3-D Road Design by Applying Differential Geometry and Conventional Design Approach criteria

THREE DIMENSIONAL ROAD DESIGN BY APPLYING

DIFFERENTIAL GEOMETRY AND CONVENTIONAL

DESIGN APPROACH CRITERIA

Kiriakos Amiridis, Graduate Student National Technical University of Athens

Basil Psarianos, Professor National Technical University of Athens

January 13, 2015

Purpose?

• Establishment of an innovative 3D

mathematical model that cam carry out the

highway alignment location problem

– Fully Automatic

– Tedious trial and error procedure no longer

required

2 Three Dimensional Road Design by Applying Differential

Geometry and Conventional Design Approach Criteria

K. Amiridis

B. Psarianos

Why 3D Design ?

• A direct, physical and realistic design approach in 3D space fully utilizing modern computational tools

• An incorporation of a plethora of geometric elements concerning road safety that have not been reclaimed until now

• Overcoming inaccuracies or even deficiencies when implementing geometric design criteria with respect to the conventional 2D design approach. Instead, they are associated substantially with the 3D configuration of the road



K. Amiridis

B. Psarianos

Comparative Advantages In Relation To

Other Suggested 3D Methodologies

• It has the potential of a fully automatic geometric design process including all geometric design controls in a single effort.

• Since the problem is solved in space with all geometric design parameters controlled concurrently

• The method proposed is fully automated and no user interference is required

• The time consuming process of applying a trial and error procedure for satisfying all parameters of the design criteria is no longer required

• It can be applied to the entire road that is under study and not just for a section of it

• The implementation of this methodology allows for universal adjustment that is not constricted by regional idiosyncrasies



K. Amiridis

B. Psarianos

Mathematical Concepts Used

Regarding The Road Centerline

• Fundamental Theorem Of Curves

– Each curve (road centerline) is defined uniquely by its

space curvature and torsion through the Frenet differential

equations.

• Rigid Body Transformation

– Geometric transformation that places the road centerline in

the correct position in space.

• Procrustes Transformation

– Straightforward algorithm that calculates the required input

for the rigid body transformation.



K. Amiridis

B. Psarianos

How Are The Mathematical Concepts

Applied In Highway Geometric Design?

FUNDAMENTAL THEOREM OF CURVES

Mathematics Application in the Highway Geometric Design

Unique definition of the shape of a curve

by its space curvature and torsion

Definition of the modified shape of the road

centerline by imposing thresholds in the space

curvature

RIGID BODY TRANSFORMATION

Mathematics Application in the Highway Geometric

Design

Calculation of the coordinates from one

reference system to another

Definition of the correct absolute position

of the three-dimensional curve in space

through the Procrustes algorithm



K. Amiridis

B. Psarianos

Basic Geometric Concepts Used

Regarding The Road Centerline

• Pseudogeodesic Curvature

• Pseudonormal Curvature

• B-spline Interpolation Curves



K. Amiridis

B. Psarianos

Pseudogeodesic Curvature Vectors



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B. Psarianos

Pseudonormal Curvature Vectors



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B. Psarianos

B-Spline Interpolation Curves



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B. Psarianos

B-Spline Interpolation Curves



K. Amiridis

B. Psarianos

Basic Geometric Concepts Used

Regarding The Road Surface

• Ruled Surface

– Surface that is formed by straight lines called

rulings

• B-spline Interpolation Surface



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B. Psarianos

Ruled Surface



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B. Psarianos

Ruled Surface



K. Amiridis

B. Psarianos

Main Analogies Regarding The

Conventional Approach

CURVES

2-D Alignment 3-D Alignment

Horizontal Alignment

line segment

clothoid

circular arc Bezier curves

Vertical Alignment line segment

parabola

CURVATURE ANALOGUE

2-D Alignment 3-D Alignment

Curvature of the horizontal (plane) curves Pseudogeodesic curvature

Curvature of the vertical (plane) curves Pseudonormal curvature



K. Amiridis

B. Psarianos

Main Analogies Regarding The

Conventional Approach

SIGN OF THE PSEUDOGEODESIC CURVATURE

Mathematics Application in the Methodology

Positive value Right turn of the steering wheel (as the length

(stationing) of the road centerline increases)

Negative value Left turn of the steering wheel (as the length

(stationing) of the road centerline increases)

SIGN OF THE PSEUDONORMAL CURVATURE

Mathematics Application in the Methodology

Positive value 3D curves over crests

Negative value 3D curves in sags



K. Amiridis

B. Psarianos

What do highway elements represent

in terms of geometry?

K. Amiridis

B. Psarianos

Three Dimensional Road Design by Applying Differential

Geometry and Conventional Design Approach Criteria 17

Highway Engineering Geometry

Road Surface Ruled surface

Road Centerline Directrix of the ruled surface

Longitudinal Slope

The tangent of the angle which is formed between

the unit tangent vector of the road centerline at

each point and the horizon

Road superelevation rate Rotation of the rulings of the ruled surface around

its directrix (road centerline)

Right borderline Right boundary of the surface

Left borderline Left boundary of the surface

How Is The 3D Model Applied ?

