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2 Computer Graphics 2 Computer Graphics What You Will Learn: The objectives of this chapter are quite ambitious; you should refer to the references cited in each Section to get a deeper explanation of the topics presented. By the end of this chapter you will be familiar with the main building blocks of a 3D graphics interactive application and will be able to use this knowledge to build your own VR projects. In particular, we will give a brief overview of the following topics: • Mathematics: coordinate systems, vectors, and transformation matrices. These are used to represent, visualize, and animate 3D objects, characters, and the like. • 3D modeling: 3D primitives and geometric representations. This is how we actually create objects with a particular 3D shape. • Image rendering: illumination models, lighting, colors and textures. These are used to visualize 3D shapes in a realistic and believable way. This knowledge will give you the foundation required to understand any 3D graphics system, be it a software API like OpenGL or DirectX, a model- ing/animation tool such as Maya, Blender 3D, or 3DS MAX, or a game engine like Torque or Unreal. 2.1 Mathematics 2.1.1 Coordinate Systems A coordinate system is a way to determine the position of a point by a set of numbers (coordinates) that can be distances from a set of reference planes, angles subtended with an origin, or a combination of both [7]. Different coordinate systems have been defined, each of them allows rep- resentation of 3D shapes; however, depending on the shape, it may be easier to represent using one coordinate system than another. The most commonly 12 2 Computer Graphics used coordinate systems are the cartesian and the polar or spherical coordinate systems. Both allow for specifying the position of a point in a 3D Euclidean space, a space in three dimensions just like the world we know. It was named Euclidean after Euclid of Alexandria, a Greek mathematician whose treatise The Elements defined the principles of geometry. There are also non-euclidean geometries, but the kind of shapes we will work with will be represented in a 3D Euclidean space. Cartesian Coordinate Systems This is the most commonly used coordinate system in computers. The position of a point is determined by three distances, from each of the planes. The axes are at right angles (orthogonal) and are typically used to specify spatial dimensions: width, height, and depth (x, y, z). Cartesian coordinates are the most common data to represent real, theoretical, and artistic worlds. Spherical Coordinate Systems A spherical coordinate system is a called polar coordinate system when used in two dimensions. The position of a point is specified using one distance (ρ from the origin) and two angles (θ and φ) measured from the reference planes at right angles (see Figure 2.1a). Spherical coordinates are commonly used in applications that need to define positions relative to the surface of the Earth, such as meteorology, geography, and oceanography. In such cases, the angles are better known as latitude and longitude. The origin is at the center of the planet and the reference planes are the Equator and Greenwich meridian. Spherical coordinates are also used in astronomy to specify the position of celestial objects. In this case, the angles are called right ascension and declination. (a) A point P in spherical (b) A cardioid function in polar co- coordinates (ρ, θ, φ) ordinates Fig. 2.1: Point and shapes in polar coordinates 2.1 Mathematics 13 As shown in Figure 2.1b, some shapes are easier to specify and plot us- ing polar coordinates than using Cartesian coordinates. However, most com- puter visualization systems, which are programmed using graphics APIs like OpenGL, use Cartesian coordinates. Thus, it is necessary to convert positions defined in polar/spherical coordinates to Cartesian coordinates through the following formulas: x = ρ sin φ cos θy= ρ sin φ sin θz= ρ cos φ In the rest of the book we will use Cartesian coordinates; as mentioned earlier, they are the most commonly used and are better “understood” by 3D APIs and other computer graphics-related tools. In general, 3D virtual worlds are created out of 3D polygons, usually trian- gles or squares arranged in 3D space to form 3D surfaces that comprise more complex objects; this is illustrated in Figure 2.2. The simplest polygon and the most efficient to represent, process, and visualize with current graphics hardware is a triangle. A triangle is defined by specifying the three points or vertices that compose it (see Figure 2.2). Commonly available hardware (graphics cards) is able to take these three vertices and use them to draw a colored triangle on the screen at very fast speeds. Currently, graphics cards can draw hundreds of millions of colored triangles per second; for example, the NVIDIA Quadro FX 4600 card can render 250 million triangles per second (March 2007). The complete “drawing” process is called rendering; it involves computing the effects of color and textures assigned to each triangle in addi- tion to lights and possibly shadows and reflections. We explain this process in Section 2.3.4. Fig. 2.2: Triangle-based 3D shape Triangles are the basic primitive for creation of 3D shapes, the building block of virtual worlds. In order to understand how to visualize and animate virtual entities in 3D space we must be able to visualize and animate each triangle comprising the environment in a coherent way. We will provide more details on animation in Chapter 3. 