1.7.1 Class Design
1.7.1 类设计

A key part of any graphics program is to have good classes or routines for geometric entities such as vectors and matrices, as well as graphics entities such as RGB colors and images. These routines should be made as clean and efficient as possible. A universal design question is whether locations and displacements should be separate classes because they have different operations; e.g., a location multiplied by one-half makes no geometric sense while one-half of a displacement does (Goldman, 1985; DeRose, 1989). There is little agreement on this question, which can spur hours of heated debate among graphics practitioners, but for the sake of example, let’s assume we will not make the distinction.
任何图形程序的关键部分都是要有好的类或例程来处理几何实体（例如矢量和矩阵）以及图形实体（例如 RGB 颜色和图像）。这些例程应尽可能简洁高效。一个普遍的设计问题是位置和位移是否应该作为单独的类，因为它们具有不同的操作；例如，位置乘以一半没有几何意义，而位移的一半有几何意义（Goldman，1985；DeRose，1989）。这个问题几乎没有一致意见，这可能会在图形从业者中引发数小时的激烈争论，但为了举例说明，我们假设我们不会做出区分。

This implies that some basic classes to be written include
这意味着要编写的一些基本类包括

• vector2. A 2D vector class that stores an x- and y-component. It should store these components in a length-2 array so that an indexing operator can be well supported. You should also include operations for vector addition, vector subtraction, dot product, cross product, scalar multiplication, and scalar division.
  vector2。存储x和y分量的 2D 向量类。它应将这些分量存储在长度为 2 的数组中，以便能够很好地支持索引运算符。您还应包括向量加法、向量减法、点积、叉积、标量乘法和标量除法的运算。

• vector3. A 3D vector class analogous to vector2.
  vector3.类似于 vector2 的 3D 矢量类。

• hvector. A homogeneous vector with four components (see Chapter 8).
  hvector。具有四个分量的齐次向量（参见第 8 章）。

• rgb. An RGB color that stores three components. You should also include operations for RGB addition, RGB subtraction, RGB multiplication, scalar multiplication, and scalar division.
  rgb。存储三个分量的 RGB 颜色。您还应包括 RGB 加法、RGB 减法、RGB 乘法、标量乘法和标量除法的运算。

• transform. A 4 × 4 matrix for transformations. You should include a matrix multiply and member functions to apply to locations, directions, and surface normal vectors. As shown in Chapter 7, these are all different.
  变换。用于变换的 4 × 4 矩阵。您应该包含矩阵乘法和成员函数，以应用于位置、方向和表面法向量。如第 7 章所示，这些都是不同的。

• image. A 2D array of RGB pixels with an output operation.
  图像。具有输出操作的 RGB 像素的二维数组。

In addition, you might or might not want to add classes for intervals, orthonormal bases, and coordinate frames.
此外，您可能想要或不想添加区间、正交基和坐标框架的类。

1.7.2 Float vs. Double
1.7.2 Float 与 Double

Modern architecture suggests that keeping memory use down and maintaining coherent memory access are the keys to efficiency. This suggests using single-precision data. However, avoiding numerical problems suggests using double-precision arithmetic. The tradeoffs depend on the program, but it is nice to have a default in your class definitions.
现代架构表明，降低内存使用率和保持一致的内存访问是提高效率的关键。这意味着使用单精度数据。但是，避免数值问题意味着使用双精度算法。权衡取决于程序，但在类定义中有一个默认值是很好的。

1.7.3 Debugging Graphics Programs
1.7.3 调试图形程序

If you ask around, you may find that as programmers become more experienced, they use traditional debuggers less and less. One reason for this is that using such debuggers is more awkward for complex programs than for simple programs. Another reason is that the most difficult errors are conceptual ones where the wrong thing is being implemented, and it is easy to waste large amounts of time stepping through variable values without detecting such cases. We have found several debugging strategies to be particularly useful in graphics.
如果你四处打听，你可能会发现，随着程序员经验的增加，他们使用传统调试器的次数越来越少。其中一个原因是，使用这种调试器调试复杂程序比调试简单程序更麻烦。另一个原因是，最难发现的错误是概念性错误，即执行了错误的事情，很容易浪费大量时间单步执行变量值而没有发现这种情况。我们发现几种调试策略在图形方面特别有用。

if x = 126 and y = 247 then
   print “blarg!”

2.1.1 Inverse Mappings
2.1.1 逆映射

If we have a function f : A ⟼ B, there may exist an inverse function f^–1: B ⟼ A, which is defined by the rule f^–1(b) = a where b = f (a) . This definition only works if every b ∈ B is an image of some point under f (i.e., the range equals the target) and if there is only one such point (i.e., there is only one a for which f (a) = b). Such mappings or functions are called bijections. A bijection maps every a ∈ A to a unique b ∈ B, and for every b ∈ B, there is exactly one a ∈ A such that f (a) = b (Figure 2.1). A bijection between a group of riders and horses indicates that everybody rides a single horse, and every horse is ridden. The two functions would be rider (horse) and horse (rider). These are inverse functions of each other. Functions that are not bijections have no inverse (Figure 2.2).
假设有一个函数f : A ⟼ B ，那么可能存在一个反函数f ^–1 : B ⟼ A ，其定义规则为f ^–1 ( b ) = a ，其中b = f ( a )。此定义仅当每个b ∈ B都是某个点在f下的像（即范围等于目标）并且仅有一个这样的点（即只有一个a使得f ( a ) = b ）时才成立。这样的映射或函数称为双射。双射将每个a ∈ A映射到唯一的b ∈ B ，并且对于每个b ∈ B ，都有且仅有一个a ∈ A使得f ( a ) = b （图 2.1 ）。一组骑手与马之间的双射表示每个人都骑一匹马，并且每匹马都被骑过。这两个函数分别是骑手（马）和马（骑手） 。它们是互为反函数。非双射函数没有逆（图 2.2 ）。

An example of a bijection is f : ℝ ⟼ ℝ, with f (x) = x³. The inverse is . This example shows that the standard notation can be somewhat awkward because x is used as a dummy variable in both f and f^–1. It is sometimes more intuitive to use different dummy variables, with y = f (x) and x = f^–1(y) . This yields the more intuitive y = x³ and . An example of a function that does not have an inverse is sqr : ℝ ⟼ ℝ, where sqr(x) = x². This is true for two reasons: first x² = (–x)², and second no members of the domain map to the negative portions of the target. Note that we can define an inverse if we restrict the domain and range to R⁺. Then, is a valid inverse.
双射的一个例子是f : ℝ ⟼ ℝ，其中f ( x ) = x ³ 。逆函数是f − 1 （十） =十3 。此示例表明，标准符号可能有些不方便，因为x在f和f ^–1中都用作虚拟变量。有时使用不同的虚拟变量更为直观，例如y = f ( x ) 和x = f ^–1 ( y )。这会产生更直观的y = x ³和十=是3 。没有逆函数的一个例子是sqr : ℝ ⟼ ℝ，其中sqr ( x ) = x ² 。这是正确的，原因有二：首先x ² = (– x ) ² ，其次，域中没有成员映射到目标的负部分。请注意，如果我们将域和范围限制为 R ⁺ ，我们可以定义逆函数。然后，十是有效的逆。

