Procedural Modeling in Archaeology:
Approximating Ionic Style Columns for Games
Richard Konečný
Stella Syllaiou
Fotis Liarokapis
Utrecht University
Utrecht, Netherlands
[email protected]
Hellenic Open University
Patra, Greece
[email protected]
Masaryk University
Brno, Czech Republic
[email protected]
Abstract— This paper demonstrates how procedural
modeling and computer graphics techniques can be
combined for creating fast, accurate and realistic modeling
of archaeological parts of buildings which can be used for
games and virtual environments. As a case study the Ionic
order of Greek and Roman temples is examined. In
particular, the temple of Portunus in Rome, Italy was used
as a case study. A set of geometric algorithms for
computing shape points based on curved surfaces are
presented, which allows creating the majority of Ionic
temples. Evaluation of the results is performed based on the
Hausdorff and Fréchet distances and results indicate that
the proposed method provides a very good approximation
for generating accurate Ionic order temples. However,
ornaments that usually decorate the column capital have
not been taken into account in this paper.
Keywords – procedural modeling, geometric algorithms,
computer games, cultural heritage.
I.
INTRODUCTION
Computer visualisation of archaeological sites and
remains provide valuable information about the past and
have a wide application to several areas, such as
archaeology, architecture, restoration, conservation,
cultural tourism, creative industries, etc. A major target
in virtual archeology is to recreate large and complex
ancient areas. This is not an easy task mainly due to large
resources and execution time required [1]. Real-time
visualisation of cultural heritage has become an
interesting and useful tool for demonstration, as well as
for research and educational purposes [2] and in
particular for the computer games industry. However,
various issues mainly related to accuracy and precision
come along with the attempt to create a visualisation of a
particular architectural order.
On the one hand, not every structure has been
preserved in a satisfying condition. Many parts of the
archaeological remains have been either destroyed or
significantly damaged and no evidence about their
original state has been preserved. Therefore, one cannot
be certain about how these used to look like in the past.
This uncertainty poses a problem for people trying to
reconstruct and visualize ancient buildings to their
original condition [3], since -in most of the cases- the
archaeologists can be certain only for the existing parts
of the buildings. On the other hand, visualizing specific
details of the high-quality archaeological models of the
Roman and Greek architecture contains a lot of very
detailed artistic elements and decorations. Thus, to create
a realistic virtual city, a lot of details need to be taken
into account. A third issue concerns the evolvement of
the buildings over time. Most of the historic buildings
change over time, whereas a digital model captures only
one point of their history. The different building and
reconstruction phases are under continuous reading,
endless renegotiation and open interpretation. Therefore,
the reconstruction technologies must permit and be
adaptable to modification and interrogation of multiple
iterations of a digital model [4].
Manual modeling a certain part of a Roman or Greek
temple with high level of detail requires large amount of
artwork and takes certain time to do. This is crucial for
the gaming industry since a significant amount of money
is spent on content creation. Manual modeling may be
also misleading for the following purposes. Public may
take very convincing reconstructions of parts, which are
based only on one’s estimate or even only on one’s
guess, as authentic reconstructions. Time required for
defining rule sets is constant and therefore, it is also
faster and more efficient method for larger scenes with
larger number of models, which are needed to be created
[5]. An attempt to solve both these problems on specific
buildings by using procedural generation methods and
techniques for games and virtual environments is
presented in this paper. In particular, the paper is focused
on the accurate approximation of Greek and Roman
Ionic order columns based on a set of geometric
algorithms for computing shape points. The rest elements
of the temple, such as ornaments, that usually decorate
the column capital, were not taken into consideration.
In the next sections, the set of geometric parameters
implemented will be introduced and described in detail.
The parametric shape grammar has been developed with
the purpose to demonstrate flexibility and simplicity of
column procedural modeling and to prove the benefit of
procedural generation for creation of Roman and Greek
temples. It consists of the following components: the
base generator, the shaft generator and the capital
generator. Each of these parts works with several various
parameters to provide enough flexibility to applications
requiring high architectural accuracy. Evaluation of the
resulted columns is performed based on the Hausdorff
and Fréchet distances and results indicate that the
proposed method is very accurate (more than 98%
accuracy).
