Application of mathematics in computer graphics

Application of mathematics in computer graphics

Mathematics for Computer Graphics
Greg Turk, August 1997.

"How much mathematics does it take to learn computer graphics ?" This is the most frequently asked question for beginners. The answer depends on what you want
Deep research in the field of computer graphics. You do not need to know that
Mathematical knowledge. If you want to learn more about computer graphics, I suggest you read the first two
Chapter (algebra, triangle and linear algebra ). If you want to be a researcher in graphics, you can learn mathematics.
It will be old and old.

If you are not particularly fond of mathematics, do you still have the opportunity to work in the computer graphics field? Yes, computer Diagram
There are indeed some aspects of morphology that do not need to be considered too many mathematical questions. You shouldn't give up because of poor math scores.
It. 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 requirements in this field
Understanding different mathematical knowledge may determine your direction. The following describes
Graph learning useful mathematics. Don't think that a researcher in graphics must be proficient in mathematics! To
I have a comprehensive view on the mathematics of graphics, and I have listed many aspects. However, many researchers never need to take an exam.
Consider the mathematics mentioned below.

After reading this article, you should have some knowledge about the application of mathematics in computer graphics.
These ideas are entirely my own. Maybe you should read more such articles, or at least engage in computing from others
People who work in Machine Graphics learn different learning points. Now let's start with the question.

 

Algebra and triangle
For beginners of computer graphics, algebra and triangle in high school may be the most important mathematics. Day 1
I work out one or more roots from simple equations. I often want to solve the problem of seeking the side lengths of some geometric figures.
Simple triangle learning. 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 ry
Learning is not often used. The reason is that many schools teach a course on how to build mathematical proofs of geometry. Although
However, it is obvious that the question is effective for improving intelligence, but for computer graphics, the questions related to geometric lessons
Processing and proof are not often used. If you graduate from mathematics (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. Next important use
In computer graphics mathematics-linear algebra, most of these books contain at least a brief introduction to Linear Algebra
Introduction.
Recommended reference books:
Computer Graphics: Principles and Practice
James Foley, Andries Van Dam, Steven Feiner, John Hughes
Addison-Wesley
[Although heavy, I like it very much]

 

 

Linear Algebra
The idea of linear algebra runs through computer graphics. In fact, as long as the geometric numeric representation is involved
Such as X, Y, and zcoordinate. We call it a vector. From beginning to end, graphics are inseparable from vectors and moments.
Array. It is better to describe rotation, translation, or scaling using vectors and matrices. Both High School and University have linear generation
Number of 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 of graphics.
Course.
Recommended reference books:
Linear Algebra and Its Applications
Gilbert Strang
Academic Press

 

 

Calculus
Calculus is an important component of advanced computer graphics. If you want to study graphics, I strongly recommend that you
Calculus has a preliminary understanding. The reason is not only that calculus is a very useful tool, but also used by many researchers.
Calculus terminology to describe their problems and solutions. In addition, in many important mathematical fields, calculus
Learning is a prerequisite for further learning. After learning basic algebra, calculus is another way to open the majority for you.
Computer Graphics and subsequent Mathematical Learning courses.
Calculus is the last middle school course I introduced. Almost all of the subjects mentioned below are university courses.

 

 

Differential Geometry
The Study of differential geometry controls the equations of smooth curves and surfaces. If you want to calculate a point that passes through a certain distance from the surface
The vector perpendicular to the surface (normal vector) will use the differential geometry. Let a car curve at a specific speed
Moving on also involves differential geometry. There is a general graphics technology used to draw smooth surfaces. It is called "Concave and convex Stickers ".
Figure ", this technology uses the differential geometry. If you want to start using curves and surfaces to create shapes (in graphics
(Modeling) You should at least learn the basis of differential geometry.
Recommended reference books:
Elementary differential geometry
Barrett o'neill
Academic Press

 

 

Numerical Method
In almost any time, we use an approximate value instead of an exact value in the computer to represent and operate the value, so the calculation process
There is always an error. In addition, there are often multiple solutions to the given numerical problem, and some methods will be more fragmented,
More accurate or less memory needs. The research objects of numerical methods include "calculation method" and "scientific calculation method ".
And so on. This is a very broad field, and the other mathematics I will mention is actually one of the numerical methods.
Some branches. 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]

 

 

Theory of sampling and signal processing
In computer graphics, we repeatedly use a number set stored in a regular two-dimensional array to represent some objects.
Slice and surface. In this case, we need to use the sampling method to represent these objects. If you want to control the products of these objects
Quality, sampling theory has become particularly important. A common example of applying sampling method to graphics is that when an object is drawn on a screen
On the screen, its contour shows a jagged edge. This jagged edge (considered as a "obfuscation" phenomenon) is not
It is often distracting, using well-known techniques in sampling, such as maneuver, Fourier transformation, spatial and frequency functions.
The number indicates that this phenomenon can be minimized. 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