2.1.2 Intervals
2.1.2 间隔

Often, we would like to specify that a function deals with real numbers that are restricted in value. One such constraint is to specify an interval. An example of an interval is the real numbers between zero and one, not including zero or one. We denote this (0, 1) . Because it does not include its endpoints, this is referred to as an open interval. The corresponding closed interval, which does contain its endpoints, is denoted with square brackets: [0, 1]. This notation can be mixed; i.e., [0, 1) includes zero but not one. When writing an interval [a, b], we assume that a ≤ b. The three common ways to represent an interval are shown in Figure 2.3. The Cartesian products of intervals are often used. For example, to indicate that a point x is in the unit cube in 3D, we say x ∈ [0, 1]³.
我们经常会想说明一个函数所处理的实数在值上有限制。指定一个区间就是这样一个约束。区间的一个例子是零到一之间的实数，不包括零或一。我们将其表示为(0, 1)。因为它不包括其端点，所以这被称为开区间。相应的闭区间包含其端点，用方括号表示：[0, 1]。这种表示法可以混合使用；即 [0, 1) 包括零但不包括一。当写区间 [ a, b ] 时，我们假设a ≤ b 。图 2.3显示了表示区间的三种常用方式。通常使用区间的笛卡尔积。例如，为了表示点x位于三维中的单位立方体中，我们说x ∈ [0, 1] ³ 。

Intervals are particularly useful in conjunction with set operations: intersection, union, and difference. For example, the intersection of two intervals is the set of points they have in common. The symbol ∩ is used for intersection. For example, [3, 5)∩[4, 6] = [4, 5) . For unions, the symbol ∪ is used to denote points in either interval. For example, [3, 5) ∪ [4, 6] = [3, 6]. Unlike the first two operators, the difference operator produces different results depending on argument order. The minus sign is used for the difference operator, which returns the points in the left interval that are not also in the right. For example, [3, 5) – [4, 6] = [3, 4) and [4, 6] – [3, 5) = [5, 6]. These operations are particularly easy to visualize using interval diagrams (Figure 2.4).
区间与集合运算结合使用特别有用：路口，联盟，以及差。例如，两个区间的交集是它们共同的点集。符号 ∩ 用于表示交集。例如，[3, 5)∩[4, 6] = [4, 5) 。对于并集，符号∪用于表示任一区间内的点。例如，[3, 5) ∪ [4, 6] = [3, 6]。与前两个运算符不同，差运算符根据参数顺序产生不同的结果。差运算符使用减号，它返回左侧区间中不在右侧区间的点。例如，[3, 5) – [4, 6] = [3, 4) 和 [4, 6] – [3, 5) = [5, 6]。使用区间图可以特别容易地可视化这些运算（图 2.4 ）。

2.1.3 Logarithms
2.1.3 对数

Although not as prevalent today as they were before calculators, logarithms are often useful in problems where equations with exponential terms arise. By definition, every logarithm has a base a. The “log base a”of x is written_a x and is defined as “the exponent to which a must be raised to get x,” i.e.,
尽管如今对数不像计算器出现之前那样流行，但在涉及指数项的方程式的问题中，对数通常很有用。根据定义，每个对数都有一个底数a 。x的“底数为a的对数”写为_a x ，其定义为“必须将a提升到哪个指数才能得到x ”，即

Note that the logarithm base a and the function that raises a to a power are inverses of each other. This basic definition has several consequences:
请注意，对数底数a和对a求幂的函数互为逆。这个基本定义有几个后果：

When we apply calculus to logarithms, the special number e = 2.718... often turns up. The logarithm with base e is called the natural logarithm. We adopt the common shorthand ln to denote it:
当我们将微积分应用于对数时，经常会出现特殊数字e = 2.718...。以e为底的对数称为自然对数。我们采用常见的简写 ln 来表示它：

Note that the “≡” symbol can be read “is equivalent by definition.” Like π, the special number e arises in a remarkable number of contexts. Many fields use a particular base in addition to e for manipulations and omit the base in their notation, i.e., log x. For example, astronomers often use base 10 and theoretical computer scientists often use base 2. Because computer graphics borrows technology from many fields, we will avoid this shorthand.
请注意，“ ≡ ”符号可以理解为“根据定义等价”。与π一样，特殊数字e出现在大量上下文中。许多领域除了使用e之外还使用特定底数进行运算，并在其符号中省略底数，即log x 。例如，天文学家通常使用10底数，而理论计算机科学家通常使用2底数。由于计算机图形学借鉴了许多领域的技术，我们将避免这种简写。

The derivatives of logarithms and exponents illuminate why the natural logarithm is “natural”:
对数和指数的导数解释了为什么自然对数是“自然的”：

The constant multipliers above are unity only for a = e.
上述常数乘数仅当a = e时才是 1。

2.3.1 Angles
2.3.1 角度

Although we take angles somewhat for granted, we should return to their definition so we can extend the idea of the angle onto the sphere. An angle is formed between two half-lines (infinite rays stemming from an origin) or directions, and some convention must be used to decide between the two possibilities for the angle created between them as shown in Figure 2.6. An angle is defined by the length of the arc segment it cuts out on the unit circle. A common convention is that the smaller arc length is used, and the sign of the angle is determined by the order in which the two half-lines are specified. Using that convention, all angles are in the range [–π, π].
尽管我们在某种程度上认为角度是理所当然的，但我们应该回到它们的定义，这样我们就可以将角度的概念扩展到球面上。两个半线（从原点发出的无限射线）或方向之间会形成一个角度，必须使用一些约定来决定它们之间形成的两种角度的可能性，如图 2.6所示。角度由它在单位圆上切出的圆弧段的长度定义。常见的约定是使用较小的弧长，并且角度的符号由指定两个半线的顺序决定。使用该约定，所有角度都在 [-π, π] 范围内。

Each of these angles is the length of the arc of the unit circle that is “cut” by the two directions. Because the perimeter of the unit circle is 2π, the two possible angles sum to 2π. The unit of these arc lengths is radians. Another common unit is degrees, where the perimeter of the circle is 360°. Thus, an angle that is π radians is 180°, usually denoted 180°. The conversion between degrees and radians is
这些角度中的每一个都是被两个方向“切割”的单位圆的弧长。因为单位圆的周长是 2 π ，所以两个可能的角度之和为 2 π 。这些弧长的单位是弧度。另一个常见单位是度，其中圆的周长是 360°。因此， π弧度的角度为 180°，通常表示为 180°。度和弧度之间的转换是

2.3.2 Trigonometric Functions
2.3.2 三角函数

Given a right triangle with sides of length a, o, and h, where h is the length of the longest side (which is always opposite the right angle), or hypotenuse, an important relation is described by the Pythagorean theorem:
给定一个直角三角形，其边长分别为a 、 o和h ，其中h是最长边的长度（始终与直角相对），或者斜边，一个重要的关系由勾股定理描述：

You can see that this is true from Figure 2.7, where the big square has area (a+o)², the four triangles have the combined area 2ao, and the center square has area h².
从图 2.7中可以看出情况确实如此，其中大正方形的面积为 ( a + o ) ² ，四个三角形的面积合计为 2 ao ，中心正方形的面积为h ² 。

Because the triangles and inner square subdivide the larger square evenly, we have 2ao + h² = (a + o)², which is easily manipulated to the form above.
因为三角形和内部正方形将大​​正方形均匀地细分，所以我们有 2 ao + h ² = ( a + o ) ² ，它可以很容易地处理成上面的形式。

We define sine and cosine of ϕ, as well as the other ratio-based trigonometric expressions:
我们定义正弦和ϕ 的余弦，以及其他基于比率的三角表达式：