978-1-5090-2722-4/16/$31.00 ©2016 IEEE
The rest of the paper is structured as follows. Section
II presents background work and section III the main
architectural principles of temples. Sections IV, V and
VI present the generation of the column. Section VII
illustrates how the rest of the temple was generated and
measures the generated error of the columns. Finally,
section VIII presents conclusions and future work.
II.
BACKGROUND
Procedural modeling techniques for urban
environments are usually based on three issues including
the well-known L-systems [6], [7], shape grammars [8],
[9], [10], as well as stack-based languages [11], [12]. A
recent survey paper for procedural modeling techniques
for virtual worlds has been recently published [13].
Previously, a number of algorithms and techniques have
been used to procedurally generate cities. Researchers
used the ‘CityEngine’ [7] to procedurally generate both
residential buildings and temples on the basis of
architectural principles. In terms of building
reconstruction, Wonka et al. presented a new method for
the automatic modeling of architecture using split
grammars [9]. An attribute matching system and a
separate control grammar, which offers the flexibility
required to model buildings using a large variety of
different orders and design ideas was also introduced. A
few years later, a novel shape grammar for the
procedural modeling of computer graphics architecture
was proposed [10]. First, the grammar generates
procedural variations of the building mass model using
volumetric shapes and then proceeds to create facade
detail consistent with the mass model. Context sensitive
rules ensured that entities like windows or doors do not
intersect with other walls.
Van Gool et al. [14] presented an approach of
creating a realistic 3D representation of Antonine
nymphaeum (temple) at the Sagalassos based on various
techniques such as: photogrammetry; advanced lighting
and post-processing scans to improve the final model. A
procedural approach for fast Chinese ancient architecture
modeling based on a set of rules has also been researched
[15]. The primary features of buildings were
parameterized to control the modeling process and more
than one hundred different buildings have been generated
by a rule set comprising of about 120 rules. Another
study presented procedural interactive modeling tools for
creating medieval castles [16]. Special emphasis was laid
on creating generic modeling tools that increase the
usability with a unified 3D user interface, by combining
the Generative Modeling Language (GML) and the
OpenSG scene graph engine.
Architectural shape grammars were also used to
procedurally generate 3D reconstructions of Puuc-style
buildings found in Xkipché, Mexico [17]. To reconstruct
the whole site various input data was used including:
building footprints, architectural information, and
elevation models. Moreover, the first version of “Rome
Reborn” [18] presented an initiative of virtual
reconstruction of ancient Rome city allowing the large
physical model to be scanned first and then the rest of the
objects to be manually created. An improved version
called “Rome Reborn 2.0” [19] was based on both
manual and procedural modeling methods. Manual
modeling was used for objects, which parameters and
characteristics are well-known by archaeologists, while
procedural modeling covered elements, which are known
with lower accuracy, to keep high level of detail for these
elements.
In another study, an efficient way to visualize cultural
heritage despite uncertainty about credibility of objects
that do not longer exist was proposed [4]. This
uncertainty comes from the fact, that some of the
historical buildings have not been preserved in the
original condition because they have been either
destroyed or damaged so much, that one cannot be sure
about how these looked like before. Omission of
uncertain parts was described as even worse practice
than estimation. A similar approach was followed [1] to
recreate virtual cities based on automatic procedures that
make use of randomness for variety and credibility.
Another approach was based on the case study of Doric
temple [20]. Finally, another study [21] described several
rules needed for creation of a shape grammar for
classical Roman buildings. These rules were inspired by
Vitruvius’ architectural rules described in his books and
they included rules for columns, temple steps, temple
bases and cellae. However, most of the approaches
presented in this section did not focus on fine detail.
