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Section 1 - Introduction

IFC 4.3.2 finalized the adoption of alignments for infrastructure works. Since the concept of alignment is relatively new to IFC, few details and examples are provided in the available literature. This guide aims to document the relevant mathematic concepts and equations a software developer would need to implement alignment-based geometry. It also provides example calculations.

Horizontal Alignment

A horizontal alignment typically consists of straight lines called tangents that are connect by curves. The curves are usually circular arcs. Easement or transition curves in the form of spirals can be used to provide gradual transitions between tangents and circular curves. The horizontal alignment is a plan view curvilinear path in a Cartesian coordinate system aligned with East and North. Positions along an alignment, measured along the plan view projection of the 3D alignment curve, are denoted with stations or chainage.

Vertical Alignment

A vertical alignment consists of straight sections of grade lines connected by vertical curves. The vertical curves are typically parabolas, though sometimes circular arcs or clothoid curves are used. A vertical alignment is defined along the curvilinear path of a horizontal alignment in a "Distance Along, Elevation" coordinate system. Combined, the horizontal and vertical alignments define a 3D curve. This combination of two 2D curves is sometimes referred to as a 2.5D geometry.

Cant Alignment

A cant alignment defines the cross-section rotation and superelevation of a rail line. The superelevation is a deviation from the vertical alignment caused by rotation about one of the rails.

Semantic Definition (Business Logic)

The semantic definition of alignment allows for the alignment to be described as close as possible to the terminology and concepts used in a business context. Examples include horizontal, vertical and cant layouts, stationing, and anchor points for domain specific properties such as design speed, cant deficiency, superelevation transitions, and widenings.

Specialized applications can analyze alignments using the semantic information. Alignments can be evaluated against design criteria such as speed requirements, sight distances, and maximum gradient, to name a few.

The IfcAlignment semantic definition is composed of an instance of IfcAlignmentHorizontal and IfcAlignmentVertical through an IfcRelNests relationship. The IfcAlignmentHorizontal and IfcAlignmentVertical are in turn composed of one or more IfcAlignmentSegment entities through IfcRelNests relationships. The IfcAlignmentSegment models the segment design parameters in a subtype of IfcAlignmentParameterSegment, IfcAlignmentHorizontalSegment and IfcAlignmentVerticalSegment in this case. The semantic definition model is shown in Figure 1.

Figure 1 IFC 4.3.2 semantic alignment model

IfcRelNests.RelatedObjects is an ordered collection. That means there is a specific order for the alignment layouts and the alignment segments. The IFC specification doesn't explicitly define a ranking for ordering. The ordering of the alignment layous can be infered from the images in the IfcAlignment documentation 5.4.3.1.5 as horizontal preceeds vertical, which preceeds cant. The ordering of IfcAlignmentSegment in an alignment layout is equally nebulus. For IfcAlignmentHorizontalSegment the only logical ordering is start to end. For IfcAlignmentVerticalSegment and IfcAlignmentCantSegment ordering in IfcRelNests.RelatedObjects could be anything because positional ordering can be derived from the StartDistAlong attribute. Such an approach is unnecessarily difficult. Segments should always be ordered in IfcRelNests.RelatedObjects in a start to end sequence.

An interesting caveat of IfcAlignmentHorizontal and IfcAlignmentVertical (and IfcAlignmentCant) is that the last IfcAlignmentSegment.DesignParameters must be zero length. If a geometric representation is provided, then its corresponding last segment must also be zero length. For a continuous composition of segments, the end of one segment is at the same location as the start of the next segment. The zero-length segment is intended to provide the end point of the last segment.

Geometric Representation

Overview

Section 4.1.7.1.1 of the IFC 4x3 specification indicates that the valid representations of IfcAlignment are:

  • IfcCompositeCurve as a 2D horizontal alignment (represented by its horizontal alignment segments), without a vertical layout.

  • IfcGradientCurve as a 3D horizontal and vertical alignment (represented by their alignment segments), without a cant layout.