These definitions allow us to set up polar coordinates, where a point is coded as a distance from the origin and a signed angle relative to the positive x-axis (Figure 2.8). Note the convention that angles are in the range ϕ ∈ (–π, π], and that the positive angles are counterclockwise from the positive x-axis. This convention that counterclockwise maps to positive numbers is arbitrary, but it is used in many contexts in graphics so it is worth committing to memory.
这些定义使我们能够建立极坐标，其中点被编码为与原点的距离和相对于正x轴的带符号角度（图 2.8 ）。请注意惯例：角度在 ϕ ∈ (-π, π ] 范围内，并且正角度从正x轴逆时针旋转。逆时针映射到正数的惯例是任意的，但它在图形学的许多情况下都有使用，因此值得记住。

Trigonometric functions are periodic and can take any angle as an argument. For example, sin(A) = sin(A +2π) . This means the functions are not invertible when considered with the domain R. This problem is avoided by restricting the range of standard inverse functions, and this is done in a standard way in almost all modern math libraries (e.g., Plauger (1991)). The domains and ranges are
三角函数是周期性的，可以以任意角度作为参数。例如，sin( A ) = sin( A +2 π )。这意味着当用域 R 考虑时，函数不可逆。通过限制标准反函数的范围可以避免这个问题，几乎所有现代数学库（例如 Plauger (1991)）都以标准方式做到这一点。域和范围是

The last function, atan2(s, c) is often very useful. It takes an s value proportional to sin A and a c value that scales cos A by the same factor and returns A. The factor is assumed to be positive. One way to think of this is that it returns the angle of a 2D Cartesian point (s, c) in polar coordinates (Figure 2.9).
最后一个函数 atan2( s, c ) 通常非常有用。它采用与 sin A成比例的s值和将 cos A按相同因子缩放的c值并返回A 。假定因子为正。一种思考方式是，它返回极坐标中二维笛卡尔点 ( s, c ) 的角度（图 2.9 ）。

2.3.3 Useful Identities
2.3.3 有用的身份

This section lists without derivation a variety of useful trigonometric identities.
本节列出了各种有用的三角恒等式（但不作推导）。

Shifting identities:
身份转变：

Pythagorean identities:
毕达哥拉斯恒等式：

Half-angle identities:
半角恒等式：

Half-angle identities:
半角恒等式：

Product identities:
产品标识：

The following identities are for arbitrary triangles with side lengths a, b, and c, each with an angle opposite it given by A, B, C, respectively (Figure 2.10),
以下恒等式适用于边长为a、b和c的任意三角形，每个三角形的对角分别由A、B、C给出（图 2.10 ），

The area of a triangle can also be computed in terms of these side lengths:
三角形的面积也可以根据以下边长来计算：

2.3.4 Solid Angles and Spherical Trigonometry
2.3.4 立体角和球面三角学

Traditional trigonometry in this section deals with triangles on the plane. Triangles can be defined on non-planar surfaces as well, and one that arises in many fields, astronomy, for example, is triangles on the unit-radius sphere. These spherical triangles have sides that are segments of the great circles (unit-radius circles) on the sphere. The study of these triangles is a field called spherical trigonometry and is not used that commonly in graphics, but sometimes, it is critical when it does arise. We wont discuss the details of it here, but want the reader to be aware that area exists for when those problems do arise, and there are a lot of useful rules such as a spherical law of cosines and a spherical law of sines. For an example of the machinery of spherical trigonometry being used, see the paper on sampling triangle lights (which project to a spherical triangle) (Arvo, 1995b).
本节中的传统三角学涉及平面上的三角形。三角形也可以在非平面表面上定义，在许多领域（例如天文学）中出现的三角形是单位半径球面上的三角形。这些球面三角形的边是球面上的大圆（单位半径圆）。这些三角形的研究领域称为球面三角学，在图形学中并不常用，但有时，当它出现时，它至关重要。我们不会在这里讨论它的细节，但希望读者知道当这些问题确实出现时，存在面积，并且有很多有用的规则，例如球面余弦定律和球面正弦定律。有关正在使用的球面三角学机制的示例，请参阅关于采样三角形光（投射到球面三角形）的论文（Arvo，1995b）。

Of more central importance to computer graphics are solid angles. While angles allow us to quantify things like “what is the separation of those two poles in my visual field,” solid angles let us quantify things like “how much of my visual field does that airplane cover.” For traditional angles, we project the posts onto the unit circle and measure arc length between them on the unit circle. We work with angles often enough that many of us can forget this definition because it is all so intuitive to us now. Solid angles are just as simple, but they may seem more confusing because most of us learn about them as adults. For solid angles, we project the visible directions that “see” the airplane and project it onto the unit sphere and measure the area. This area is the solid angle in the same way the arc length is the angle. While angles are measured in radians and sum to 2π (the total length of a unit circle), solid angles are measured in steradians and sum to 4π (the total area of a unit sphere).
对计算机图形学来说，立体角更为重要。虽然角度使我们能够量化诸如“我的视野中两极之间的距离是多少”之类的事物，但立体角使我们能够量化诸如“那架飞机覆盖了我的视野的多少”之类的事物。对于传统角度，我们将柱子投影到单位圆上，并在单位圆上测量它们之间的弧长。我们经常使用角度，以至于我们中的许多人可能会忘记这个定义，因为它现在对我们来说非常直观。立体角同样简单，但它们可能看起来更令人困惑，因为我们大多数人都是成年后才了解它们的。对于立体角，我们将“看到”飞机的可见方向投影到单位球面上并测量面积。这个面积就是立体角，就像弧长就是角度一样。虽然角度以弧度为单位测量，总和为 2 π （单位圆的总长度），但立体角以球面度，总和为 4 π （单位球体的总面积）。

2.4.1 Vector Operations
2.4.1 向量运算

Vectors have most of the usual arithmetic operations that we associate with real numbers. Two vectors are equal if and only if they have the same length and direction. Two vectors are added according to the parallelogram rule. This rule states that the sum of two vectors is found by placing the tail of either vector against the head of the other (Figure 2.12). The sum vector is the vector that “completes the triangle” started by the two vectors. The parallelogram is formed by taking the sum in either order. This emphasizes that vector addition is commutative:
向量具有与实数相关的大多数常见算术运算。当且仅当两个向量具有相同的长度和方向，它们才相等。两个向量相加符合平行四边形规则。该规则指出，两个向量的和是通过将任一向量的尾部靠在另一个向量的头部上来获得的（图 2.12 ）。和向量是“完成由两个向量组成的三角形”的向量。以任意顺序求和可形成平行四边形。这强调了向量加法是交换的：

Note that the parallelogram rule just formalizes our intuition about displacements. Think of walking along one vector, tail to head, and then walking along the other. The net displacement is just the parallelogram diagonal. You can also create a unary minus for a vector: –a (Figure 2.13) is a vector with the same length as a but opposite direction. This allows us to also define subtraction:
请注意，平行四边形规则只是形式化了我们对位移的直觉。想象一下沿着一个向量行走，从尾部到头部，然后沿着另一个向量行走。净位移就是平行四边形的对角线。你也可以创建一个向量的一元减法： – a （图 2.13 ）是与a长度相同但方向相反的向量。这使我们能够定义减法：

You can visualize vector subtraction with a parallelogram (Figure 2.14). We can write
可以用平行四边形来直观地展示向量减法（图 2.14 ）。我们可以这样写