Also, to our knowledge none of them tried to measure
the error (or how accurate) the final outcome was
compared to the real representation. In the following
section, the main principles of Ionic temples, as well as
the architecture of the proposed technique, are illustrated.
III.
BASIC PRINCIPLES
Architecture played a major role in ancient Rome.
Roman architecture has developed many aspects of the
ancient Greek architecture. Roads, bridges and aqueducts
provided communication network and supply routes
across the empire, city walls protected inhabitants of
cities, which served as administrative centres and also
symbolized the Roman power. In the past, Roman
temples served political and religious purposes, which
were closely connected to the Roman world. Cults
celebrated outside played very important role for the
community and public life. Temples were especially
significant to emperors because they represented a
testament of their commitment to the conventional
customs and indicated the empire membership. Roman
temples have several distinctive components like raised
podium, staircase in the front and columns arranged
along the sides.
The architecture of ancient Greece has determined
the future form of most subsequent European art [22] and
influenced building styles until today. It is best known
from its temples, structures specifically designed for
serving and the worship of the deities [23]. The history
and development of Greek sanctuaries and temples
reflects the development of Greek society [24]. The
temples were used by the city-states to worship and
honour the Gods, to display the power and glorify the
city-states, to embellish their sanctuaries [25]. Greek
architects made use of the principles of proportion, or
otherwise called ‘dynamic symmetry’ [26]. Several
attempts to explain the harmony of proportions has been
made [27]. Vitruvius has analysed issues concerning the
symmetry, the classification, and the proportions of
temples [27], [28].
The ancient Greek temples have been adapted to the
natural relief, conceived as a sculptural entity within the
landscape, most often raised on high ground so that the
elegance of its proportions and the effects of light on its
surfaces might be viewed from all angles [29]. The
entrance to the temple was usually from the east side of
the building. The rectangular temple is the most common
form of ancient Greek public architecture. The temple
was conformed to an architectural order, a set of
principles and rules that define in different ways the
symmetry and harmony. Ancient Greek architecture is
divided in three distinct architectural systems or ‘orders’:
the Doric order, the Ionic order and the Corinthian order
that ‘was produced out of the other two orders’.
The architectural orders describe the style of a
building. The most obvious difference was their
columns’ capitals. Initially, the temples were smaller in
size and their columns are made mostly of wood and
mud-bricks. From the 6th century B.C. most of the
temples became more ‘monumental’, were increased in
size and made of durable materials, usually of local
stone, like soft limestone (poros), or marble. The
changes in Greek temples were not connected to changes
in cult practices, ‘but rather a decision to monumentalise’
[24]. For the upper parts of the temple the ancient Greeks
used mudbrick and timber. It is worth mentioning, that
some of the wooden forms of the first temples have
influenced the Greek stone temples. At the end of the 7th
century there is a typical ground plan [25].
The Doric order is probably originated between 1200
and 1000 BC [30]. The Doric column evolved from the
earlier wooden pillars into column in stone. They had a
capital made of a circle and topped by a square. The
column shaft was slightly tapered, vertically fluted, and
usually thinner at its top. There was no column base and
they usually rest directly on the stylobate. On the capital
there is the entablature made of three parts: the
architrave, the frieze, and the cornice. Above the column
there was the frieze. The entablature frieze carried
alternating series of metopes and triglyphs. The Ionic
order has its origins in the area of Asia Minor in the 6th
century BC. It is defined by its column type [30].
In Ionic architectural order the shafts were taller and
slimmer than the Doric ones. The bases support the
columns have vertical flutes. The abacus is narrow. The
Ionic entablature usually consists of three simple
horizontal bands often carries a frieze with richly carved
sculpture arranged in a continuous pattern around the
building. Ionic temples were more delicate than the
Doric ones and not as elaborated as the Corinthian ones.
The third order of the Greek architecture was the
Corinthian and has some similarities to the Ionic order. It
was developed in Athens in the 5th century BC and it was
quite common in the Hellenistic and the Roman period.
COLUMN SHAFT GENERATION
IV.
