curve, clothoid  CLOTHOID 
Used Symbols and their meaning
Symbol  meaning  Unit, value range 

L  full length of segment  positive length L > 0 
s  current position on segment  0 < s < L 
θ  (Greek "theta") Longitudinal slope angle (incline or decline)  rad 
g  gradient (math); g=tan(θ)  
x(s)  variable longitudinal coordinate of the projection of the alignment / track centreline into the ground plan.  length 
y(s)  variable transverse coordinate of the projection of the alignment / track centreline into the ground plan.  length 
z(s)  Variable vertical coordinate of the projection of the track centreline in plan in a Cartesian coordinate system in the vertical direction.  length 
z_{c}(s)  Ordinate of the vertical circular arc of measured away from the tangent line at position s.  length 
L_{V}  length of vertical radius radius (inverse curvature)  length 
R_{V}  radius (inverse curvature) of the track centreline at a point in the elevation diagram (longitudinal section)  length 
κ_{V}  (Greek "kappa") Vertical curvature  1/radius_{V} 
Z_{G}  Distance of the tangent intersection from the chord of the vertical circular arc  length 
Z_{M}  Distance of the centre of the vertical circular arc to the tangent intersection point (stitch height)  length 
l_{T}  Length of the tangents of the vertical circular arc  length 
References to EN 13803/2017
EN 13803/2017 covers "Track alignment design parameters". As such it is not fully compatible with definitions for IFC Alignment. Therefore rail specific terms like track have been replaced with more general terms also applicable to road design.
Referenced content of EN 13803/2017 "Table 2  Elements for vertical alignment" has been modified as follows:
Vertical curve, parabola: Derivative of gradient with respect to chainage is constant
Generalized: Derivative of gradient with respect to horizontal projection of alignment is constant
Vertical curve, circular: Derivative of vertical angle with respect to sloping length along the track is constant
Generalized: Derivative of vertical angle with respect to 3D arc length along the alignment is constant
EN13803 clause 3.5: Chainage: longitudinal distance along the horizontal projection of the track centre line.
8.7.2.3.2 Type values
Type  Description 

CIRCULARARC

Vertical alignment segment where the derivative of vertical angle with respect to sloping length along the track (3D length) is constant. The curvature for vertical circular arc segment is provided by: $$ \kappa_v = \frac{1}{R_v(s)} = \frac{d\theta}{ds} $$ The length for vertical circular arc segment is provided by: $$ l_V =\Delta s_v = \frac{\Delta \theta}{\kappa_v} = \Delta \theta \cdot R_v $$ The distance between point on segment to tangent is provided by: $$ z_c(s) = \frac{s^2}{2\cdot R_v} $$ 
CLOTHOID

Vertical alignment segment where the derivative of vertical angle with respect to sloping length along the track (3D length) obeys a linear change. The curvature equation of the vertical clothoid segment is provided by: $$ \displaylines { \xi = \frac{s}{L} \\ \kappa_v(s) = \kappa_{v1} + \xi \Delta \kappa_v } $$ 
CONSTANTGRADIENT

Vertical alignment segment with constant gradient. 
PARABOLICARC

Vertical alignment segment where the derivative of gradient with respect to distance along is constant.
General equation of the parabolic arc segment is provided by: $$ y = a x^2 + b x + c $$ The gradient (slope) of this curve at any point (first derivative) is provided by: $$ \frac{dy}{dx} = 2 a x + b $$ The rate of change of gradient of the parabolic arc segment is constant. The variation of curvature is therefore provided by: $$ \frac{d^2y}{d^2x} = 2 a $$ 
8.7.2.3.3 Formal representation
TYPE IfcAlignmentVerticalSegmentTypeEnum = ENUMERATION OF
(CIRCULARARC
,CLOTHOID
,CONSTANTGRADIENT
,PARABOLICARC);
END_TYPE;