Vectors can also be multiplied. In fact, there are several kinds of products involving vectors. First, we can scale the vector by multiplying it by a real number k.
向量也可以相乘。事实上，向量的乘法有好几种。首先，我们可以通过将向量乘以实数k来缩放该向量。

This just multiplies the vector’s length without changing its direction. For example, 3.5a is a vector in the same direction as a, but it is 3.5 times as long as a. We discuss two products involving two vectors, the dot product and the cross product, later in this section, and a product involving three vectors, the determinant, in Chapter 6.
这只是将向量的长度相乘，而不改变其方向。例如，3.5 a是一个与a 方向相同的向量，但它的长度是a的 3.5 倍。我们将在本节后面讨论涉及两个向量的两种乘积，即点积和叉积，并在第 6 章讨论涉及三个向量的乘积，即行列式。

2.4.2 Cartesian Coordinates of a Vector
2.4.2 向量的笛卡尔坐标

A 2D vector can be written as a combination of any two nonzero vectors which are not parallel. This property of the two vectors is called linear independence. Two linearly independent vectors form a 2D basis, and the vectors are thus referred to as basis vectors. For example, a vector c may be expressed as a combination of two basis vectors a and b (Figure 2.15):
二维向量可以写成任意两个不平行的非零向量的组合。这两个向量的这种性质称为线性独立。两个线性独立的向量构成一个二维基，因此这些向量被称为基向量。例如，向量c可以表示为两个基向量a和b的组合（图 2.15 ）：

Note that the weights a_c and b_c are unique. Bases are especially useful if the two vectors are orthogonal; i.e., they are at right angles to each other. It is even more useful if they are also unit vectors in which case they are orthonormal. If we assume two such “special” vectors x and y are known to us, then we can use them to represent all other vectors in a Cartesian coordinate system, where each vector is represented as two real numbers. For example, a vector a might be represented as
注意，权重a _c和b _c是唯一的。如果两个向量正交；即它们彼此成直角。如果它们也是单位向量，则更有用，在这种情况下它们是正交的。如果我们假设我们知道两个这样的“特殊”向量x和y ，那么我们可以用它们来表示笛卡尔坐标系中的所有其他向量，其中每个向量都表示为两个实数。例如，向量a可以表示为

where x_a and y_a are the real Cartesian coordinates of the 2D vector a (Figure 2.16). Note that this is not really any different conceptually from Equation (2.3), where the basis vectors were not orthonormal. But there are several advantages to a Cartesian coordinate system. For instance, by the Pythagorean theorem, the length of a is
其中x _a和y _a是二维向量a的实数笛卡尔坐标（图 2.16 ）。请注意，这与公式 (2.3) 的概念并没有什么不同，因为公式中的基向量不是正交的。但笛卡尔坐标系有几个优点。例如，根据勾股定理， a的长度为

It is also simple to compute dot products, cross products, and coordinates of vectors in Cartesian systems, as we’ll see in the following sections.
在笛卡尔系统中计算点积、叉积和向量坐标也很简单，正如我们将在以下章节中看到的。

By convention, we write the coordinates of a either as an ordered pair (x_a,y_a) or a column matrix:
按照惯例，我们将a的坐标写为有序对 ( x _a , y _a ) 或列矩阵：

The form we use will depend on typographic convenience. We will also occasionally write the vector as a row matrix, which we will indicate as a^T:
我们使用^{的形式取决于印刷方便性。我们有时也会将向量写成行矩阵，我们将其表示为 T} ：

We can also represent 3D, 4D, etc., vectors in Cartesian coordinates. For the 3D case, we use a basis vector z that is orthogonal to both x and y.
我们还可以在笛卡尔坐标系中表示 3D、4D 等向量。对于 3D 情况，我们使用与x和y正交的基向量z 。

2.4.3 Dot Product
2.4.3 点积

The simplest way to multiply two vectors is the dot product. The dot product of a and b is denoted a · b and is often called the scalar product because it returns a scalar. The dot product returns a value related to its arguments’ lengths and the angle ϕ between them (Figure 2.17):
两个向量相乘最简单的方法是点积。a和b的点积表示为a · b ，通常称为标量积，因为它返回标量。点积返回与其参数长度以及它们之间的角度 ϕ 相关的值（图 2.17 ）：

The most common use of the dot product in graphics programs is to compute the cosine of the angle between two vectors.
图形程序中点积的最常见用途是计算两个向量之间角度的余弦。

The dot product can also be used to find the projection of one vector onto another. This is the length a→b of a vector a that is projected at right angles onto a vector b (Figure 2.18):
点积也可用于求一个向量在另一个向量上的投影。这是向量 a 以直角投影到向量 b上的长度a → b （图 2.18 ）：

The dot product obeys the familiar associative and distributive properties we have in real arithmetic:
点积遵循实数算术中我们熟悉的结合律和分配律：

If 2D vectors a and b are expressed in Cartesian coordinates, we can take advantage of x · x = y · y = 1 and x · y = 0 to derive that their dot product is
如果二维向量 a 和 b 以笛卡尔坐标表示，我们可以利用x · x = y · y = 1 和x · y = 0 来推导出它们的点积为

Similarly in 3Dwe can find
同样在 3D 中我们可以找到

2.4.4 Cross Product
2.4.4 叉积

The cross product a × b is usually used only for three-dimensional vectors; generalized cross products are discussed in references given in the chapter notes. The cross product returns a 3D vector that is perpendicular to the two arguments of the cross product. The length of the resulting vector is related to sin ϕ:
叉积a × b通常仅用于三维向量；广义叉积在章节注释中给出的参考文献中讨论。叉积返回一个垂直于叉积的两个参数的三维向量。结果向量的长度与 sin ϕ 有关：

The magnitude ||a × b|| is equal to the area of the parallelogram formed by vectors a and b. In addition, a × b is perpendicular to both a and b (Figure 2.19). Note that there are only two possible directions for such a vector. By definition, the vectors in the direction of the x-, y- and z-axes are given by
幅值 || a × b || 等于由向量a和b构成的平行四边形的面积。此外， a × b与a和b都垂直（图 2.19 ）。请注意，此类向量只有两个可能的方向。根据定义， x 轴、 y 轴和z轴方向上的向量由下式给出

and we set as a convention that x × y must be in the plus or minus z direction. The choice is somewhat arbitrary, but it is standard to assume that
并且我们约定x × y必须位于正或负z方向。选择有点随意，但标准做法是假设

All possible permutations of the three Cartesian unit vectors are
三个笛卡尔单位向量的所有可能排列是

Because of the sin ϕ property, we also know that a vector cross itself is the zero vector, so x × x = 0 and so on. Note that the cross product is not commutative, i.e., x × y = y × x. The careful observer will note that the above discussion does not allow us to draw an unambiguous picture of how the Cartesian axes relate. More specifically, if we put x and y on a sidewalk, with x pointing east and y pointing north, then does z point up to the sky or into the ground? The usual convention is to have z point to the sky. This is known as a right-handed coordinate system. This name comes from the memory scheme of “grabbing” x with your right palm and fingers and rotating it toward y. The vector z should align with your thumb. This is illustrated in Figure 2.20.
由于 sin ϕ 性质，我们还知道向量叉积本身是零向量，因此x × x = 0 ，依此类推。请注意，叉积不交换，即x × y = y × x 。细心的观察者会注意到，上述讨论并未让我们明确地描绘出笛卡尔坐标轴之间的关系。更具体地说，如果我们将x和y放在人行道上， x指向东， y指向北，那么z指向天空还是地面？通常的惯例是让z指向天空。这被称为右手坐标系。这个名称来自于用右手掌和手指“抓住” x并将其向y旋转的记忆方案。向量z应该与拇指对齐。如图 2.20所示。