  • IfcSegmentedReferenceCurve as a 3D curve defined relative to an IfcGradientCurve to incorporate the application of cant.

  • IfcOffsetCurveByDistances as a 2D or 3D curve defined relative to an IfcGradientCurve or another IfcOffsetCurveByDistances.

  • IfcPolyline or IfcIndexedPolyCurve as a 3D alignment by a 3D polyline representation (such as coming from a survey).

  • IfcPolyline or IfcIndexedPolyCurve as a 2D horizontal alignment by a 2D polyline representation (such as in very early planning phases or as a map representation).

IfcGradientCurve and IfcSegmentReferenceCurve inherit from IfcCompositeCurve. These curves consist of a sequence of segments defined by IfcCurveSegment. The mathematical computations for IfcCurveSegment geometry are the primary focus of this document.

IfcPolyline and IfcIndexedPolyCurve are trivial mathematically and are not discussed in this document.

IfcOffsetCurveByDistances is treated in a dedication section.

Understanding Geometric Representation of Alignment

The semantic definition of an alignment contains a lot of geometric information, such as segment type (line, curve, spiral) and parameters such as length, radius, and gradient. The geometric definition seems redundant. While related, the semantic definition and geometric definition serve different purposes.

Since the semantic and geometric definitions of an alignment are so similar, it's easier to understand the difference by examining a different type of entity. Consider a highway sign. The semantic definition describes an object as a sign that says "Exit 101" with an arrow pointing upwards to the right and it is positioned at a specific location (station and offset) relative to an alignment. The geometric definition describes the sign as a rectangle of certain dimension and thickness and is located precisely at a point X, Y, Z and the face of the sign oriented in a certain plane. The distinction between the business and geometric information is clearer in this example, the same concept of separate business logic and geometric definition are applicable to alignments.

The semantic definition of an alignment describes its business attributes (e.g. design speed) while the geometric definition describes the precise geometry (e.g. line from point A to point B followed by an arc of radius R starting at point B and ending at point C).

The geometric representation of the IfcAlignment consists of an IfcShapeRepresentation for the plan view horizontal alignment as well as a 3D curve for the combined horizontal and vertical alignment. This is illustrated in Figure 2, Figure 2.1 and Figure 2.2.

Geometrically, the horizontal alignment is defined by an IfcCompositeCurve and the vertical alignment is defined by an IfcGradientCurve. The horizontal alignment is the basis curve for the vertical alignment.

Figure 2 IFC 4.3.2 geometric representation of horizontal and vertical alignment

Figure 2.1 Geometric representations for the three basic alignment constructs (H, H+V, H+V+C).

Figure 2.2 Geometric representations for reusing horizontal alignment.

The horizontal alignment is defined by a sequence of straight tangent runs and circular horizontal curves. Transition spirals are used as well, primarily for rail but sometimes for roads.

The geometric elements are modeled with IfcCurveSegment. IfcCurveSegment cuts a segment from a parent curve and places it in the alignment coordinate system. A horizontal circular arc is modeled with an IfcCurveSegment that cuts an arc from an IfcCircle. Figure 3 shows an alignment consisting of a tangent run (red) and a horizontal curve (green). The line and curve segments are cut from their IfcLine and IfcCircle parent curves, respectively. The process of defining a parent curve, cutting a segment from that curve, and placing the cut segment into the alignment is repeated to define the entire horizontal alignment. All the horizontal alignment IfcCurveSegment objects are then combined to create an IfcCompositeCurve to complete the plan view geometric representation of the horizontal alignment.

A similar process is used to create the vertical alignment. IfcCurveSegment objects are cut from parent curves and combined to create an IfcGradientCurve. IfcGradientCurve is a sub-type of IfcCompositeCurve with the additional attribute BaseCurve. The segments of the gradient curve are positions in a 2-dimensional plane in a "distance along the horizontal alignment, elevation" coordinate system. The IfcGradientCurve uses the horizontal alignment IfcCompositeCurve as its base curve, which means the horizontal coordinates of the vertical alignment are taken from the horizontal alignment curve as illustrated in Figures 1 and 2.

