3-D Road Design by Applying Differential Geometry and Conventional Design Approach criteria
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Transcript of 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
3 Three Dimensional Road Design by Applying Differential
Geometry and Conventional Design Approach Criteria
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
4 Three Dimensional Road Design by Applying Differential
Geometry and Conventional Design Approach Criteria
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.
5 Three Dimensional Road Design by Applying Differential
Geometry and Conventional Design Approach Criteria
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
6 Three Dimensional Road Design by Applying Differential
Geometry and Conventional Design Approach Criteria
K. Amiridis
B. Psarianos
Basic Geometric Concepts Used
Regarding The Road Centerline
• Pseudogeodesic Curvature
• Pseudonormal Curvature
• B-spline Interpolation Curves
7 Three Dimensional Road Design by Applying Differential
Geometry and Conventional Design Approach Criteria
K. Amiridis
B. Psarianos
Pseudogeodesic Curvature Vectors
8 Three Dimensional Road Design by Applying Differential
Geometry and Conventional Design Approach Criteria
K. Amiridis
B. Psarianos
Pseudonormal Curvature Vectors
9 Three Dimensional Road Design by Applying Differential
Geometry and Conventional Design Approach Criteria
K. Amiridis
B. Psarianos
B-Spline Interpolation Curves
10 Three Dimensional Road Design by Applying Differential
Geometry and Conventional Design Approach Criteria
K. Amiridis
B. Psarianos
B-Spline Interpolation Curves
11 Three Dimensional Road Design by Applying Differential
Geometry and Conventional Design Approach Criteria
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
12 Three Dimensional Road Design by Applying Differential
Geometry and Conventional Design Approach Criteria
K. Amiridis
B. Psarianos
Ruled Surface
13 Three Dimensional Road Design by Applying Differential
Geometry and Conventional Design Approach Criteria
K. Amiridis
B. Psarianos
Ruled Surface
14 Three Dimensional Road Design by Applying Differential
Geometry and Conventional Design Approach Criteria
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
15 Three Dimensional Road Design by Applying Differential
Geometry and Conventional Design Approach Criteria
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
16 Three Dimensional Road Design by Applying Differential
Geometry and Conventional Design Approach Criteria
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
18 Three Dimensional Road Design by Applying Differential
Geometry and Conventional Design Approach Criteria
K. Amiridis
B. Psarianos
How Is The 3D Model Applied ?
• 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
K. Amiridis
B. Psarianos
Three Dimensional Road Design by Applying Differential
Geometry and Conventional Design Approach Criteria 19
How Is The 3D Model Applied ?
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
20 Three Dimensional Road Design by Applying Differential
Geometry and Conventional Design Approach Criteria
K. Amiridis
B. Psarianos
How Is The 3D Model Applied ?
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).
Spline degree 3 (Cubic Bezier curves)
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.
21 Three Dimensional Road Design by Applying Differential
Geometry and Conventional Design Approach Criteria
K. Amiridis
B. Psarianos
How Is The 3D Model Applied ?
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).
Spline degree 3 (Cubic Bezier curves)
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).
22 Three Dimensional Road Design by Applying Differential
Geometry and Conventional Design Approach Criteria
K. Amiridis
B. Psarianos
How Is The 3D Model Applied ?
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
23 Three Dimensional Road Design by Applying Differential
Geometry and Conventional Design Approach Criteria
K. Amiridis
B. Psarianos
How Is The 3D Model Applied ?
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
24 Three Dimensional Road Design by Applying Differential
Geometry and Conventional Design Approach Criteria
K. Amiridis
B. Psarianos
Case Study
25 Three Dimensional Road Design by Applying Differential
Geometry and Conventional Design Approach Criteria
K. Amiridis
B. Psarianos
Case Study
26 Three Dimensional Road Design by Applying Differential
Geometry and Conventional Design Approach Criteria
K. Amiridis
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
27 Three Dimensional Road Design by Applying Differential
Geometry and Conventional Design Approach Criteria
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
28 Three Dimensional Road Design by Applying Differential
Geometry and Conventional Design Approach Criteria
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
29 Three Dimensional Road Design by Applying Differential
Geometry and Conventional Design Approach Criteria
K. Amiridis
B. Psarianos
Hydroplaning Calculation
30 Three Dimensional Road Design by Applying Differential
Geometry and Conventional Design Approach Criteria
K. Amiridis
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
31 Three Dimensional Road Design by Applying Differential
Geometry and Conventional Design Approach Criteria
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
32
Three Dimensional Road Design by Applying Differential
Geometry and Conventional Design Approach Criteria
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
Three Dimensional Road Design by Applying Differential
Geometry and Conventional Design Approach Criteria 33
No More Approximations !!
• Direct And Accurate Solutions In All Aspects
• Fully Automatic Geometric Design
34 Three Dimensional Road Design by Applying Differential
Geometry and Conventional Design Approach Criteria
K. Amiridis
B. Psarianos
THANK YOU
FOR LISTENING!
35 Three Dimensional Road Design by Applying Differential
Geometry and Conventional Design Approach Criteria
K. Amiridis
B. Psarianos