International Journal of Geosciences, 2011, 2, 689-694
doi:10.4236/ijg.2011.24070 Published Online November 2011 (
Copyright © 2011 SciRes. IJG
Simulation Topographical Surfaces Geographical and
Geological Using Differential Geometry
Mohammedi Ferhat, Bensaada Said
Laboratory L.A.R.H.Y.S.S, University Med Khider B.P, Biskra, Algerie
Received June 9, 2011; revised August 21, 2011; accepted October 23, 2011
By applying differential geometry to analogue models developed such a model is calculated for the geomet-
rical shape. Dip measurements are critical data for geologists, and in particular for structural studies. They
enable quantifying geologic features observed across the surface in order to model the sub-surface. Dip
measurements are provided by direct or indirect sources: geological maps, fieldwork data, Digital Elevation
Model (DEM). This quantification then allows for comparison of such models to measured field data and
supplants the use interferometry Radar describes and compares 3-D deformations. This example supplements
and is based on the material found in L.S.S.I.T. Theory as well as some of the experimental results with the
new method are delineated.
Keywords: Structural Geology, Topography Surfaces, Differential Geometry, Modeling Images Optically
1. Introduction
We present a mathematical analysis of interference phe-
nomena for shape recognition. The basic theoretical concept
—and tool—will be the contour interferences fringes
function. We show that the mathematical analysis is greatly
simplified by the systematic recourse to this tool. By
deforming analogue theoretical models made of starting
from the models carried out at the laboratory (e.g., labo-
ratory-test), it is possible to create intricate 3D surfaces
for comparison with optically fold surfaces. If the model
surface resembles the topographical surface, inferences
on the sequence and directions of Loading can be drawn
for the deformation of the assembly experimental surface
from the known boundary conditions of the optical images
of the assembly model. However, establishing geometric
similarity of the model and optically surfaces requires both
surfaces to be mapped in three dimensions. Surfaces quanti-
fied using differential geometry can be compared precisely
for various geometric similarities and differences as a pre-
requisite for kinematic and dynamic comparisons. (Figure 1)
The purpose of this paper is to report an alternative method
for analyzing the derivative of the surface deformation is
then retrieved from a series of frames of the sheared images
collected as the topographical surface is deformed, the sur-
face slope is extracted. Some limitations of this method
along with possible solutions are also presented.
Figure 1. Topographical surfaces with curves opposed lo-
cally 3D.
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2. Generation of the Analogue Surface
Mathematically, pattern recognition is a classification
problem. It usually denotes classification and/or descrip-
tion of a set of process or events. The set of processes or
events to be classified could be a set of physical objects.
The mathematical approach of topographic surfaces ori-
ginals from the optical interferences with the differential
geometry was simulated with exactitudes on the basis of
the models of laboratory. The topographically surfaces
considered in this example (Figure 2) was created by
Laboratory-test, for the 3-D visualization of interfer-
ence patterns of multiply-folded surfaces using an ex-
perimental setup described in [2,5]. Prior to bench top
deformation, the layer of paraffin wax was shaped ap-
proximately like a several topographical surfaces. The
deformed surface was scanned in 3-D with sub-mil-
limeter accuracy.
3. Topographical Surfaces Illustrates
Severally Applications
Simulations of the analogue mathematically surfaces are
obtained using quantities derived from the fundamental
forms. Using the concept of topographical curvature (Fig-
ure 2), the model surface can be decomposed into the
several local surface shapes in three dimensional 3-D
shapes (Figure 3).
3.1. Parameters for Topography Simulations
In order to correctly describe the properties involved in
light triangulation measurements a number of parameters
must be defined before implementation in the MATLAB
simulation model. As a starting point, the light source
requires to be shaped as a line and to also have a width
distribution, describing the luminance along the width of
Figure 2. Classification the mathematically cu rvatures of surfaces [2].
(a) (b)
(c) (d)
Figure 3. (a) Surface representing a tangent plan 3D; (b) Numerical simulation of a surface of revolution3D; (c) Numerical
simulation of a regulated surface 3D; (d) Numerical simulation of a circular surface of cylinder 3D.
Copyright © 2011 SciRes. IJG
the line. The width distribution should be easy to alter, in
order to enable simulations using different light sources.
The surface hit by the emitted light should be fed to the
simulation model as a matrix of height values, enabling
any surface, which is not merely restricted to object-
surface to be simulated. Reflectance is also a parameter
that requires to be input into the model. In the majority
of cases, the reflectance of a surface can be said to be
constant over a topographic-surface, so this parameter
can be set using a constant value rather than a distributed
function. Background light must also be considered,
which will manifest itself as a noise level added to the
reflected image seen by the camera. Since the line of
light is projected onto the surface at a particular angle and
is picked up by a camera at a different angle, these angles
are also vital parameters for simulation. In order to de-
termine the actual height of the surface at a given point,
the centre of gravity of the reflected line of light must be
calculated. The center of gravity of the line represents the
average dislocation due to the surface topography and is
used to calculate the surface height at each position of
the surface matrix. The height values produced will be
dependant on all of the parameters listed above.
3.2. Computer Simulation and Numerical
Results of Surfaces-Geometrical Constraints
We give some examples of surfaces and we can establish
the mathematical relations, to position our topographic
surfaces in the field of fringes with the appearance of the
fringe of order zero, or central fringe. The simulation is
then controlled from a standard MATLAB function Win-
dow, in which the simulation model core file is included
as a path. The following program example provides a
brief outline regarding how the model can be used and
controlled (see Figure 4(a) and (b)).
3.3. Experimental Procedure and Results
By analogy between topographic surfaces, and the sur-
faces plunged or immersed in 3-D map of the differential
geometry one notes:
- Surfaces with null curves
- Surfaces with average curves
- Surfaces of Weingarten
- Surfaces like envelopes of plan.
3.4. Physical Analysis
The physical concept is based on the functions of trans-
missions of the gratings by interferences and the mathe-
Figure 4. (a) Controller program example; (b) Matlab in-
terfaces with camera scientific using image Acquisition
matical aspect associated the objects and real topog-
raphic surfaces and of the examples to the models of the
laboratory. The general expression of interference effect
of the function of transmission is given by:
() exp2π
exp 2π
From this relation comes fundamental equation of in-
terferences gratings we can expressed by:
  (2)
As a result, an expression for the intensity can be ex-
pected to follow a sinusoidal waveform whose phase is a
function of x, y, i.e.,
Copyright © 2011 SciRes. IJG
0sin ,
Ixy (3)
When gratings are overlapped on surfaces objects, the
resultant interferences effect reveals the functional form
of the variation. The intensity of the image can thus be
(, )exp
m nhs
hs mn
xyaaccU (4)
Especially in our method of analysis, the relative tem-
poral displacement of the two objects points are compared.
At the plane of the sensor, the intensity at a point on the
image can be expressed:
(, ,)(, )1cos((, )
IxytI xyxy
,,Uxyt is the displacement at an object point (x,y),
,  Ux xy y, and U is the z-component of the dis-
placement. The relative displacements and the slope
along the x-y axis at the point (x,y) is
The phase is then unwrapped as in phase shifting in-
terferometry and thereby a 3-D map of the time depend-
ent phase be generated. The final phase can be expressed
by this equation:
 