The shaft generator creates a column shaft based on
the following 9 parameters as illustrated in Table 1.
ID
$a
Name
Shaft height
$b
Channel
height
$c
Shaft
top
radius
Shaft bottom
radius
Channels
count
Channel
angle
$d
$e
$f
arc
$g
Channel deep
$h
Channel
slices count
$i
Channel
slices per one
fillet slice
Description
Height of the column shaft without the height
of channel arcs (Figure 1 - A)
Height of channel arcs at the top and bottom of
the columns shaft expressed as a ratio of the
height to the radius of a channel (Figure 1 - A)
Radius of the circle at the top of the column
shaft (Figure 1 - B)
Radius of the circle at the bottom of the
column shaft (Figure 1 - B)
Number of channels in the column shaft
(Figure 1 - C)
Central angle that delimits the diameter of a
channel ellipse parallel to a tangent line of the
circular cross-section (Figure 1 - C)
Diameter of a channel ellipse perpendicular to
a tangent line of the circular cross-section
defined by the ratio of the diameter d1 to the
diameter d2 (Figure 1 - C)
Number of vertical slices which constitute one
channel. This value also defines the number of
horizontal rows used in every channel arc and
controls the number of polygons (Figure 1 - D)
Number of channel slices needed for one fillet
slice (Figure 1 - D).
Table 1 Parameters for the column shaft
All the components of the column shaft are also
presented in Figure 1. The algorithm provides a simple
way to create a custom triangulation of a surface area
between two sets of vertices. It has been used to generate
the correct triangulation between sets of vertices
representing certain levels in the column shaft and uses
the following parameters. Two arrays of vertices, the
direction parameter, the “isClosed” parameter and the
sequence parameter. Each array represents a set of
vertices lying on the edges of a surface area to be
triangulated. These arrays contain at least one vertex and
don't contain the same vertices.
Figure 1 Column shaft generation
The “isClosed” parameter can also be applied to
arrays of vertices, where the last vertex is connected to
the first one and defines whether the surface area is
either “closed” or “unclosed”. Triangulation of such
geometric figures can be visualized using “closed rings”.
The sequence parameter represents a way of triangle
distribution between two arrays of vertices. Each number
in the sequence parameter represents a number of
connections of a particular node with nodes from the
other array. The order of numbers in the sequence is
determined by visiting nodes along their most-right
connections until they reach the first node or a node with
only one connection. Moreover, the variables $h and $i
that allow for dynamic level of detail make the solution
more interesting for the real-time applications.
V.
COLUMN BASE GENERATION
Bases of columns in the Ionic (and Doric) order
usually consist of set of several cylinders and torus. For
the purpose of this paper, these parts will be called
floors. Each floor is defined by 4 parameters: the bottom
radius, the top radius, the height and the curvature. The
curvature may be equal either 1 for a convex edge curve,
or -1 for a concave edge curve (Figure 2 - A). The
grammar constructs floors one on the top of the previous
one according to series of modules specified by user.
Parameters
Module
Example
Representation
r ∈ ℝ, h ∈ ℝ, c ∈ {1,-1}
M(r1, h1, c1, r2, h2, c2, … , rn, hn, cn, rn+1)
G = M(0, 0.1, -1, 0.1), M(0.1, 0.1, 1, 0.2, 0.1, 1, 0.1)
Table 2 Parametric shape grammar structure for column base
The “r” parameter represents an addition to the
bottom radius of the shaft ($d), the “h” parameter
represents the height of a floor and the “c” parameter
represents a floor curvature. In comparison to L-systems,
this approach does not work recursively but the series of
modules is pre-defined. One module may contain any
number of floors (but at least one), in which case vertices
of their adjacent bases are welded together (Figure 2 B). For better clarity, the actual structure used in the
code is slightly different: instead of modules, only
module parameter lists separated by a comma are used;
parameters are separated by a space and one of three
indicators (R, H, C) is inserted before each parameter to
indicate its meaning.
