# 1.2: Elements of 3D Graphics

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)When we turn to 3D graphics, we find that the most common approaches have more in common with vector graphics than with raster graphics. That is, the content of an image is specified as a list of geometric objects. The technique is referred to as * geometric modeling*. The starting point is to construct an “artificial 3D world” as a collection of simple geometric shapes, arranged in three-dimensional space. The objects can have attributes that, combined with global properties of the world, determine the appearance of the objects. Often, the range of basic shapes is very limited, perhaps including only points, line segments, and triangles. A more complex shape such as a polygon or sphere can be built or approximated as a collection

of more basic shapes, if it is not itself considered to be basic. To make a two-dimensional image of the scene, the scene is projected from three dimensions down to two dimensions. Projection is the equivalent of taking a photograph of the scene. Let’s look at how it all works in a little more detail.

**First, the geometry.** . . . We start with an empty 3D space or “world.” Of course, this space exists only conceptually, but it’s useful to think of it as real and to be able to visualize it in your mind. The space needs a coordinate system that associates each point in the space with three numbers, usually referred to as the x, y, and z coordinates of the point. This coordinate system is referred to as “world coordinates.”

We want to build a scene inside the world, made up of geometric objects. For example, we can specify a line segment in the scene by giving the coordinates of its two endpoints, and we can specify a triangle by giving the coordinates of its three vertices. The smallest building blocks that we have to work with, such as line segments and triangles, are called * geometric primitives*. Different graphics systems make different sets of primitive available, but in many cases only very basic shapes such as lines and triangles are considered primitive. A complex scene can contain a large number of primitives, and it would be very difficult to create the scene by giving explicit coordinates for each individual primitive. The solution, as any programmer should immediately guess, is to chunk together primitives into reusable components. For example, for a scene that contains several automobiles, we might create a geometric model of a wheel. An automobile can be modeled as four wheels together with models of other components. And we could then use several copies of the automobile model in the scene. Note that once a geometric model has been designed, it can be used as a component in more complex models. This is referred to as

*.*

**hierarchical modeling**Suppose that we have constructed a model of a wheel out of geometric primitives. When that wheel is moved into position in the model of an automobile, the coordinates of all of its primitives will have to be adjusted. So what exactly have we gained by building the wheel? The point is that all of the coordinates in the wheel are adjusted in the same way. That is, to place the wheel in the automobile, we just have to specify a single adjustment that is applied to the wheel as a whole. The type of “adjustment” that is used is called a geometric transform (or geometric transformation). A * geometric transform* is used to adjust the size, orientation, and position of a geometric object. When making a model of an automobile, we build one wheel. We then apply four different transforms to the wheel model to add four copies of the wheel to the automobile. Similarly, we can add several automobiles to a scene by applying different transforms to the same automobile model.

The three most basic kinds of geometric transform are called * scaling*,

*, and*

**rotation***. A scaling transform is used to set the size of an object, that is, to make it bigger or smaller by some specified factor. A rotation transform is used to set an object’s orientation, by rotating it by some angle about some specific axis. A translation transform is used to set the position of an object, by displacing it by a given amount from its original position. In this book, we will meet these transformations first in two dimensions, where they are easier to understand. But it is in 3D graphics that they become truly essential.*

**translation****Next, appearance.** . . . Geometric shapes by themselves are not very interesting. You have to be able to set their appearance. This is done by assigning attributes to the geometric objects. An obvious attribute is color, but getting a realistic appearance turns out to be a lot more complicated than simply specifying a color for each primitive. In 3D graphics, instead of color, we usually talk about * material*. The term material here refers to the properties that determine the intrinsic visual appearance of a surface. Essentially, this means how the surface interacts with light that hits the surface. Material properties can include a basic color as well as other properties such as shininess, roughness, and transparency.

One of the most useful kinds of material property is a * texture*. In most general terms, a texture is a way of varying material properties from point-to-point on a surface. The most common use of texture is to allow different colors for different points. This is done by using a 2D image as a texture, which can be applied to a surface so that the image looks like it is “painted” onto the surface. However, texture can also refer to changing values for things like transparency or “bumpiness.” Textures allow us to add detail to a scene without using a huge number of geometric primitives; instead, you can use a smaller number of textured primitives.

A material is an intrinsic property of an object, but the actual appearance of the object also depends on the environment in which the object is viewed. In the real world, you don’t see anything unless there is some light in the environment. The same is true in 3D graphics: you have to add simulated * lighting* to a scene. There can be several sources of light in a scene. Each light source can have its own color, intensity, and direction or position. The light from those sources will then interact with the material properties of the objects in the scene. Support for lighting in a graphics system can range from fairly simple to very complex and computationally intensive.

**Finally, the image.** . . . In general, the ultimate goal of 3D graphics is to produce 2D images of the 3D world. The transformation from 3D to 2D involves * viewing* and

*. The world looks different when seen from different points of view. To set up a point of view, we need to specify the position of the viewer and the direction that the viewer is looking. It is also necessary to specify an “up” direction, a direction that will be pointing upwards in the final image. This can be thought of as placing a “virtual camera” into the scene. Once the view is set up, the world as seen from that point of view can be projected into 2D. Projection is analogous to taking a picture with the camera.*

**projection**The final step in 3D graphics is to assign colors to individual pixels in the 2D image. This process is called * rasterization*, and the whole process of producing an image is referred to as

*the scene.*

**rendering**In many cases the ultimate goal is not to create a single image, but to create an * animation*, consisting a sequence of images that show the world at different times. In an animation, there are small changes from one image in the sequence to the next. Almost any aspect of a scene can change during an animation, including coordinates of primitives, transformations, material properties, and the view. For example, an object can be made to grow over the course of an animation by gradually increasing the scale factor in a scaling transformation that is applied to the object. And changing the view during an animation can give the effect of moving or flying through the scene. Of course, it can be difficult to compute the necessary changes. There are many techniques to help with the computation. One of the most important is to use a “physics engine,” which computes the motion and interaction of objects based on the laws of physics. (However, you won’t learn about physics engines in this book.)