ZZ mathematical knowledge required for Computer Graphics

Original post http://www.cnblogs.com/ufoym/archive/2009/01/21/1379529.html

The Application of mathematics in computer graphics Greg Turk, August 1997 "How much mathematics is required to learn computer graphics ?" This is the most frequently asked question for beginners. The answer depends on how deep you want to study in the computer graphics field. You don't need to know much about mathematics if you just use the graphic software that is at your fingertips. To learn more about computer graphics, I suggest you read the first two chapters (algebra, triangle, and linear algebra ). If you want to be a researcher in graphics, learning mathematics will become old. If you are not particularly fond of mathematics, do you still have the opportunity to work in the computer graphics field? Yes, there are indeed some aspects of computer graphics that do not need to be considered too many mathematical problems. You shouldn't give up on it because of poor scores. However, if you learn more about mathematics, it seems that you will have more options on research topics. There is no clear answer to which mathematics is important in computer graphics. Different aspects in this field require different mathematical knowledge. Maybe interest will determine your direction. The following describes the mathematics that I think is useful for computer graphics. Don't think that a researcher in graphics must be proficient in mathematics! In order to have a comprehensive view on the mathematics used in graphics, I have listed many aspects. However, many researchers never need to consider the mathematics mentioned below. Finally, although I have read this article, you should have some knowledge about the application of mathematics in computer graphics, but these ideas are completely my own. Maybe you should read more such articles, or at least learn different learning points from other people engaged in computer graphics. Now let's start with the question. For beginners of computer graphics, algebra and triangle in high school may be the most important mathematics. Day after day, I work out one or more roots from simple equations. I often want to solve simple triangle learning problems similar to the side length of some geometric figures. Algebra and triangle learning are the most basic knowledge of computer graphics. So what about high school geometry? It may be surprising, but in most computer graphics, high school geometry is not often used. The reason is that many schools teach a course on how to build mathematical proofs of geometry. Although it is obvious that the question proves to improve intelligence, for computer graphics, theorem and proofs related to geometric lessons are not often used. If you graduated from mathematics-related fields (including computer graphics), you will find that although you are proving the theorem, it is not necessary to start learning graphics. If you are proficient in algebra and triangle learning, you can start to read an entry book on computer graphics. The next important mathematics used in computer graphics is linear algebra. Most of these books contain at least a brief introduction to linear algebra. Recommended reference books: Computer Graphics: Principles and Practice James Foley, Andries Van Dam, Steven Feiner, John Hughes Addison-Wesley [Although heavy, however, I like it very much] the idea of linear algebra runs through computer graphics. In fact, as long as the geometric numeric representation is involved, we often abstract the values such as X, Y, and Z, which we call a vector. Graphics are inseparable from vectors and matrices from beginning to end. It is better to describe rotation, translation, or scaling using vectors and matrices. Both High School and University have linear algebra courses. As long as you want to work in the computer graphics field, you should lay a solid foundation for linear algebra. As I mentioned earlier, many graphics books have a brief introduction to linear algebra-enough to teach you the first course in graphics. Recommended reference books: Linear Algebra and its applications Gilbert Strang Academic Press calculus micropoints are an important component of advanced computer graphics. If you want to study graphics, I strongly recommend that you have a preliminary understanding of calculus. The reason is not only that calculus is a very useful tool, but also that many researchers use calculus terminology to describe their problems and solutions. In addition, calculus is a prerequisite for further learning in many important mathematical fields. After learning basic algebra, calculus is a course that can open the door to most computer graphics and subsequent Mathematical Learning. Calculus is the last middle school course I introduced. Almost all of the subjects mentioned below are university courses. The Study of differential geometry controls the equations of smooth Curves and curved surfaces. If you want to calculate a vector (normal vector) that goes through a point away from the surface and perpendicular to the surface, it will use the differential geometry. Let a car go on a curve at a specific speed also involves a differential geometry. There is a general graphics technology used to draw smooth surfaces, called a "concave-convex post image". This technology uses differential geometry. If you want to start using curves and surfaces to create a shape (known as Modeling in graphics), you should at least learn the basis of differential geometry. Recommended reference book: elementary differential geometry Barrett o 'Neill Academic Press numerical method. At almost any time, we use approximate values in the computer to represent and manipulate values, therefore, there is always an error in the calculation process. In addition, there are often multiple solutions for given numeric problems. Some methods may be more accurate or require less memory. The research objects of numerical methods include "calculation method" and "scientific calculation. This is a very broad field, and the other mathematics I will mention is actually a branch of the numerical method. These branches include the sampling theory, matrix equations, numerical differential equations, and optimization. Recommended reference books: Numerical recipes in C: The art of scientific computing William press, Saul teukolsky, william vetterling and Brian Flannery Cambridge University Press [This reference book is very valuable but rarely used as a teaching material] pumping theory and signal processing in computer graphics we have repeatedly used A set of numbers to represent some objects, examples and surfaces. In this case, we need to use the sampling method to represent these pairs. To control the quality of these objects, the sampling theory becomes particularly important. A common example of applying sampling method to graphics is that when an object is drawn on a screen, its contour shows a jagged edge. This jagged edge (which is considered to be a "obfuscation" phenomenon) is very distracting, using well-known techniques in sampling, such as maneuver, Fourier transform, the function representation of space and frequency can minimize this phenomenon. These