For instance, the above mentioned example (Figure 2
- Ca) would be written as: G1 = “R0 H0.1 C-1 R0.1,
R0.1 H0.1 C1 R0.2 H0.1 C1 R0.1”. An alternative way
which provides much more detail (Figure 2 - Cb) is: G2
= “R0.12 H0.12 C1 R0.12, R0.12 H0 C1 R0.074,R0.074
H0.046 C1 R0.12 H0.046 C1 R0.074, R0.074 H0.015 C1
R0.074, R0.074 H0.027 C-1 R0.026 H0.018 C-1 R0.044,
R0.044 H0.015 C1 R0.044, R0.044 H0.038 C1 R0.082
H0.038 C1 R0.044, R0.044 H0.030 C1 R0.044, R0.044
H0.053 C-1 R0”.
Figure 2 Column base generation (A) column base subdivided into
floors, a is convex and b is concave curvatures (B) column base
subdivided into modules (Ca) output using G1 (Cb) output using G2
Although the above method produces curved shapes;
these are limited only to circle or ellipse segments. To
avoid this, curved surfaces were used since they provide
additional flexibility but also produce more accurate
results. Bézier curve is a parametric curve described by
polynomials based on control points and was first
introduced by Pierre Bézier and Paul de Casteljau [31],
[32]. Bézier curves provide only pseudo-local control,
which means that any change of the control point
position moves every single vertex in the curve in the
same direction [33]. The basic concept of creation of the
column base by constructing one floor above each other
remains unchanged, although there is not any difference
between a floor and a module because each module
symbolises exactly one floor. The parametric equation
with Bernstein basis polynomial has been used to
calculate the positions of vertices lying on the curve:
Bin (t ) =
n!
t i (1 − t ) n−i where ϵ [0, 1]
i!( n − i )!
(1)
Parametric equation:
n
n
i =0
i =0
x(t ) =  xi Bin (t ) and y (t ) =  yi Bin (t )
(2)
The parametric shape grammar structure is illustrated in
Table 3. The “r” and “h” parameters have similar
meanings like in the first version, but they refer to the
control points of a Bézier curve instead of the edges of
tori: In the sectional view of the column, with the origin
located at the point, where the distance from the column
central axis is equal to the shaft bottom radius and the
height of the current floor is equal to 0, the “r” parameter
represents a coordinate on the horizontal axis pointing
out of the column centre and the “h” parameter
represents a coordinate on the vertical axis pointing
upwards.
Representation
r ∈ ℝ, h ∈ ℝ, c ∈ {1}∨ c ∈ ∅
M(r1, h1, r2, h2, … , rn, hn, c)
G = M(1,0,2,0,2,1,1,1), M(1,0,1,1,1)
same column base. The half-transparent generated
column bases are placed over the template.
Table 3 Parametric shape grammar structure based on Bezier curves for
column base
The main part of the capital creation is the generation
of a spiral. There are several spiral sorts that have been
considered for this purpose. The Archimedes' spiral is a
special case of an Archimedean spiral and is defined by
equation r = a + bθ(1/n), where r and θ represent polar
coordinates and a, b ∈ R are parameters controlling its
orientation and the distance between turnings
respectively; and n is a constant which determines how
tightly the spiral is wrapped. The Archimedes’ spiral is
an Archimedean spiral, where n=1, and thus r = a + bθ.
The distance between its successive turnings is
considered constant. Euler spiral is a curve whose
curvature changes linearly with its curve length. Fermat's
spiral is another type of an Archimedean spiral with n =
2, described as: r = a + bθ(1/2) in polar coordinates. The
hyperbolic spiral (also called the inverse spiral) belongs
to the group of Archimedean spirals too, in this case n =
-1, and therefore the hyperbolic spiral is defined by the
following equation: r = a + b/θ. The lituus is an
Archimedean spiral with n = -2, and thus its polar
equation is given by r = a + bθ(-1/2). The logarithmic
spiral is described by the following polar equation r = a +
becθ, where r and θ represent polar coordinates and a, b, c
∈ R are parameters. For the capital generation, the
logarithmic spiral was defined as: r = a/( becθ) - d, where
a, c, d ∈ R, b ∈ R - {0} are parameters controllable by
the user, because it is most similar to the capital volute
from the above mentioned spiral types. The following
parametric equation was used to calculate positions of
the vertices in the outer spiral. The inner spiral was
created using the same equation with the radius reduced
by the spiral thickness.
