Highway Geometric Modeling Method byCurvature and Gradient Functions
Koji Makanae
Miyagi University, JapanABSTRACT: Highway alignments are designed as the sequences of elements such as lines, arcs and spirals.
This method makes the difficulty to build and edit the 3-D highway models on CAD systems, because horizontal/vertical alignments and cross-sections are defined individually and each kind of elements has different
parameters for shape definition. This paper shows that highway alignments can be determined by functions ofcurvature and gradient, and shows the schema to build the 3-D geometric models. Utilizing this method, the3-D highway geometric models can be built with fewer data systematically, and it can be applied to the highway product models for computer-based design in future.
1 INTRODUCTION
The 3-D highway geometry is the composition ofhorizontal alignments, vertical alignments and crosssections. In the current design method, a horizontalalignment is designed as a sequence of elements
such as lines, spirals, and curves (arcs), and a vertical alignment is designed as a sequence of elements
such as lines and arcs. This method makes the difficulty to build and edit the 3-D highway models on
CAD systems because horizontal/vertical alignments
and cross-sections are defined individually and eachkind of elements has different parameters for shapedefinition. Furthermore this problem is an obstacle
to build auto-designing systems for highway.
This short paper proposed the highway geometricmodeling method utilizing curvature/gradient functions.
2 FUNCTIONALIZATION OF HIGHWAY
ALIGNMENTOn the assumption of an automobile traveling at aconstant velocity, a time-velocity-curvature system
can be expressed by distance. Time-velocitycurvature and time-velocity-gradient systems can beregarded to be equivalent to distance-curvature anddistance-gradient systems, respectively. Here, functions with the horizontal distance along a highwayalignment as a parameter in each system are definedas the curvature and gradient functions. The method
proposed in this paper handles these curves as functions that determine highway alignments and thereby

tries to design highway alignments in a process reverse to conventional methods. Horizontal and curvature functions can be represented by continuous
piecewise linear functions and can be specified by
the list of turning points of curvature and gradientwith the distance.
A driver traveling along a continuous highwayalignment turns the steering wheel continuously.
Then curvature function Θ(l ) is becomes also continuous. Θ(l ) may thus be regarded as a continuous
piecewise linear function. Θ(l ) at distance l in section n (n = 1, 2, 3…) is expressed as shown in Equation 1. Curvature function Θ(l ) can be defined bygiven (ln, θ n ) as a boundary condition.
Θ(l ) = ∆θ n (l − l n −1 ) + Θ(l n −1 )θ n − θ n −1∆θ n = l − lnn −1(1)where,
ln : distance at the end of section n (where l0 =0)θ n : curvature at the end of section n ( θ 0 is theinitial curvature)∆θ n : rate of change in curvature in section n
For existing design methods, elements of a verticalalignment consist of straight lines and verticalcurves. For vertical curves, parabolic curves aregenerally used. Gradient, which is a differential ofvertical alignment, is then represented by a linear
expression. Since vertical gradient is continuous aswell as horizontal alignment, gradient function J(l)at distance l in section n ( n = 1,2,3…) is represented

Layer 1
Basic
Conditions

Layer 2
DesignConditions

design speed

max.crossfall

Layer 3
Expansion
Condition

Layer 4
DesignInformationsight distance

Layer 5
Supplementary
Information

gradient
function J(l)max. rate ofchange ingradientmin. length oftransition curve

Layer 6
Expansion

3D coordinates ofcenter line

mileage interval
for the expansion
of 3D geometry
Cuvature function
Θ(l)max. curvaturecornering force
coefficientmax. crossfall
vehicle type

Super elevation
I'(l)length ofcrossfalltransitions

3D coordinates ofthe edge of lanes

lane widthWl(l)standardlane width
standard
shoulder widthroad width(one side)W(l)standard
crossfall

3D coordinates ofthe edge of shoulders

functions for defining highway geometry

design conditions

Figure 1. Schema of highway geometric modelby equation shown in Equation 2. Thus, gradientfunction J(l) can be defined by boundary condition(ln , jn )
 J (l ) = ∆j n (l − l n −1 ) + J (l n −1 )j n − j n −1∆j n = l − lnn −1(2)

where,
jn : gradient at the end of section n ( j0 is the initial gradient)∆j n :rate of change in gradient in section n
3 SCHEMA OF GEOMETRIC DESIGN MODEL
Based on the boundary conditions for curvature andgradient functions, 3D coordinates of the highwayalignment can be defined. Crossfall and widths ofthe lane can be determined by the curvature functions automatically. Thus, a 3D highway geometric
design model can be built in the design space.
The modeling process of the 3D highway geometry
can be classified into 6 layers schematically (Figure1).
4 BENEFITS OF GEOMETRIC DESIGNSYSTEM
The highway geometric model using curvature andgradient functions produces the following benefits.
-Setting only of curvature and gradient functions enables automatic definition of geometric design andimmediately provides the 3-D geometric model.

-Curvature and gradient functions are always continuous and guarantee the continuity of the highway alignment.
-Transmission of precise design information is possible only with a few data. Data processing is alsoeasy.
5 CONCLUSIONSThis paper shows that highway geometric design canbe determined basically by curvature and gradientfunctions with distance l as a variable.
In the field of highway engineering as well as inother fields, product models are now being developed (OKSTRA, LandXML etc.). For defining highway alignments, however, existing paper-based
methods based on alignment parameters are still being used. The method proposed in this paper canbuild the 3-D highway geometric models with fewer
data, thus it can be applied to the product models forcomputer-based design in future.
REFERENCES
Autodesk. 2000. LandXML Schema Version 1.0.
http://www.landxml.org.
Makanae,K. 2002. Functionalization of Highway Alignment
for Computer-Based Design. Proceedings of the 9th International Conference on Computing in Civil and Building
Engineering: 877-882.
OKSTRA. 2001. Objektkatalog für das Straßen und Verkehrswesen. http://www.okstra.de/.