Uxy yy
 
  (7)
y is the initial phase which usually a
constant. From these extrapolations the 3-D plot of the
object slope or curvature surfaces can be extracted
 
 
Uxyx x
xy xy (8)
xy xy
 
 
 
The functions
of normal stress and For
small strains. In general solved in thought using a numeri-
cal or analogical modelling according the Hook’s law. For
each set of loading conditions and initial random perturba-
tions, the surface is deformed starting at time zero (unre-
formed aside from the initial perturbations) in time steps
At which sum to the total time of deformation, tfinal. In
order to compare models starting from the same pseudo-
random surface but subjected to different loading condi-
tions, the amount of strain induced was used as the met-
ric. For small values of t Equations (6)-(7) and Equations
(8)-(9) are approximately equal. This allows for direct
comparison of pure shear and simple shear models at the
same time steps at, as long as the total time tfinal is rela-
tively small, due to the fact that the amount of maximum
shortening is approximately equivalent for the pure shear
and simple shear cases at each time step.
3.5. Mathematical Analysis
One will limit oneself to surfaces of the Euclidean space
brought back to the coordinates Cartesian X, Y, Z. One
can interpreter the formula of transformation of the coor-
dinated geographical (λ, φ, h) by the following formula
cos cos
sin cos
ZNe h
One can reverse the system (λ, φ, h) to express in
function (X, Y, Z) by equation:
arct an
It is an iterative and convergent process, and offers an
identical numerical result.
3.6. Geometric Analysis by Segmentation with
the Analogue Geographical Surface
Image segmentation is a process that partitions an image
into its constituent regions or objects. Effective segmen-
tation of complex images is one of the most difficult tasks
in image processing. Various image segmentation algo-
rithms have been proposed to achieve efficient and ac-
curate results. Among these algorithms, watershed seg-
mentation is a particularly attractive method. The major
idea of watershed segmentation is based on the concept
of topographic representation of image intensity. If one
compares with topography these images with a moun-
tainous relief, there are parts more luminous than others
(see Figure 5(a)). As pointed already, the basic concept
of watershed is based on visualizing a gray level image
into its topographic representation, which includes three
basic notions: minima, catchment basins and watershed
lines. Figure 5(b) illustrates the meanings of these defi-
nitions. In the image of Figure 5(a), if we imagine the
bright areas have “high” altitudes and dark areas have
“low” altitudes, then it might look like the topographic
surface illustrated by Figure 5(b). In this surface, it is
natural to consider three types of points: 1) points be-
longing to the different minima; 2) points with certainty
to a single minimum; and 3) points at equally likely to fall
to more than one minimum. The first type of points forms
different minima of the topographic surface. The second
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type points which construct a gradient interior region is
called catchment basin. The third type of points form crest
lines dividing different catchment basins, which is termed
by watershed lines (see Figure 5(a)).
4. Discussion
In this paper, pattern recognition theory is introduced so
that a surface can be classified by comparison with known
surface. As a demonstration of the results that can be
acquired from the simulation model a test case is con-
sidered for geometrical constraints. In some cases the
object was deformed to over 5 - 10 mm using a high
speed CCD camera a large number of frames of the ob-
ject being deformed acquired. In principle one can use a
standard CCD camera to collect the data. The parameters
fed to the model have been identified as the surface for
simulation, the light source, the projection and imaging
Figure 5. (a),(b)—surfaces natural real representing a mountainous relief with geometrical constraints; (c) surfaces natural
real representing a mountainous relief.
Copyright © 2011 SciRes. IJG
angles, noise parameters and reflectance coefficients.
Geologists should record and use the geographic coordi-
nates (see Figures 5(b) and (c)) for every sampled point
on the structures they are mapping. Meaningful interpre-
tations of geological surfaces can then be presented
quantitatively in light of the sampling, data precision,
and motivation of the analysis.
5. Acknowledgements
The authors gratefully acknowledge the financial support
by the M.E.S.R.S Ministry for the Higher education and
the Required Scientist of Algeria Under fundamental re-
Code du Project: D 01420090011.
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