Parameters
Module
Example
The “c” parameter may be either omitted, or be
present at the end of a module. The first row of vertices
in a module will be skipped and the next row will be
connected to the last row of the previous module (it does
not take any effect when it is set in the first module).
This provides a choice to set the vertex normals in a
different way to avoid an effect of the sharp angle
between two modules (Figure 3).
Figure 3 a) the “c” parameter omitted b) the “c” parameter present
The actual structure, which the parametric shape
grammar works with, is slightly adapted in the second
version too, but it follows the same rules and principles
as in the first version. The number of control points per
module is not limited unless there are at least two of
them. Unlimited number of control points allows the user
to define very specific and highly accurate curve shapes.
If a module is defined by only two control points, the
generated curve is a straight line and thus, in order to
minimize the number of polygons, only two rows of
vertices are created regardless of the required number of
slices. In summary, both versions construct the base
using several levels (or floors) of adapted tori. However,
the difference is in the way how the edge of a torus,
visible from a side view, is calculated. The first version
of parametric shape grammar for the base generation is
able to create tori, which edges are circular or ellipse
segments. Although, it is not possible to generate these
exactly using the second version, but these can be
sufficiently approximated. Furthermore, the second
version provides incomparably better control over the
curve shape of a torus edge.
VI.
x(ϑ ) = R(
a
a
− d ) cos(θ ) , y (ϑ ) = R( cθ − d ) sin(θ ) (3)
be
becθ
Moreover, the capital volute is also defined by the
other nine adjustable parameters, as shown in Table 4.
Parameters
Start angle
Thickness
Min Thickness
Increasing max
Spiral circuits
Figure 4 A column base constructed using: a) the first b) the second
version, c) shows the positions of the control points of a Bézier curve
Figure 4 illustrates a comparison between both
versions. Each version has been used to generate the
COLUMN CAPITAL GENERATION
Capital circle
radius
Capital spiral
radius
Offset X
Offset Y
Description
the lower limit of the angle θ in degrees
the thickness of a spiral, or reduction of the radius
used for calculation of the inner spiral
the thickness of a spiral may also be linearly
increasing, in such a case, this value determines its
lower limit
the angle in degrees at which the incensement
reaches the upper limit of the thickness (which is
equal to the Thickness parameter)
the upper limit of the angle θ, the first circuit has a
value of 2.5π and every next one increases the limit
by 2π (for example: 1 circuit=2.5π, 2 circuits=4.5π,
3 circuits=6.5π etc.)
the radius of the circle in the middle of spiral
the radius used in parametric equations (the R
value)
the horizontal distance between the circle and the
spiral origins
the vertical distance between the circle and the
spiral origins
Table 4 Capital column parameters
The other five parameters (Figure 5) control general
properties of the capital including: the length of the
capital, the width of the capital, the length of the spiral
margin, the vertical position of the spiral scrolls relative
to the rest of capital and the vertical position of the
capital relative to the rest of the column.
the second with the maximum x, y and z co-ordinates.
This allows the user to define all required cuboids and
generate a temple in a short period of time. One
exception has been included for the roof creation, where
it is possible to define one extra parameter: the height
difference. The third parameter is optional, but if
defined, the four vertices with maximum z coordinate are
shifted by the specified difference towards the direction
of y axis, so inclined cuboids can be created. For
example, two inclined cuboids have been used in the roof
of the temple of Portunus (Figure 6).