• Definition of the 3D road centerline in 3 steps

– Definition of the 3D coordinates of the potential control points

– Definition of the values of the pseudogeodesic and pseudonormal curvatures by inputting thresholds that comply with current regulations and legislations

– Definition of the values of longitudinal slope by inputting thresholds that comply with current regulations and legislations



K. Amiridis

B. Psarianos


• Definition of the 3D road surface in 2 steps

– Approximation of the Road Surface as a Ruled

Surface

– Approximation of the Road Surface as a B-spline

Surface

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B. Psarianos




Road centerline 3D B-spline curve

Control Points (de Boor points) 3D coordinates corresponding to the

horizontal polygonal curve

Spline degree 3 (Cubic Bezier curves)

Absolute position in 3D space Correct

Purpose

Temporary road centerline

Enables calculations of geometrical

concepts

Calculation of the coordinate matrix

in the target system in the Procrustes

transformation

First Approximation of the

Road Centerline



K. Amiridis

B. Psarianos


Second Approximation of the

Road Centerline Road centerline 3D B-spline curve

Control Points (de Boor points)

3D coordinates that occur through the

implementation of the fundamental theorem of

curves (the curvature limits are taken under

consideration).


Absolute position in 3D space

False. The rigid body transformation must

be applied through the Procrustes

algorithm where the target system is the

matrix containing the 3-d coordinates of

the first approximation of the road

centerline.

Purpose

Imposition of limits to pseudogeodesic and

pseudonormal curvature. These two curvatures

are calculated based on the first approximation

of the road centerline.

Implementation (shape definition) Through the fundamental theorem of curves.



K. Amiridis

B. Psarianos


Third Approximation of the

Road Centerline Road centerline 3D B-spline curve

Control Points (de Boor points)

3D coordinates that occur through the implementation

of the fundamental theorem of curves (the longitudinal

slope limit is taken under consideration).


Absolute position in 3D space

False. The rigid body transformation must be

applied through the Procrustes algorithm where

the target system is the matrix containing the 3-d

coordinates of the second approximation of the

road centerline.

Purpose

Imposition of the longitudinal slope limits depending

on whether the location is an upgrade or downgrade.

The longitudinal slope is calculated from the second

approximation of the road centerline.

Implementation (shape definition)

Through the fundamental theorem of curves (the

change regarding the longitudinal slope leads to a

modification of the 3D curvature).



K. Amiridis

B. Psarianos


First Approximation of the

Road Surface

Road Surface Ruled Surface

Base Curve or Directrix Third-final approximation of the road centerline

Rulings Perpendicular to the road centerline

Width of the rulings Semi width of the road surface for the

specific road category

Rotation of the rulings in relation to the base

curve and the horizon

Superelevation rate depending on the

pseudogeodesic curvature

Absolute position in 3-D space Correct

Purpose Definition of the initial Road Surface



K. Amiridis

B. Psarianos


Second Approximation of the

Road Surface

Road Surface 3D B-Spline Surface

Control Points (de Boor points) 3D coordinates of the ruled surface

Spline degree 3

Absolute position in 3-D space Correct

Purpose Definition of the final 3D Road Surface



K. Amiridis

B. Psarianos

Case Study



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B. Psarianos

Case Study



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B. Psarianos

Is The Methodology Practical?

• Friction will be observed because the methodology seems to

differ significantly from the 2D conventional approach

• However, closer inspection reveals several analogies between

the conventional approach and the suggested one

• Moreover, nowadays this methodology is feasible due to

modern computational tools and advanced technology

regarding 3D geometric modeling



K. Amiridis

B. Psarianos

A Long-Lasting Problem Has Been

Solved

• The 3D geometric alignment has been considered a crucial

issue for years now regarding highway engineering, but no

convincing solution had been suggested until now

• The methodology has dealt with the heart of the 3D highway

alignment location problem

• It can be applied immediately and directly

• All the required restrictions and limitations are taken under

consideration

• It forms the basis of an integrated optimization model



K. Amiridis

B. Psarianos

Some Applications In 3D Space

• Hydroplaning

– Calculation of the water film thickness (WFT)

• Sight Distance Calculation

– Avoids inaccuracies derived from the conventional

approach

• 3D Cross Section Representation

• Volume Excavations

• Road Safety



K. Amiridis

B. Psarianos

Hydroplaning Calculation



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B. Psarianos

Road Safety

• This methodology can evaluate road safety for

geometric concepts pertaining to the road surface and

not just to the road centerline

• It forms new road safety indicators which clearly

have geometric orientation and refer to the whole

road surface (e.g. Gaussian curvature)

• It will substantially contribute in reinforcing road

safety



K. Amiridis

B. Psarianos

Software Implementation

• The whole methodology is implemented through the

software Mathematica

– High capabilities concerning the visualization and plotting

of curves and surfaces

– Mathematica has many built-in commands and libraries

concerning concepts of differential geometry

– B-spline curves are embedded in the software

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K. Amiridis

B. Psarianos

Software Implementation

• A user friendly integrated software environment must

be developed that will provide the user with all the

necessary help required (e.g. function navigator,

virtual book) in order to completely embrace its

significance and to fully leverage its capabilities.

• Provision of Stereoscopic View

K. Amiridis

B. Psarianos



No More Approximations !!

• Direct And Accurate Solutions In All Aspects

• Fully Automatic Geometric Design



K. Amiridis

B. Psarianos

THANK YOU

FOR LISTENING!



K. Amiridis

B. Psarianos