IfcSegmentedReferenceCurve is also a sub-type of IfcCompositeCurve. It defines a deviating elevation and rail head cross slope along the base curve. The base curve can be either an IfcGradientCurve or IfcCompositeCurve. If it is an IfcGradientCurve, the "distance along" coordinate is along the gradient curve's base curve.

Figure 3 Illustration of parent curves and their placement with IfcCurveSegment.Placement

When defining an IfcCurveSegment, the location and orientation of the parent curve are not particularly important. The Placement attribute of IfcCurveSegment defines the location of the start point of the trimmed curve as well as the orientation of the tangent at the start of the trimmed curve.

For IfcGradientCurve and IfcSegmentedReferenceCurve the segments do not need to start or end at the same location of their base curve. These curves can be shorter than the horizontal alignment. This is illustrated in Figure 4 where the blue curve is the full horizontal curve and the orange curve is the portion of the horizontal curve coinciding with the vertical curve. The IfcGradientCurve (#770) has the IfcCompositeCurve (#657) as its base curve. The placement of the first IfcCurveSegment (#771) in the IfcGradientCurve is defined by IfcAxis2Placement2D (#777) with Location as IfcCartesianPoint (#778). The first segment is located at a distance of 250.0036 along the horizontal IfcCompositeCurve at an elevation of 145.3129 m.

#657=IFCCOMPOSITECURVE((#658,#671,#684,#697,#710,#723,#736,#749,#762),.F.)
#770=IFCGRADIENTCURVE((#771,#784,#796,#809,#821,#834,#846,#859,#871),.F.,#657,#887)
#771 = IFCCURVESEGMENT(.CONTSAMEGRADIENTSAMECURVATURE., #777, FCNONNEGATIVELENGTHMEASURE(0.),
IFCNONNEGATIVELENGTHMEASURE(40.0002408172751), #780);
#777 = IFCAXIS2PLACEMENT2D(#778, #779);
#778 = IFCCARTESIANPOINT((250.003634293681, 145.312908277025));

Figure 4 - Example of IfcGradientCurve with a start location different than the start of the IfcCompositeCurve BaseCurve.

Horizontal Alignment -- IfcCompositeCurve

The geometric representation of a horizontal is accomplished with an IfcCompositeCurve. The composite curve consists of a sequence of IfcCurveSegment entities. The curve segment geometry is cut from a parent curve as discussed above.

Table 1 maps the IfcAlignmentHorizontal.PredefinedType to the appropriate parent curve type.

Business Logic (IfcAlignmentHorizontal.PredefinedType) Geometric Representation (IfcCurveSegment.ParentCurve)
LINE IfcLine
CIRCULARARC IfcCircle
CLOTHOID IfcClothoid
CUBIC IfcPolynomialCurve
HELMERTCURVE IfcSecondOrderPolynomialSpiral
BLOSSCURVE IfcThirdOrderPolynomialSpiral
COSINECURVE IfcCosineSpiral
SINECURVE IfcSineSpiral
VIENNESEBEND IfcSeventhOrderPolynomialSpiral

Table 1- Mapping of business logic to geometric representation for horizontal alignment

A general algorithm for evaluating a point on an IfcCurveSegment is as follows:

  • Evaluate parent curve equation to get a point as well as tangent vector (RefDirection) and normal vector. The tangent vector is obtained by evaluating the derivative of the parent curve equation. The normal vector is a 90-degree counter-clockwise rotation of the tangent vector.

  • Translate the point so it is relative to (0,0). This is accomplished by subtracting the trimmed curve point at 0.0 from the start of the trimmed curve from the point under consideration.

  • Apply a rotation in the opposite direction of the parent curve axes so the tangent at the start of the trimmed curve is in the direction (1,0).