Figure 6 Temple of Portunus by Piranesi [34]
Figure 5 Column capital generation a) Capital general parameters b)
Capital height parameters
Next, a wholly new parametric shape grammar has
been developed for generation of the surface of the spiral
scrolls (Figure 5 - B). The grammar (Table 5) allows the
user to specify any number of rings, which should be
connected and so constitute the scroll body.
Alphabet
Parameters
Module
Example1
Example2
Representation
R (radius), P (position)
r ∈ [0, 2], p ∈ [0, 1] (representing 0% - 200% and 0%
- 100%)
Rr1 Pp2
R0.7 P0.05, R0.7 P0.4, R0.8 P0.5, R0.7 P0.6, R0.7
P0.95
R0.7 P0.05, R0.6 P0.1, R0.6 P0.3, R0.7 P0.35, R0.6
P0.4, R0.6 P0.45, R0.7 P0.5, R0.6 P0.55, R0.6 P0.6,
R0.7 P0.65, R0.6 P0.7, R0.6 P0.9, R0.7 P0.95
Table 5 Parametric shape grammar for the capital of the column
Each module of the grammar represents a ring at a
specified position with a specified radius. The radius
parameter represents the percentage of the spiral total
radius and the position is determined by the percentage
of the length between the front and rear spirals, where
0% means the front and 100% means the rear part of a
spiral scroll. In the series of modules, each position
percentage should be greater than or equal to the position
percentage of the previous module.
VII.
TEMPLE GENERATION AND EVALUATION
The methodology can also be set to the mode for
generation of temple. In this paper, a model of temple is
generated instead of model of column. This has been
created with the intention of providing a tool for
generation of very simple temples, which could be used
for better demonstration of generated columns. The
temple generation is only able to construct a building
using cuboids, which edges parallel to the scene axes.
For each cuboid, the user only needs to define two
vertices: the first with the minimum x, y and z values and
To test the procedural output, the temple of Portunus
has been chosen to be generated. The temple is in
relatively good state of preservation (Figure 7) and it has
been quite well documented in drawings from an Italian
artist Giovanni Battista Piranesi [34]. His drawings
contain detailed drafts along with proportions of
particular column elements, such as capital volute,
column base, shaft cross-section, temple plot, temple
side views etc.; hence they provide sufficient foundation
for generation. Procedural generation of the temple of
Portunus and its Ionic columns was based on the
following four drawings available in high resolution (cca
4500 × 6400 px) (Figure 6).
Figure 7 Real temple of Portunus (left) and the procedural (right)
The model of the column and the model of the temple
have been generated and exported separately. Both
models have been then put together in a standard
modeling tool (3ds Max). The columns have been
positioned appropriately and textures have been applied
to all the models as shown in Figure 7. To evaluate the
similarity between the generated 3D model and the
original column, three most relevant parts of the column
were compared to their corresponding drafts documented
by Piranesi [34]. These include the column base, the
capital volute and the plot of shaft (Figure 8). Two
methods were used: the Hausdorff distance and the
Fréchet distance. The Hausdorff distance is a measure of
mismatch between two subsets of a metric space. The
Fréchet distance is a measure that provides more
accurate comparison of similarity of two curves than the
Hausdorff distance because it also takes into account the
continuity of shapes and ordering of the points [35], [36].
Figure 8 Column parts to be evaluated
Since the definitions of continuous curves describing
the real objects are not known and they are only to be
seen on the picture, polygonal curves with a constant
distance between two adjacent nodes have been created
for all three drafts. The distance between two adjacent
nodes was the same in all of them. The curves describing
the generated 3D model have been created in the same
way from screenshots. The curves created according to
drafts have been then compared with the curves created
according to screenshots in order to compute appropriate
measurements describing a similarity between these
curves. Taking into account the unique nature of these
curves: high visual similarity and the special striking
analogy between each two of them, the following
procedures have been used to calculate particular
measurements.
To determine the weak Fréchet distance between two
curves, a circle has been drawn around every node with
such a radius that a circle touches the closest point of
another curve, including points on edges between nodes.