ideas are equally important in the field of image and audio processing. Recommended reference books: The Fourier Transform and Its Applications Ronald N. Bracewell McGraw Hill moment array equations many computer graphics problems need to use the numerical solution of matrix equations. Some matrix-related problems include: finding the best position and direction for objects to match each other (Least Squares) and creating a surface that overwrites the given point set, and make the minimum degree of wrinkle (thin-board spline algorithm), as well as material simulation, such as water and clothes. In Graphics, matrix expressions are quite popular. Therefore, in mathematics used in graphics, I have a high rating on matrix equations. Recommended reference book: matrix computations gene Golub and Charles Van Loan Johns Hopkins University Press ry physics is clearly not a branch of mathematics, it is a self-developed discipline. However, physics and mathematics are closely related in some computer graphics fields. In Graphics, physics problems are involved, including how light and object surfaces affect each other, how people and animals move, and how water and air flow. Physics knowledge is essential to simulate these natural phenomena. This is closely related to the settlement of the differential equations. I will refer to the differential equations in the next section. I believe that for computer graphics, the technique of solving differential equations is very important. As we discussed earlier, computer graphics is committed to simulating physical systems originating in the real world. How waves are formed in water and how animals walk on the ground are two examples of simulating physical systems. The problem of simulating physical systems is often how to solve the numerical solution of the differential equation. Note that the Numeric Solution of the differential equation is very different from the Symbolic solution of the differential equation. The Symbolic solution is used to find solutions without errors and is often used only for some very simple equations. Sometimes the "differential equations" in university courses only teach symbolic solutions, but this does not help most computer graphics problems. In the simulation of physical systems, we divide the world into many small elements that represent vectors. Then the relationship between these elements can be described in a matrix. Although the matrix equations to be processed often do not have a very accurate solution, instead, a series of calculations are executed, which generate an approximate solution expressed as a series of values. This is the numerical solution of the differential equation. Note that the solution of matrix equations is closely related to the solution of the differential equations. Optimization in computer graphics, we often seek a suitable method to describe an object or an object set for the desired purpose. For example, if you want to arrange the lighting position so that the lighting in the room looks like a special "feeling", how do the characters in the animation move their limbs to realize a special action, how to typeset will not confuse the page. The above examples can be attributed to the optimization problem. There was almost no literature on optimization technology in computer graphics a decade ago, but recently this field has paid more and more attention to optimization theory. I think optimization will become increasingly important in computer graphics. Probability Theory and statistics are used in many fields of computer graphics. When researchers are involved in anthropology, they certainly need statistics to analyze data. Graphics-related fields involve human disciplines, such as virtual reality and Human-Computer Interaction (HCI ). In addition, many questions that use computers to depict the real world involve the probability of various unknown events. Two examples: a growth tree, the probability of its branches splitting, and how Virtual animals decide its route. Finally, some techniques for solving difficult equations use random numbers to estimate the solutions of the equations. Important example: the Monte Carlo method is often used for how light is transmitted. The above is only part of the methods that use probability theory and statistics in computer graphics. Computational ry computational ry studies how to efficiently represent and operate ry with a computer. Typical problems include collision detection. polygon is divided into triangles to find the point closest to a specific position. This discipline includes calculation rules, data structures, and mathematics. As long as you are involved in creating a shape (modeling), you need to use a lot of computational geometry. Recommended reference books: Computational Geometry in C Joseph o 'Rourke Cambridge University Press [University textbook] computational geometry: An Introduction Franco Preparata and Michael shamos Springer-Verlag [very classic, but it's a bit old.] Conclusion: Mathematical applications and mathematical theories for graphics, many mathematical disciplines mentioned above share one thing in common: compared with the theoretical values of these mathematical theories, we are more inclined to explore their application values. Don't be surprised. Many issues in graphics are closely related to the issues that physics researchers and engineers study. The mathematical tools used by physics and engineers are exactly what graphics researchers use. Most disciplines that study pure mathematical theory are never used in computer graphics. However, this is not absolute. Note these special cases: Molecular Biology is using a theory to study DNA subdynamics, and sub-atomic physics uses Abstract group theory. Maybe one day, pure mathematical theory can also promote the development of computer graphics. Who knows? Some seemingly important mathematics are not often used in computer graphics. Maybe topology is the most interesting in this type of mathematics. In a word, it is used to describe topology. It is used to investigate why the fried donut is essentially the same as the coffee cup. The answer is that they all have only one hole surface. Let's discuss the idea of topology. Although the surface is an important component of computer graphics, the course of differential geometry has covered most topology knowledge useful to graphics. Differential Geometry studies the shape of a surface, but topology studies the adjacent relationship of a surface. I think topology is almost useless for graphics. This is because topology is concerned with abstract things, and topology is far from the core of most graphics-the concept of three-dimensional Euclidean space. For graphics, the form of topology (symbolic Notation) is a simple way to express ideas, but graphics seldom use practical tools for abstract topology. For graphics, topology is like a beautiful vase, but don't expect it to bring you back immediately. Someone once asked me if computer graphics has used abstract algebra (group theory, rings, etc ....) Or number theory. I have never met anyone. Like topology, these disciplines have many beautiful ideas. Unfortunately, these ideas are rarely used in computer graphics. -- The end --