  • Apply the rotation of the curve segment placement.

  • Apply the transitional of the curve segment placement

This algorithm can be represented as follows:

  1. Evaluate the parent curve at IfcCurveSegment.SegmentStart. This results in a 4x4 matrix, $M_{PCS}$, for the tangent and normal vectors as well at the coordinate point.

  2. Create a 4x4 identity matrix. Assign -X, -Y, -Z from $M_{PCS}$, found in column 4, to column 4 of the new matrix. Call this $M_{T}$.

  3. Copy $M_{PCS}$ to a new matrix, $M_{R}$. Assign (0,0,0,1) to column 4. Multiply the values in (1,2) and (2,1) by -1. This is the "dy" value of the vectors. This operation is rotation of the curve to make the tangent direction (1,0).

  4. Represent the IfcCurveSegment.Placement in matrix form, $M_{CSP}$. This information is given in the IfcCurveSegment definition.

  5. For each point to be evaluated along the IfcCurveSegment

    a. Evaluate parent curve at distance from start of trimmed parent curve. Represent this point in matrix form, $M_{PC}$

    b. Map to curve segment, $M_{h} = M_{CSP}M_{R}M_{T}M_{PC}$

A numerical example of these calculations will be illustrated below.

Vertical Alignment -- IfcGradientCurve

The geometric representation of a vertical alignment is accomplished with IfcGradientCurve. This defines the vertical alignment in a "distance along, elevation" coordinate system. Distance along is a distance measured along IfcGradientCurve.BaseCurve=IfcCompositeCurve.

Business Logic (IfcAlignmentVertical.PredefinedType) Geometric Representation (IfcCurveSegment.ParentCurve)
CONSTANTGRADIENT IfcLine
CIRCULARARC IfcCircle
CLOTHOID IfcClothoid
PARABOLICARC IfcPolynomialCurve

Cant Alignment -- IfcSegmentedReferenceCurve

Cant alignment geometry is defined by a IfcSegmentedReferenceCurve. This defines a "Distance along, deviating elevation" coordinate system. Distance along is that of the base curve (IfcCompositeCurve) and deviating elevation is a deviation from the BaseCurve elevation (typically IfcGradientCurve, but could be IfcCompositeCurve). The curve segments of an IfcSegmentReferenceCurve also define railhead cross slope.

Business Logic (IfcAlignmentCant.PredefinedType) Geometric Representation (IfcCurveSegment.ParentCurve)
CONSTANTCANT IfcLine
LINEARTRANSITION IfcClothoid
HELMERTCURVE IfcSecondOrderPolynomialSpiral
BLOSSCURVE IfcThirdOrderPolynomialSpiral
COSINECURVE IfcCosineSpiral
SINECURVE IfcSineSpiral
VIENNESEBEND IfcSeventhOrderPolynomialSpiral

Mapping of Business Logic to Geometric Representation

The major topic in the sections that follow is mapping of the business logic (IfcAlignmentSegment) to its geometric representation (IfcCurveSegment). In general, the mapping formulas are given, not derived. The formulas have been taken from the reference implementation, EnrichIFC4x3, published at IFC-Rail-Unit-Test-Reference-Code/EnrichIFC4x3/EnrichIFC4x3/business2geometry at master · bSI-RailwayRoom/IFC-Rail-Unit-Test-Reference-Code (github.com).

In some cases, the EnrichIFC4x3 program uses incorrect formulas. These issues will be highlighted in this document, so the reader understands why the information presented herein deviates from the reference implementation.

The example calculations use the alignment models defined with the reference implementation repository at IFC-Rail-Unit-Test-Reference-Code/alignment_testset/IFC-WithGeneratedGeometry at master · bSI-RailwayRoom/IFC-Rail-Unit-Test-Reference-Code. A source document will be cited for each calculation example.

Units of measure

Unless otherwise specified, the unit of length measure all the calculations in this document is meter.