The radius of such a circle equals the shortest distance
between a node and another curve, or the shortest leash a
man would need to remain connected with his dog if his
position was at this node. The longest radius of all these
circles represents the weak Fréchet distance. This
process is only applicable with very similar curves,
which are approximately parallel. Although the Fréchet
distance is the most suited measurement for this
comparison and provides the most relevant results, to
provide more details about the curve shapes, some
additional measurements have also been computed.
These include the Hausdorff distance, the average
error and the percentages relative to the length of the
curve (or the average length of the curves, in case the
curve lengths are not equal). The Hausdorff distance has
been determined in a similar way to Fréchet distance, but
the point lying on the edges between nodes have not
been taken into account during the measurement of the
distance from a node to another curve. The average error
has been calculated as the average of all the node-curve
distances, and the percentage of a measurement equals
the measured value divided by the average of two curve
lengths.
Figure 9 Left Image: Comparison between generated (red) and original
(yellow) curves: a) column base, b) column shaft, c) capital spiral,
Right Image: Circles with radii equal to node-curve distance between
generated (red) and original (yellow) curves: a) column base, b)
column shaft, c) capital spiral
The distance between each two adjacent nodes equals
approximately 7mm. Figure 9 (left image) shows a
comparison between a curve describing the original
object and the curve describing the generated model
drawn in the same measuring scale. Figure 9 (right
image) shows circles drawn to determine the weak
Fréchet distance. An overview of the measured values
for all three parts of the column is shown in Table 6.
Number of
nodes
Curve
length
Fréchet
distance
Hausdorff
distance
Average
error
Fréchet
dist. (%)
Hausdorff
dist. (%)
Average
error (%)
Base
97 (original) +
97 (generated)
96 (original) +
96 (generated)
1.7664
(25.23 mm)
1.8257
(26.08 mm)
0.2797
(4.00 mm)
1.84%
Shaft
564 (original) +
564 (generated)
563 (original) +
563 (generated)
2.3741
(33.92mm)
2.3741
(33.92mm)
0.5649
(8.07mm)
0.42%
Spiral
352 (original) +
355 (generated)
351 (original) +
354 (generated)
1.2913
(18.45mm)
1.2913
(18.45mm)
0.3355
(4.79mm)
0.37%
1.90%
0.42%
0.37%
0.29%
0.10%
0.09%
Table 6 Results based on the Hausdorff and Fréchet distances
Results indicate the level of detail and similarity
achievable using this software solution. However, these
results correspond to the current settings and parameters.
Therefore, they do not necessarily represent the limits of
the developed software solution. Although these settings
have been determined after several exhaustive
adjustments with intent to get as realistic output as
possible, the possibility of existence of better settings
and parameters, which would result in more accurate
output using this solution, cannot be excluded.
VIII.
CONCLUSIONS AND FUTURE WORK
The aim of this research was to investigate how
realistically the Roman and Greek Ionic temples can be
created using procedural generation and if this method is
also advantageous for generation of more detailed parts
so that it can be used in games and virtual environments.
The paper addressed two main problems: uncertainty
coming from condition of preserved historic sites and
time-consuming visualisation of details. A hybrid
solution for generation of an Ionic column and a temple
was developed. To determine the accuracy of the
geometric algorithms implemented, a 3D model of the
temple of Portunus, which contains several Ionic
columns, has been generated and its similarity to the
drawings of the original temple, documented by Piranesi
[34], was measured. The results indicate that very high
similarity between the generated 3D models and the real
objects has been achieved, although the results of the
evaluated parameters do not represent exactly the same
level of detail. In particular, it is possible to generate
Ionic columns procedurally with more than 98%
accuracy; however, ornaments that usually decorate the
column capital have not been taken into account in this
study. In the future, emphasis will be given on other
parts of the temple for example ornaments that were not
modeled in this work. Also, real columns will be laser
scanned and the produced procedural columns will be
compared with the digitised models.
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