Üç boyutlu transformasyonlar paralel projeksiyonlar aydınlatma modelleri

Abstract

Bu tezde, üç boyutlu grafikler üzerinde; ölçekleme, deformasyon, döndürme, yansıma, öteleme gibi üç boyutlu transformasyonlar; ortografik ve aksonometrik paralel projeksiyonlar; basit, Gouraud, Phong, ışın izleme gibi aydınlatma modelleri, ayrıntılı bir şekilde, hem teorik ve karşılaştırmalı, hem de 3D-Professional isimli (3D-PRO) örnek program üzerinde uygulamalı olarak irdelenmiştir. Birinci bölümde, temel üç boyutlu transformasyonlardan yola çıkılarak çoklu transformasyonlar ve kompleks transformasyon teknikleri ele alınmıştır. İkici bölümde, düzlem geometrik projeksiyonlar tanıtılarak, ortografik ve aksonometrik projeksiyonlar ayrıntılı bir şekilde incelenmiştir. Üçüncü bölümde, aydınlatma işleminin temelleri ve en basitinden daha karmaşık ve gerçekçilerine kadar aydınlatma modelleri karşılaştırmalı olarak ele alınmıştır. Dördüncü bölümde, tez ile birlikte sunulan 3D-PRO örnek programının tanıtımı, derleme ve çalıştırma esasları ile gelecekteki gelişim süreci planlaması yer almaktadır. Beşinci bölümde ise, tez bünyesinde geliştirilen yeni yöntem ve algoritmalar tanıtılarak, benzerleri ile karşılaştırmalı olarak incelenmiştir.

In this thesis, three dimensional transformations such as scaling, shearing, rotation, reflection and translation; orthographic and axonometric parallel projections; basic, Gouraud, Phong and Ray tracing rendering models on three dimensional graphics, are covered both theoretically and comparatively. These methods are also implemented, writing an example program named 3D-Professional. In first section, beginning from basic transformations, multiple transformations and more complex transformation methods are examined in detail. In second section, introducing the plane geometric projections, orthographic and axonometric projections are explained both theoretically and practically. In third section, the basics of the rendering and the common rendering models from easy ones to complex and photo realistic ones are discussed theoretically and comparatively. In fourth section, the example program 3D-PRO, given with this thesis is introduced, compilation and execution steps are clearly explained. Additionally, the future improvement planning of the program is also included. In fifth section, newly developed methodologies and algorithms are introduced, and compared with common techniques, in detail. Transforming and displaying the three-dimensional objects, are the main two work areas on three-dimensional modelling. Transformations, such as translation, scaling, rotation and reflection makes the three-dimensional objects more understandable. The three- dimensional transformations are mostly similar to the two-dimensional ones, but require more complex arithmetical operations. To implement three-dimensional transformations, a 4x4 transformation matrix is used. Basic three-dimensional transformations, implemented in this thesis are listed below: - Tree dimensional scaling - Three dimensional shearing - Three dimensional rotation - Three dimensional reflection - Three dimensional translation Complex transformations can be build using simple transformations. Transformations to be applied can be expressed using simple transformations, and a multipple transformation matrix is created. Some covered multipple transformations are: - Rotation about an axis parallel to a coordinate axis - Rotation about an arbitrary axis in space. - Reflection through an arbitrary plane. Geometric theorems have been developed for both perspective and affine geometry. The theorems of affine geometry are identical to those for Euclidean geometry. In both affine and Euclidean geometry parallelism is an important concept. In perspective geometry, lines are generally nonparallel. XI Both, affine and perspective transformations are three- dimensional. They are transformations from one three space to another three space. However, viewing the results on a two-dimensional surface requires a projection from three space to two space. The result is called a "Plane geometric projection". The projection matrix from three space to two space always contains a column of zeros. Consequently the determinant of a projective transformation is always zero. In this thesis, parallel plane geometric projections are examined. The included sub-projections are: - Orthographic projections: The simplest of the parallel projections is the orthographic projection, commonly used for engineering drawings. They accurately show the correct or true size and shape of a single plane face of an object. Orthographic projections are projections onto one of the coordinate planes x=0, y=0, z=0. - Axonometric projections: An orthographic projection fails to illustrate the general three-dimensional shape of an object. Axonometric projections overcome this limitation. An axonometric projection is constructed by manipulating the object, using rotations and translations, such that at least three adjent faces are shown. The result is then projected from a center of projection at infinity onto one of the coordinate planes. Unless a face is parallel to the plane of projection, an axonometric projection does not show its true shape. However, the relative lengths of originally parallel lines remain constant, parallel lines are equally foreshortened. The foreshortening factor is the ratio of the projected length of a line to its true length. The three axonometric projection types are: - Trimetric projections - Dimetric projections - Isometric projections XII Simply defined, rendering is the process of producing realistic images or pictures. Producing realistic images involves both physics and psychology. Light, electromagnetic energy, reaches the eye after interacting with the physical environment. In the eye, physical and chemical changes take place that generate electrical pulses that are interpreted, perceived, by the brain. Another characteristic of the eye, which has implications for computer graphics is that the brightness perceived by the eye tends to overshoot at the boundaries of regions of constant intensity. This characteristic results in areas of constant intensity beeing perceived as having varying intensity. The phenomenon is called the "Mach band effect" after the Austrian physicist Ernst Mach, who first observed it. The Mach band effect occurs whenever the slope of the light intensity curve changes abruptly. When light energy falls on a surface, it can be absorbed, reflected, or transmitted. Some of the light energy incident on the surface is absorbed and converted to heat. The rest is either reflected or transmitted. It is the reflected or transmitted light that makes an object visible. If all the incident light energy is absorbed,, the object is invisible. The amount of energy absorbrd, reflected or transmitted depends on the wavelength of the light. If the intensity of incident light is reduced nearly equally for all wavelengths, then an object, illuminated with white light, which contains all wavelengths appears gray. If nearly all the light is absorbed, the object appaers black. If only a small fraction is absorbed, the light leaving the object has a different energy distribution. The object appaers colored. The color of the object is determined by the wavelengths selectively absorbed. xm The character of the light reflected from the surface of an object depends on the composition, direction and geometry of the light source, the surface orientation, and the surface properties of the object. The light reflected from an object is also characterized by being either diffusely or specularly reflected. Diffusely reflected light can be considered as light that has penetrated below the surface of an object, been absorbed, and than reemitted. Diffusely reflected light is scattered equally in all directions. Hence, the position of the observer is unimportant. Specularly reflected light is reflected from the outer surface of the object. Common rendering techniques, examined in this thesis and implemented in application program 3D-PRO, are listed below: - Simple illumination - Gouraud shading - Phong shading - Ray tracing The three-dimensional transformations and parallel projections mentioned above are also implemented in application program 3D Professional (3D-PRO). Besides, a new rendering model "Fifik shading", new techniques and algorithms are also included. The application program comes in three installation disks, including C and ASM source codes, build in help documents, development tools, sample images and three-dimensional object file examples. In section 4, the 3D-PRO program features are covered in detail. Additionally, section 5 includes a list of the newly developed techniques and algorithms.