14 2 Computer Graphics 2.1.2 Vectors, Transformations and Matrices To illustrate the basic mathematical principles behind visualization of 3D shapes, we start with a single triangle. In order to represent a triangle shape in a computer we use a Cartesian coordinate system and three points in the space to delimit the shape; we call them vertices. A vertex v is composed of three cartesian coordinates (vx,vy,vz). This set of three coordinates is a 3D vector, an array of three values, which are the vector components. A triangle can thus be defined by three 3D vectors that specify the coordinates of each vertex in the space: v1, v2, v3. In a 3D environment, triangles and any other 3D shape can suffer three basic coordinate transformations: they can be translated, rotated, or scaled. When combined, these three basic transformations allow us to produce a great variety of animation, from simple linear motion to complex trajectories and surface deformations. Some important questions to answer are: how can we translate/rotate/scale a 3D shape? what happens to the vertices forming the shape? Translating a 3D shape means adding a vector T to each of its composing vectors. The addition of two vectors consists of summing each of its compo- nents: a+b = c, where the components of vector c are: (ax+bx,ay+by,az +bz). Scaling affects the components of each vertex by multiplying them by a scale factor s. The components of vector v when scaled by a given factor are (vx × sx,vy × sy,vz × sz). The rotation transformation is a combination of addition and multiplica- tion operations that result from basic trigonometry. It is easier and computa- tionally efficient to use a matrix representation and matrix multiplication to perform a rotation transformation over a vector. In fact, the most convenient way of processing coordinate transformations is to set up matrices containing the parameters of the transformations and use matrix multiplication to reduce a series of transformations to a combined single transformation matrix. Homogeneous Coordinates In order to process transformations in a uniform way, it is necessary to add a component to the vectors and increase the dimensions of the matrices. For 3D space, 4D vectors are used, the four components are called homogeneous coordinates. The components of a 4D homogeneous vector are reduced to three dimensions by the following normalization operation: x, y, z, h −→ x , y , z h h h Any vertex in 3D space can be represented as x, y, z, 1 ;4× 4 matrices should be set up in such a way that multiplying a transformation matrix M by a vector v will give as the result a vector v whose components x,y,z, 1 correspond to the coordinates of v after applying the transformations param- eterized in M. 2.1 Mathematics 15 The fourth dimension (fourth component) will normally remain 1. Most of the graphical transformations can be broken down into products of the three basic ones. Transformation matrices are defined as follows. Translation Vector v , which is the position of v, after a translation of tx units over the X axis, ty units on Y ,andtz units on the Z axis is calculated as follows: ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 100tx x x + tx ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 010ty ⎥ ⎢ y ⎥ ⎢ y + ty ⎥ v = T · v = ⎣ ⎦ · ⎣ ⎦ = ⎣ ⎦ 001tz z z + tz 000 1 1 1 When applied to all the vectors (vertices) that comprise a 3D shape, this transformation will move the shape by the magnitude and direction of vector tx,ty,tz . Note that translation cannot be achieved using 3 × 3 matrix mul- tiplication with a 3D vector. This is one of the most important reasons for using homogeneous coordinates: they allow composite transformations to be handled by a single matrix. Scale The scaled version of v is calculated as follows: ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ sx 000 x sx · x ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 sy 00⎥ ⎢ y ⎥ ⎢ sy · y ⎥ v = S · v = ⎣ ⎦ · ⎣ ⎦ = ⎣ ⎦ 00sz 0 z sz · z 0001 1 1 A scaled 3D shape normally changes both its size and its location. Only shapes centered on the origin will remain in the same position. Scaling may also be done with reference to a point Po =[xo,yo,zo], which may be the center of the 3D shape.
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