The cross product has the nice property that
叉积具有良好的性质

and
和

However, a consequence of the right-hand rule is
然而，右手定则的结果是

In Cartesian coordinates, we can use an explicit expansion to compute the cross product:
在笛卡尔坐标中，我们可以使用显式展开来计算叉积：

So, in coordinate form,
因此，以坐标形式，

2.4.5 Orthonormal Bases and Coordinate Frames
2.4.5 正交基和坐标系

Managing coordinate systems is one of the core tasks of almost any graphics program; the key to this is managing orthonormal bases. Any set of two 2D vectors u and v form an orthonormal basis provided that they are orthogonal (at right angles) and are each of unit length. Thus,
管理坐标系是几乎所有图形程序的核心任务之一；其中的关键是管理正交基。任何两个二维向量u和v的集合构成一个正交基，前提是它们正交（成直角）且每个向量的长度为单位。因此，

and
和

In 3D, three vectors u, v,and w form an orthonormal basis if
在三维空间中，三个向量u 、 v和w构成一个正交基，若

and
和

This orthonormal basis is right-handed provided
该正交基为右手系，条件是

and otherwise, it is left-handed.
否则，它就是左撇子。

Note that the Cartesian canonical orthonormal basis is just one of infinitely many possible orthonormal bases. What makes it special is that it and its implicit origin location are used for low-level representation within a program. Thus, the vectors x, y, and z are never explicitly stored and neither is the canonical origin location o. The global model is typically stored in this canonical coordinate system, and it is thus often called the global coordinate system. However, if we want to use another coordinate system with origin p and orthonormal basis vectors u, v, and w, then we do store those vectors explicitly. Such a system is called a frame of reference or coordinate frame. For example, in a flight simulator, we might want to maintain a coordinate system with the origin at the nose of the plane, and the orthonormal basis aligned with the airplane. Simultaneously, we would have the master canonical coordinate system (Figure 2.21). The coordinate system associated with a particular object, such as the plane, is usually called a local coordinate system.
请注意，笛卡尔正则标准正交基只是无限多个可能的正交基之一。它的特殊之处在于，它及其隐式原点位置用于程序中的低级表示。因此，向量x 、 y和z从未明确存储，正则原点位置o也是如此。全局模型通常存储在这个正则坐标系中，因此它通常被称为全局坐标系。但是，如果我们想使用另一个以原点p和正交基向量u 、 v和w为原点的坐标系，那么我们确实会明确存储这些向量。这样的系统称为参考系或坐标系。例如，在飞行模拟器中，我们可能希望维护一个坐标系，其原点位于飞机机头，正交基与飞机对齐。同时，我们将拥有主标准坐标系（图 2.21 ）。与特定对象（例如飞机）相关联的坐标系通常称为局部 坐标 系 报告 错误.

At a low level, the local frame is stored in canonical coordinates. For example, if u has coordinates (x_u,y_u,z_u) ,
在低层次上，局部框架存储在标准坐标中。例如，如果u的坐标为 ( x _u , y _u , z _u )，

A location implicitly includes an offset from the canonical origin:
位置隐式包含与规范原点的偏移量：

where (x_p,y_p,z_p) are the coordinates of p.
其中（ _xp ， _yp ， _zp ）是p的坐标。

Note that if we store a vector a with respect to the u-v-w frame, we store a triple (u_a,v_a,w_a) which we can interpret geometrically as
请注意，如果我们存储相对于u - v - w框架的向量a ，我们存储一个三元组 ( u _a ,v _a ,w _a )，我们可以将其几何解释为

To get the canonical coordinates of a vector a stored in the u-v-w coordinate system, simply recall that u, v,and w are themselves stored in terms of Cartesian coordinates, so the expression u_au + v_av + w_aw is already in Cartesian coordinates if evaluated explicitly. To get the u-v-w coordinates of a vector b stored in the canonical coordinate system, we can use dot products:
要获取存储在u - v - w坐标系中的向量a的标准坐标，只需记住u 、 v和w本身是以笛卡尔坐标形式存储的，因此如果明确求值，表达式u _a u + v _a v + w _a w已经是笛卡尔坐标。要获取存储在标准坐标系中的向量b的u - v - w坐标，我们可以使用点积：

This works because we know that for some u_b, v_b,and w_b,
这是可行的，因为我们知道对于某些u _b 、 v _b和w _b ，

and the dot product isolates the u_b coordinate:
点积分离出u _b坐标：

This works because u, v,and w are orthonormal.
这是可行的，因为u 、 v和w是正交的。

Using matrices to manage changes of coordinate systems is discussed in Sections 7.2.1 and 7.5.
使用矩阵来管理坐标系的变化在第 7.2.1 和7.5节中讨论。

2.4.6 Constructing a Basis from a Single Vector
2.4.6 从单个向量构造基

Often we need an orthonormal basis that is aligned with a given vector. That is, given a vector a, we want an orthonormal u, v, and w such that w points in the same direction as a (Hughes & Möller, 1999), but we don’t particularly care what u and v are. One vector isn’t enough to uniquely determine the answer; we just need a robust procedure that will find any one of the possible bases.
我们经常需要与给定向量对齐的正交基。也就是说，给定向量a ，我们需要正交u 、 v和w ，使得w指向与a相同的方向（Hughes & Möller，1999），但我们并不特别关心u和v是什么。一个向量不足以唯一地确定答案；我们只需要一个可以找到任何可能基的强大程序。

This can be done using cross products as follows. First, make w a unit vector in the direction of a:
这可以使用交叉积来实现，如下所示。首先，使w成为a方向上的单位向量：

Then, choose any vector t not collinear with w, and use the cross product to build a unit vector u perpendicular to w:
然后，选择任何与w不共线的向量t ，并使用叉积构建一个垂直于w 的单位向量u ：

If t is collinear with w, the denominator will vanish, and if they are nearly collinear, the results will have low precision. A simple procedure to find a vector sufficiently different from w is to start with t equal to w and change the smallest magnitude component of t to 1. For example, if then . Once w and u are in hand, completing the basis is simple:
如果t与w共线，分母将消失，如果它们几乎共线，结果的精度将很低。找到与w足够不同的矢量的一个简单方法是从t等于w开始，然后将t的最小幅度分量更改为 1。例如，如果 w=(1/2−1/20)，则 t=(1/2−1/21)。一旦掌握了w和u ，完成基础就很简单：

An example of a situation where this construction is used is surface shading, where a basis aligned to the surface normal is needed but the rotation around the normal is often unimportant.
使用这种构造的情况的一个例子是表面着色，其中需要与表面法线对齐的基础，但围绕法线的旋转通常并不重要。

For serious production code, recently researchers at Pixar have developed a rather remarkable method for constructing a vector from two vectors that is impressive in its compactness and efficiency (Duff et al., 2017). They provide battle-tested code, and readers are encouraged to use it as there are not “gotchas” that have emerged as it used throughout the industry.
对于严肃的生产代码，皮克斯的研究人员最近开发了一种相当出色的方法，用于从两个向量构建一个向量，其紧凑性和效率令人印象深刻（Duff 等人，2017 年）。他们提供了经过实战检验的代码，鼓励读者使用它，因为它在整个行业中使用时没有出现“陷阱”。

