Notation.md

Notation

Scalars and Parameters

Arc-Length and Distance

Symbol	Definition
$s$	Arc-length parameter along the parent curve
$s_0$	Arc-length at the start of the trim; equals SegmentStart
$L$	Segment length; equals SegmentLength
$\ell$	Horizontal distance from the segment start; evaluation variable in vertical and cant sections
$d$	Distance measured along the horizontal IfcCompositeCurve
$d_0$	Distance along at the trim start; $d_0 = x(s_0)$
$h_l$	Horizontal length of a vertical segment; equals HorizontalLength

Angles

Symbol	Definition
$\theta(s)$	Bearing angle at arc-length $s$ in the horizontal plane
$\theta(\ell)$	Tangent angle at horizontal distance $\ell$; grade angle $\tan^{-1}(g(\ell))$ in the vertical plane; slope of deviating elevation $\tan^{-1}(D'(\ell))$ in the cant plane
$\theta_0$	Tangent angle at the trim start $s_0$
$\theta_p$	Tangent angle of IfcCurveSegment.Placement.RefDirection
$\theta_v$	Grade direction angle; $\theta_v = \tan^{-1}(dy_v / dx_v)$
$\phi$	Cross-slope angle; angle from the transverse $y$-axis to the Axis vector
$\phi_s,\ \phi_e$	Cross-slope angle at the start and end of a cant segment
$\phi_p$	Cross-slope angle of IfcCurveSegment.Placement.Axis
$\Delta$	Sweep angle of a circular arc; angle subtended at the center by the arc from the trim start to an evaluation point
$\Delta_0$	Angular position of the trim-start point on the parent circle; angle from the circle's positive $x$-axis to the radius vector at the trim start (Section 3.4)
$\Delta_{pc}$	Radial angle at the evaluation point on the parent circle; $\Delta_{pc} = \Delta_0 - \Delta$ for a clockwise segment, $\Delta_0 + \Delta$ for counter-clockwise (Section 3.4)

Curvature and Radius

Symbol	Definition
$\kappa(s)$	Curvature at arc-length $s$; $\kappa = 1/R$
$\kappa_s$	Curvature at the segment start; $\kappa_s = 1/R_s$
$\kappa_e$	Curvature at the segment end; $\kappa_e = 1/R_e$
$R$	Radius of curvature
$R_s$	Radius at the segment start; equals StartRadiusOfCurvature
$R_e$	Radius at the segment end; equals EndRadiusOfCurvature
$f$	Cumulative change in curvature over the segment; $f = L(1/R_e - 1/R_s)$

Grade

Symbol	Definition
$g(s)$	Gradient (slope) at arc-length $s$
$g_s$	Gradient at the segment start; equals StartGradient
$g_e$	Gradient at the segment end; equals EndGradient

Position and Elevation

Symbol	Definition
$x(s),\ y(s)$	Coordinates of the parent curve at arc-length $s$
$x_0,\ y_0$	Parent curve position at the trim start $s_0$
$x_p,\ y_p$	Placement location; equals IfcCurveSegment.Placement.Location
$z$	Elevation (vertical $y$-coordinate mapped to 3D $z$)
$z_0$	Elevation at the trim start; equals StartHeight
$C_x,\ C_y$	Center coordinates of a circular arc parent curve
$c$	Chord length from the trim start to an evaluation point

Cant (Deviating Elevation)

