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首页Vector Math for 3D Computer Graphics
This is a tutorial on vector algebra and matrix algebra from the viewpoint of computer graphics. It covers most vector and matrix topics needed for college-level computer graphics text books. Most graphics texts cover these subjects in an appendix, but it is often too short. This tutorial covers the same material at greater length, and with many examples
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Vector Math Tutorial for 3D Computer Graphics
Vector Math for 3D Computer Graphics
An Interactive Tutorial
Second Revision, July 2000
This is a tutorial on vector algebra and matrix algebra from the viewpoint of computer
graphics. It covers most vector and matrix topics needed for college-level computer graphics
text books. Most graphics texts cover these subjects in an appendix, but it is often too short.
This tutorial covers the same material at greater length, and with many examples.
·
Chapter 1
--- Vectors, Points, and Displacement
·
Chapter 2
--- Vector Addition
·
Chapter 3
--- Displacement Vectors
·
Chapter 4
--- Length of Vectors
·
Chapter 5
--- Direction of Vectors
·
Chapter 6
--- Scaling and Unit Vectors
·
Chapter 7
--- The Dot Product
·
Chapter 8
--- Length and the Dot Product
·
Chapter 9
--- The Angle between two Vectors.
·
Chapter 10
--- The Angle between 3D Vectors.
·
Chapter 11
--- Projecting one Vector onto Another.
·
Chapter 12
--- Vector Cross Product.
·
Chapter 13
--- Matrices and Simple Matrix Operations.
·
Chapter 14
--- Matrix-Column Matrix Multiplicaton.
·
Chapter 15
--- Matrix-Matrix Multiplication
·
Chapter 16
--- Identity Matrix and Matrix Inverse
·
Index
Although primarily aimed at computer science students, this tutorial is useful to all
programmers interested in 3D computer graphics or 3D computer game programming. In spite
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Vector Math Tutorial for 3D Computer Graphics
of their appealing blood-and-gore covers, mass trade books on game programming require the
same understanding of vectors and matrices as more staid text books (and usually defer these
topics to the same skimpy mathematical appendix).
This tutorial is useful for more than computer graphics. Vectors and matrices are used in all
scientific and engineering fields, and any other field that uses computers (are there any that
don't?) In many fields, the vocabulary used for vectors and matrices does not match that used
in computer graphics. But the ideas are the same, and reading these notes will take only a
slight mental adjustment.
These notes assume that you have studied plane geometry and trigonometry sometime in the
past. Notions such as "point", "line", "plane", and "angle" should be familiar to you. Other
notions such as "sine", "cosine", "determinant", "real number", and the common trig identities
should at least be a distant memory.
These pages were designed at 800 by 600 resolution with "web safe" colors. They have been
(somewhat) tested with not-too-old versions of Netscape Navigator and Internet Explorer, using
"Times Roman" font (the usual browser default font). Many pages require Javascript, and some
pages require Java. If you lack these (or are behind a firewall that does not allow these inside)
you will be able to read most pages, but the interactive features will be lost.
Some sections are more than three years old and have been used in several lecture sections
(and hence are "classroom tested" and likely to be technically correct and readable). Other
sections have just been written and might fall short of both goals.
This tutorial may be freely downloaded and used as long as copyright and authorship
information is not removed. (They are contained in HTML comments on each page.) People
who wish to reward this effort may do so by going to their local public library and checking out
any long neglected, lonely book from the stacks.
● Return to the list of courses.
● Return to CCSU Computer Science Department home page.
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CHAPTER 1 --- Vectors, Points, and Column Matrices
created 08/19/97; revised 08/01/00; bug (ant) fixes 09/14/00
CHAPTER 1 --- Vectors, Points, and
Column Matrices
This chapter discusses the objects of computer graphics---vectors and points---and how they are
represented in a computer---as column matrices. A column matrix is a mathematical object that has
many uses besides its use in computer graphics. These notes discuss only the aspects that are used in
computer graphics.
Chapter Topics:
● Computer graphics as modeling and viewing.
● Geometrical points and vectors.
● Column and row matrices.
● Calculating displacements.
● Equality of column matrices.
● Names for column matrices.
● Representing points with column matrices.
Computer graphics books use one of two ways to represent points and vectors. Some books use row
matrices; other books use column matrices. The two methods are exactly equivalent (although some
formulae have to be adjusted). These notes use column matrices.
Some graphics books use the term "column vector" for the object that these notes call a "column
matrix." This is just a variation in terminology and does not affect the concepts or formulae presented
here.
QUESTION 1:
(Review: ) What two types of objects are represented with column matrices?
Click Here after you have answered the question
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Virtual Tourist
A good answer might be:
(1) Points, and (2) Vectors.
Virtual Tourist
By "geometrical point" these notes will mean the "point" of plane geometry and solid geometry.
Hopefully you have studied geometry sometime in the past.
Computer graphics consists of two activities: (1) Creating an imaginary world inside a computer, and
(2) producing two dimensional images of that world from various viewpoints. A graphics program is
like a tourist wandering through a fantastic landscape taking pictures with a camera. With computer
animation, the virtual tourist is equipped with a movie camera.
The imaginary landscape is built of objects in three dimensional space. Each object is a set of points
and connections between points that form the edges of a 3D solid. Here, for example is a set of points
that model a teapot (from the Open GL Utility Toolkit):
It may be hard to see, but the figure consists of points and the line segments that connect them. To
make a realistic picture of a teapot many operations are needed to fill in the area between line
segments, apply texture and lighting models, and other rendering operations. But the fundamental
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