2.4.7 Constructing a Basis from Two Vectors
2.4.7 从两个向量构造基

The procedure in the previous section can also be used in situations where the rotation of the basis around the given vector is important. A common example is building a basis for a camera: it’s important to have one vector aligned in the direction the camera is looking, but the orientation of the camera around that vector is not arbitrary, and it needs to be specified somehow. Once the orientation is pinned down, the basis is completely determined.
上一节中的过程还可用于基线围绕给定向量的旋转很重要的情况。一个常见的例子是为相机构建基线：重要的是让一个向量与相机所看的方向对齐，但相机围绕该向量的方向不是任意的，需要以某种方式指定。一旦确定了方向，基线就完全确定了。

A common way to fully specify a frame is by providing two vectors a (which specifies w) and b (which specifies v). If the two vectors are known to be perpendicular, it is a simple matter to construct the third vector by u = b × a.
完整指定框架的常用方法是提供两个向量a （指定w ）和b （指定v ）。如果已知这两个向量垂直，则通过u = b × a构造第三个向量是很简单的事情。

To be sure that the resulting basis really is orthonormal, even if the input vectors weren’t quite, a procedure much like the single-vector procedure is advisable:
为了确保最终得到的基础确实是正交的，即使输入向量不完全正交，建议采用类似于单向量过程的过程：

In fact, this procedure works just fine when a and b are not perpendicular. In this case, w will be constructed exactly in the direction of a,and v is chosen to be the closest vector to b among all vectors perpendicular to w.
事实上，当a和b不垂直时，此过程也很好用。在这种情况下， w将精确地沿a的方向构造，并且v被选为所有垂直于w的向量中最接近b的向量。

This procedure won’t work if a and b are collinear. In this case, b is of no help in choosing which of the directions perpendicular to a we should use: it is perpendicular to all of them.
如果a和b共线，则此过程将不起作用。在这种情况下， b对于选择我们应该使用哪个垂直于a 的方向毫无帮助：它垂直于所有方向。

In the example of specifying camera positions (Section 4.3), we want to construct a frame that has w parallel to the direction the camera is looking, and v should point out the top of the camera. To orient the camera upright, we build the basis around the view direction, using the straight-up direction as the reference vector to establish the camera’s orientation around the view direction. Setting v as close as possible to straight up exactly matches the intuitive notion of “holding the camera straight.”
在指定相机位置的示例中（第 4.3 节），我们希望构建一个框架，其中w与相机观察的方向平行，并且v应指向相机的顶部。为了使相机直立，我们围绕视线方向构建基础，使用直立方向作为参考向量来确定相机围绕视线方向的方向。将v设置为尽可能接近直立方向完全符合“保持相机直立”的直观概念。

2.4.8 Squaring Up a Basis
2.4.8 计算基数

Occasionally, you may find problems caused in your computations by a basis that is supposed to be orthonormal but where error has crept in—due to rounding error in computation, or to the basis having been stored in a file with low precision, for instance.
有时，您可能会发现计算中出现问题，问题是由本应是正交的基引起的，但其中却出现了错误 - 例如由于计算中的舍入误差，或者由于基存储在精度较低的文件中。

The procedure of the previous section can be used; simply constructing the basis anew using the existing w and v vectors will produce a new basis that is orthonormal and is close to the old one.
可以使用上一节的步骤；只需使用现有的w和v向量重新构建基，就会产生一个正交且接近旧基的新基。

This approach is good for many applications, but it is not the best available. It does produce accurately orthogonal vectors, and for nearly orthogonal starting bases, the result will not stray far from the starting point. However, it is asymmetric: it “favors” w over v and v over u (whose starting value is thrown away). It chooses a basis close to the starting basis but has no guarantee of choosing the closest orthonormal basis. When this is not good enough, the SVD (Section 6.4.1) can be used to compute an orthonormal basis that is guaranteed to be closest to the original basis.
这种方法适用于许多应用，但并非最佳方法。它确实能产生精确正交的向量，并且对于几乎正交的起始基，结果不会偏离起始点太远。但是，它是不对称的：它“偏爱” w而不是v ，偏爱 v而不是u （其起始值被丢弃）。它选择一个接近起始基的基，但不能保证选择最接近的正交基。当这还不够好时，可以使用 SVD（第 6.4.1 节）来计算保证最接近原始基的正交基。

2.5.1 Averages and weighted averages
2.5.1 平均值和加权平均值

Integrals compute the total of things. Lengths, areas, volumes, etc. But they are often used to compute averages. For example, we can compute the total volume of a region by integrating the elevation over a region (like a country).
积分计算事物的总和。长度、面积、体积等。但它们通常用于计算平均值。例如，我们可以通过对某个区域（如一个国家）的海拔进行积分来计算该区域的总体积。

float volume = integrate(elevation(), country)

But we could also compute the average elevation:
但我们也可以计算平均海拔：

float averageElevation = integrate(elevation(),
      country) / integrate(1, country)

This is basically “divide the volume by the area.” This can be abstracted as
这基本上就是“用体积除以面积”。这可以抽象为

Float averageElevation = average(elevation, country)

We can also take a weighted average. Here, we add a weighting function to emphasize some points in the average more than others. For example, if we want to emphasize a parts of the region by the temperature (this is pretty arbitrary, and we will see more graphics relevant examples in the next section):
我们也可以取加权平均值。在这里，我们添加一个加权函数来强调平均值中的某些点。例如，如果我们想通过温度来强调该区域的某些部分（这非常随意，我们将在下一节中看到更多与图形相关的示例）：

float weightedAverageElevation =
      integrate(temperature()*elevation(),
      country) / integrate(temperature(), country)

It’s a good idea to keep an eye out for this form; often integrals contain a weighted average without explicitly pointing that out and it can sometimes help intuition.
留意这种形式是个好主意；积分通常包含加权平均值而没有明确指出这一点，有时它可以帮助直觉。

2.5.2 Integrals over solid angle
2.5.2 立体角上的积分

One example of a type of integral we see a lot is one of these forms or something related:
我们经常看到的一种积分类型的一个例子是这些形式之一或相关形式：

float shade = integrate(cos()*f*(),
   unit-hemisphere)

Note that since integrate(cos(), unit-hemisphere) = pi, the weighted average version is just
请注意，由于integrate(cos(), unit-hemisphere) = pi ，加权平均版本只是

float shade = integrate((1/pi)*cos()*f*(),
       unit-hemisphere)

The more traditional form of this integral is
这个积分的更传统形式是

Or with spherical coordinates as we might use to solve such integrals algebraically:
或者使用球坐标，我们可以用代数方法求解此类积分：

The sine term if an area-correction factor for spherical coordinates. Note that in graphics, we will rarely need to write that all out and will use simpler forms without explicit coordinates as we numerically solve the integrals
正弦项是球面坐标的面积校正因子。请注意，在图形中，我们很少需要将其全部写出，并且在我们以数字方式求解积分时，我们将使用没有明确坐标的更简单的形式

The particular integral above is the shade of a perfectly reflective matte (dif-fuse) surface, and it is also a weighted average of all incident colors. This structure can be great for intuition; the color of a surface is usually related to a weighted average of incident colors.
上面的特定积分是完全反射的哑光（漫反射）表面的色调，也是所有入射颜色的加权平均值。这种结构对于直觉来说非常有用；表面的颜色通常与入射颜色的加权平均值相关。