Symbol	Definition
$D(s)$	Deviating elevation at arc-length $s$; vertical offset of the track centerline from the gradient curve
$D_0$	Deviating elevation at the trim start; $D_0 = D(s_0)$
$D_s,\ D_e$	Deviating elevation at the segment start and end; $D_s = (D_{sl} + D_{sr})/2$
$D_{sl},\ D_{sr}$	Left and right cant elevation at the segment start; equals StartCantLeft, StartCantRight
$D_{el},\ D_{er}$	Left and right cant elevation at the segment end; equals EndCantLeft, EndCantRight
$D_p$	Deviating elevation component of IfcCurveSegment.Placement.Location
$\Delta D$	Change in deviating elevation over the segment; $\Delta D = D_e - D_s$
$D_{rh}$	Rail head distance; center-to-center distance between railheads; equals RailHeadDistance
$\beta_s$	Cross-slope ratio at the segment start; $\beta_s = (D_{sr} - D_{sl}) / D_{rh}$
$\beta_e$	Cross-slope ratio at the segment end; $\beta_e = (D_{er} - D_{el}) / D_{rh}$
$h_{cg}$	Gravity centerline height; height of the vehicle center of gravity above the rail plane; equals GravityCenterLineHeight
$cf$	Cant factor; $cf = -420,(h_{cg}/L)(\beta_e - \beta_s)$; scales the Viennese Bend polynomial coefficients

Spiral Curve Coefficients

Symbol	Definition
$A$	Clothoid constant
$u$	Clothoid unit parameterization parameter; $u = s,/,\lvert A\sqrt{\pi}\rvert$
$A_i$	Dimensional polynomial coefficient (with units) for spiral and parabolic curves
$a_i$	Normalized (dimensionless) polynomial coefficient prior to scaling
$A_{i1},\ A_{i2}$	Coefficient $A_i$ for the first and second halves of a Helmert segment

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Vectors

Symbol	Definition
$dx,\ dy$	Horizontal tangent direction components; $dx = \cos\theta$, $dy = \sin\theta$
$dx_p,\ dy_p$	Tangent direction components at the placement; $dx_p = \cos\theta_p$, $dy_p = \sin\theta_p$
$dx_v,\ dy_v$	Grade direction components from $M_v$; $dx_v = \cos\theta_v$, $dy_v = \sin\theta_v$
$\mathbf{RefDir}_p$	Reference direction of IfcCurveSegment.Placement; equals Placement.RefDirection
$\mathbf{Axis}_p$	Axis direction of IfcCurveSegment.Placement; equals Placement.Axis
$\mathbf{X}_p,\ \mathbf{Y}_p,\ \mathbf{Z}_p$	Orthonormal basis vectors of the placement frame

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Matrices

All matrices are $4 \times 4$ homogeneous transformation matrices. Column 4 carries position; columns 1–3 carry the frame orientation.

Symbol	Definition
$M_{CSP}$	Curve segment placement matrix; constructed from IfcCurveSegment.Placement; maps the trimmed segment into the alignment coordinate system
$M_N$	Normalization matrix; translates the trim-start point to the origin and rotates the tangent to align with the positive $x$-direction
$M_{PC}$	Parent curve matrix at the evaluation point; encodes $x(s)$, $y(s)$, and $\theta(s)$
$M_{PCS}$	Cant parent curve matrix evaluated at the trim start $s_0$
$M_{PC\ell}$	Cant parent curve matrix evaluated at the point under consideration $\ell$
$M_h$	Horizontal placement matrix; $M_h = M_{CSP},M_N,M_{PC}$
$M_v$	Vertical placement matrix; $M_v = M_{CSP},M_N,M_{PC}$ in the (distance-along, elevation) plane
$M'_v$	Modified vertical matrix; rows 2 and 3 of $M_v$ swapped, distance-along component zeroed, for multiplication with $M_h$
$M''_v$	Orientation-only form of $M'_v$; column 4 set to $(0,0,0,1)^T$ for use in the cant 3D composition
$M_c$	Cant placement matrix; $M_c = M_{CSP},M_N,M_{PC}$ in the (distance-along, deviating elevation) plane
$M'_c$	Modified cant matrix; deviating elevation moved from row 2 to row 3, distance-along component zeroed, for multiplication
$M''_c$	Orientation-only form of $M'_c$; column 4 set to $(0,0,0,1)^T$ for use in the cant 3D composition
$M_{3D}$	Full 3D placement matrix combining horizontal and vertical; $M_{3D} = M_h,M'_v$
$M_{3Dcant}$	Full 3D placement matrix combining horizontal, vertical, and cant; $M_{3Dcant} = M_h,M''_v,M''_c$
$I$	$4 \times 4$ identity matrix