 

 

Matrix Equations
Many problems in computer graphics require the numerical solution of matrix equations. Some Problems Related to matrices include:
Obtain the best position and direction to make the objects match each other (Least Square Method), and create a curve that overwrites the given point set.
Surface, and make the minimum degree of wrinkle (thin slice algorithm), as well as material simulation, such as water and clothes. In Graphics
Matrix expressions are quite popular. Therefore, in the mathematics used in graphics, I have a high rating on matrix equations.
Recommended reference books:
Matrix computations
Gene Golub and Charles Van Loan
Johns Hopkins University Press

 

 

Physics
Physics is clearly not a branch of mathematics. It is a self-developed discipline. However, in some areas of computer graphics,
Physics and mathematics are closely related. In Graphics, physics problems are involved, including how light and the surface of an object are
The movement of people and animals, and the flow of water and air. To simulate these natural phenomena, physical
Knowledge is essential. This is closely related to the settlement of the differential equations. I will refer to the differential equations in the next section.

 

 

Numerical Solution of Differential Equations
I believe that for computer graphics, the technique for solving differential equations is very important. As we discussed earlier,
Computer Graphics is designed to simulate physical systems originating in the real world. How waves are formed in water, animals are
How to walk on the ground is an example of two simulated physical systems. The problem with simulating physical systems is often
How to solve the numerical solution of the differential equation. Note that the numerical solution of the differential equation and the symbolic solution of the differential equation are very good.
Big difference. The Symbolic solution is used to find solutions without errors and is often used only for some very simple equations. Sometimes big
The "differential equation" in the course only teaches the Symbolic solution, but it does not help most computer graphics problems.
Help.
In the simulation of physical systems, we divide the world into many small elements that represent vectors. Then
The relationship between them can be described in a matrix. Although the matrix equations to be processed are often not accurate
Instead, a series of calculations are executed, which generate an approximate solution represented as a series. This is differentiation.
The numerical solution of the 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 set of objects for the desired purpose.
Method. For example, if you want to place a light to make the room lighting look like a special "feeling ",
We can only perform a special action on the active limbs. The above examples can be:
It comes down to the optimization problem. There was almost no literature on Optimization Technology for computer graphics a decade ago, but recently this
More and more attention is paid to optimization theory in the field. In my opinion, in computer graphics, optimization will become increasingly important.
Add.

 

 

Probability Theory and statistics
Probability Theory and statistics are used in many fields of computer graphics. When researchers are involved in anthropology
However, statistics are required to analyze the 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
Child: A growth tree, the probability of its branches splitting; how Virtual animals decide its route. Finally,

Some Techniques for solving difficult equations use random numbers to estimate the solution of the equations. Important example: Monte Carlo
The method is often used for how light is transmitted. The above is only part of the use of probability theory and statistics in computer graphics
Learning method.

 

 

Computational Geometry
Computational Geometry studies how to use computers to efficiently represent and operate ry. Typical problems such as Collision Detection
Edges are decomposed into triangles to find the points closest to a certain position. This discipline includes the 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 teaching materials]
Computational Geometry: An Introduction
Franco Preparata and Michael shamos
Springer-Verlag
[Very classic, but a little old]

 

 

Conclusion: Mathematical applications and mathematical theories
For graphics, many mathematical disciplines mentioned above have one thing in common: compared with the theoretical value of these mathematical,
We are more inclined to explore their application values. Don't be surprised. Many graphics problems and physics scholars and engineers
Their research problems are closely related, and the mathematical tools used by physics scholars and engineers are just the research of graphics.
. Most disciplines that study pure mathematical theory are never used in computer graphics. But this is not absolute
. Note these special cases: Molecular Biology is using a theory to study DNA molecular dynamics, sub-atomic physics.
Abstract group theory is used. 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. It is possible that topology is the best in this class of mathematics.
Interesting. A word is used to describe topology. It studies 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 a gauge
An important component of computer graphics, but the course of differential geometry already covers most topology useful to graphics
Knowledge. Differential Geometry studies the shape of a surface, but topology studies the adjacent relationship of a surface. I think topology is correct.
It is almost useless in graphics, because topology is concerned with abstract things, and topology is far away from most graphics.
The core of learning is the concept of three-dimensional Euclidean space. For graphics, the topology form (symbolic Notation) is
A simple way to express ideas, however, graphics seldom use practical tools for abstract topology. For graphics
Learning 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
IDEA is rarely used in computer graphics.
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