The integrals over solid angle are almost always the same but use a wide variety of notations. Key is to recognize this is just notations and map the notations you see to one you are most comfortable with. This is much like reading pseudocode!
立体角上的积分几乎总是相同的，但使用各种各样的符号。关键是要认识到这只是符号，并将您看到的符号映射到您最熟悉的符号。这很像阅读伪代码！

2.7.1 2D Implicit Curves
2.7.1 二维隐式曲线

Intuitively, a curve is a set of points that can be drawn on a piece of paper without lifting the pen. A common way to describe a curve is using an implicit equation. An implicit equation in two dimensions has the form
直观地讲，曲线是一组点，可以不用抬起笔就画在一张纸上。描述曲线的常用方法是使用隐式方程。二维隐式方程的形式为

The function f (x, y) returns a real value. Points (x, y) where this value is zero are on the curve, and points where the value is nonzero are not on the curve. For example, let’s say that f (x, y) is
函数f ( x, y ) 返回一个实数值。该值为零的点 ( x, y ) 在曲线上，而该值非零的点不在曲线上。例如，假设f ( x, y ) 是

where (x_c,y_c) is a 2D point and r is a nonzero real number. If we take f (x, y) = 0, the points where this equality holds are on the circle with center (x_c,y_c) and radius r. The reason that this is called an “implicit” equation is that the points (x, y) on the curve cannot be immediately calculated from the equation and instead must be determined by solving the equation. Thus, the points on the curve are not generated by the equation explicitly, but they are buried somewhere implicitly in the equation.
其中 ( x _c , y _c ) 是二维点， r是非零实数。如果取f ( x, y ) = 0，则该等式成立的点位于以 ( x _c , y _c ) 为圆心、半径为r 的圆上。之所以将其称为“隐式”方程，是因为曲线上的点 ( x, y ) 不能立即从方程中计算出来，而是必须通过求解方程来确定。因此，曲线上的点不是由方程明确生成的，而是隐式地埋在方程的某个地方。

It is interesting to note that f does have values for all (x, y) . We can think of f as a terrain, with sea level at f = 0 (Figure 2.24). The shore is the implicit curve. The value of f is the altitude. Another thing to note is that the curve partitions space into regions where f > 0, f < 0, and f = 0. So you evaluate f to decide whether a point is “inside” a curve. Note that f (x, y) = c is a curve for any constant c,and c = 0 is just used as a convention. For example, if f (x, y) = x² + y² – 1, varying c just gives a variety of circles centered at the origin (Figure 2.25).
值得注意的是， f对于所有的 ( x, y ) 都有值。我们可以把f想象成地形，在f = 0 处为海平面（图 2.24 ）。海岸是隐式曲线。f 的值是海拔。另外要注意的是，曲线将空间划分为f > 0、 f < 0 和f = 0 的区域。因此，通过计算f可以判断某个点是否在曲线“内部”。请注意， f ( x, y ) = c是任意常数c 的^曲线，而c = 0 只是作为惯例。例如，如果f ( x, y ) = x2 + y2 – 1，则改变^c只会给出以原点为中心的各种圆（图 2.25 ）。

We can compress our notation using vectors. If we have c = (x_c,y_c) and p = (x, y) , then our circle with center c and radius r is defined by those position vectors that satisfy
我们可以使用向量来压缩符号。如果我们有c = ( x _c , y _c ) 和p = ( x, y ) ，那么以c为圆心、 r为半径的圆由满足以下条件的位置向量定义：

This equation, if expanded algebraically, will yield Equation (2.9), but it is easier to see that this is an equation for a circle by “reading” the equation geometrically. It reads, “points p on the circle have the following property: the vector from c to p when dotted with itself has value r².” Because a vector dotted with itself is just its own length squared, we could also read the equation as, “points p on the circle have the following property: the vector from c to p has squared length r².”
如果用代数方式展开这个方程，就会得到方程 (2.9)，但通过用几何方式“解读”这个方程，更容易看出这是一个圆的方程。它是这样理解的：“圆上的点p具有以下属性：从c到p的向量加自身点后，其值为r ² 。”因为加自身点的向量就是其自身长度的平方，所以我们也可以将方程解读为：“圆上的点p具有以下属性：从c到p的向量的长度平方为r ² 。”

Even better, is to observe that the squared length is just the squared distance from c to p, which suggests the equivalent form
更好的是，观察到长度的平方就是从c到p的距离的平方，这表明等效形式

and, of course, this suggests
当然，这意味着

The above could be read “the points p on the circle are those a distance r from the center point c,” which is as good a definition of circle as any. This illustrates that the vector form of an equation often suggests more geometry and intuition than the equivalent full-blown Cartesian form with x and y. For this reason, it is usually advisable to use vector forms when possible. In addition, you can support a vector class in your code; the code is cleaner when vector forms are used. The vector-oriented equations are also less error prone in implementation: once you implement and debug vector types in your code, the cut-and-paste errors involving x, y,and z will go away. It takes a little while to get used to vectors in these equations, but once you get the hang of it, the payoff is large.
上述内容可以理解为“圆上的点p是与中心点c距离为r的点”，这与任何圆的定义一样好。这说明方程的矢量形式通常比等效的具有x和y 的完整的笛卡尔形式更能体现几何学和直觉。因此，通常建议尽可能使用矢量形式。此外，您可以在代码中支持矢量类；使用矢量形式时代码会更简洁。面向矢量的方程在实现时也更不容易出错：一旦您在代码中实现并调试矢量类型，涉及x 、 y和z 的剪切粘贴错误就会消失。需要一点时间来适应这些方程中的矢量，但一旦掌握了窍门，回报将是巨大的。

2.7.2 The 2D Gradient
2.7.2 二维梯度

If we think of the function f (x, y) as a height field with height = f (x, y) , the gradient vector points in the direction of maximum upslope, i.e., straight uphill. The gradient vector ∇f(x, y) is given by
如果我们将函数f ( x, y ) 视为高度 = f ( x, y ) 的高度场，则梯度向量指向最大上坡方向，即直线上坡。梯度向量 ∇ f ( x, y ) 由下式给出

The gradient vector evaluated at a point on the implicit curve f (x, y) = 0 is perpendicular to the tangent vector of the curve at that point. This perpendicular vector is usually called the normal vector to the curve. In addition, since the gradient points uphill, it indicates the direction of the f (x, y) > 0 region.
在隐式曲线f ( x, y ) = 0 上某一点求得的梯度向量垂直于该点处曲线的切向量。这个垂直向量通常称为曲线的法向量。此外，由于梯度指向上坡，它指示了f ( x, y ) > 0 区域的方向。

In the context of height fields, the geometric meaning of partial derivatives and gradients is more visible than usual. Suppose that near the point (a, b) , f (x, y) is a plane (Figure 2.26). There is a specific uphill and downhill direction. At right angles to this direction is a direction that is level with respect to the plane. Any intersection between the plane and the f (x, y) = 0 plane will be in the direction that is level. Thus, the uphill/downhill directions will be perpendicular to the line of intersection f (x, y) = 0. To see why the partial derivative has something to do with this, we need to visualize its geometric meaning. Recall that the conventional derivative of a 1D function y = g(x) is
在高度场中，偏导数和梯度的几何意义比平常更加明显。假设在点 ( a, b ) 附近， f ( x, y ) 是一个平面（图 2.26 ）。有一个特定的上坡和下坡方向。与此方向垂直的方向是与平面水平的方向。平面与f ( x, y ) = 0 平面之间的任何交点都将在水平方向上。因此，上坡/下坡方向将垂直于交线f ( x, y ) = 0。要了解偏导数为何与此有关，我们需要将其几何意义形象化。回想一下，一维函数y = g ( x ) 的传统导数为

This measures the slope of the tangent line to g (Figure 2.27).
测量结果为g的切线斜率（图 2.27 ）。

The partial derivative is a generalization of the 1D derivative. For a 2D function f (x, y) , we can’t take the same limit for x as in Equation (2.10), because f can change in many ways for a given change in x. However, if we hold y constant, we can define an analog of the derivative, called the partial derivative (Figure 2.28):
偏导数是一维导数的推广。对于二维函数f ( x, y ) ，我们不能像公式 (2.10) 中那样对x取相同的极限，因为f会随着x的给定变化而发生多种变化。但是，如果我们保持y不变，我们可以定义导数的类似物，称为偏导数（图 2.28 ）：

Why is it that the partial derivatives with respect to x and y are the components of the gradient vector? Again, there is more obvious insight in the geometry than in the algebra. In Figure 2.29, we see the vector a travels along a path where f does not change. Note that this is again at a small enough scale that the surface height (x, y) = f (x, y) can be considered locally planar. From the figure, we see that the vector a = (Δx, Δy) .
为什么关于x和y 的偏导数是梯度向量的分量？同样，几何学比代数更能说明这个问题。在图 2.29中，我们看到向量a沿着f不变的路径行进。请注意，这又是在一个足够小的尺度上，表面高度 ( x, y ) = f ( x, y ) 可以被认为是局部平面的。从图中，我们看到向量a = (Δ x, Δ y )。

Because the uphill direction is perpendicular to a, we know the dot product is equal to zero:
因为上坡方向垂直于a ，所以我们知道点积等于零：

We also know that the change in f in the direction (x_a,y_a) equals zero:
我们还知道f在方向 ( x _a , y _a ) 上的变化等于零：

Given any vectors (x, y) and (x,y) that are perpendicular, we know that the angle between them is 90 degrees, and thus, their dot product equals zero (recall that the dot product is proportional to the cosine of the angle between the two vectors). Thus, we have xx + yy = 0. Given (x, y) , it is easy to construct valid vectors whose dot product with (x, y) equals zero, the two most obvious being (y,–x) and (–y, x) ; you can verify that these vectors give the desired zero dot product with (x, y). A generalization of this observation is that (x, y) is perpendicular to k(y,–x) where k is any nonzero constant. This implies that
给定任何垂直向量 ( x, y ) 和 ( x,y )，我们知道它们之间的角度是 90 度，因此它们的点积等于零（回想一下，点积与两个向量之间角度的余弦成比例）。因此，我们有xx + yy = 0。给定 ( x, y )，很容易构造与 ( x, y ) 的点积等于零的有效向量，最明显的两个是 ( y,–x ) 和 (– y, x ) ；您可以验证这些向量与 ( x, y ) 的点积是否为零。这一观察的概括是 ( x, y ) 垂直于k ( y ,–x)，其中k是任何非零常数。这意味着

Combining Equations (2.11) and (2.12) gives
结合方程 (2.11) 和 (2.12) 可得

where k is any nonzero constant. By definition, “uphill” implies a positive change in f , so we would like k > 0,and k = 1 is a perfectly good convention.
其中k是任何非零常数。根据定义，“上坡”意味着f发生正变化，因此我们希望k > 0,和k = 1 是一个非常好的惯例。

As an example of the gradient, consider the implicit circle x² + y² – 1 = 0 with gradient vector (2x, 2y) , indicating that the outside of the circle is the positive region for the function f (x, y) = x² + y² – 1. Note that the length of the gradient vector can be different depending on the multiplier in the implicit equation. For example, the unit circle can be described by Ax² + Ay² – A = 0 for any nonzero A. The gradient for this curve is (2Ax, 2Ay) . This will be normal (perpendicular) to the circle, but will have a length determined by A. For A > 0, the normal will point outward from the circle, and for A < 0, it will point inward. This switch from outward to inward is as it should be, since the positive region switches inside the circle. In terms of the height-field view, h = Ax² + Ay² – A, and the circle is at zero altitude. For A > 0, the circle encloses a depression, and for A < 0, the circle encloses a bump. As A becomes more negative, the bump increases in height, but the h = 0 circle doesn’t change. The direction of maximum uphill doesn’t change, but the slope increases. The length of the gradient reflects this change in degree of the slope. So intuitively, you can think of the gradient’s direction as pointing uphill and its magnitude as measuring how uphill the slope is.
以梯度为例，考虑隐式圆x ² + y ² – 1 = 0，其梯度向量为 (2 x, 2 y ) ，表示圆外部是函数f ( x, y ) = x ² + y ² – 1 的正区域。请注意，梯度向量的长度可能因隐式方程中的乘数而不同。例如，对于任何非零的A ，单位圆可以用Ax ² + Ay ² – A = 0 来描述。此曲线的梯度为 (2 Ax, 2 Ay )。这将垂直于圆，但长度由A决定。对于A > 0，法线将指向圆外，而对于A < 0，它将指向圆内。这种从外到内的转变是理所当然的，因为正区域在圆内切换。从高度场角度看， h = Ax ² + Ay ² – A ，圆位于零高度。对于A > 0，圆包围一个凹陷，对于A < 0，圆包围一个凸起。随着A变得越来越负，凸起的高度会增加，但h = 0 圆不会改变。最大上坡方向不会改变，但坡度会增加。坡度的长度反映了坡度的变化。 因此，直观地讲，你可以将坡度的方向视为指向上坡，将其大小视为测量斜坡的上坡程度。

Implicit 2D Lines
隐式二维线

The familiar “slope-intercept” form of the line is
熟悉的“斜率截距”直线形式是

This can be converted easily to implicit form (Figure 2.30):
这可以很容易地转换为隐式形式（图 2.30 ）：

Here, m is the “slope” (ratio of rise to run), and b is the y value where the line crosses the y-axis, usually called the y-intercept. The line also partitions the 2D plane, but here “inside” and “outside” might be more intuitively called “over” and “under.”
这里， m是“斜率”（上升与下降的比率）， b是直线与y轴相交处的y值，通常称为y 截距。该线也将二维平面进行划分，但这里的“内部”和“外部”可能更直观地称为“上方”和“下方”。

Because we can multiply an implicit equation by any constant without changing the points where it is zero, kf (x, y) = 0 is the same curve for any nonzero k. This allows several implicit forms for the same line, for example,
因为我们可以将隐式方程乘以任何常数而不改变其为零的点，所以对于任何非零k来说， kf ( x, y ) = 0 都是同一条曲线。这允许同一条直线有几种隐式形式，例如，

One reason the slope-intercept form is sometimes awkward is that it can’t represent some lines such as x = 0 because m would have to be infinite. For this reason, a more general form is often useful:
斜率截距形式有时不方便的一个原因是它不能表示某些直线，例如x = 0，因为m必须是无限的。因此，更一般的形式通常很有用：

for real numbers A, B, C.
对于实数A 、 B 、 C 。

Suppose we know two points on the line, (x₀, y₀) and (x₁,y₁) . What A, B, and C describe the line through these two points? Because these points lie on the line, they must both satisfy Equation (2.15):
假设我们知道直线上的两个点， _（ x0 ， y0 ）和（ x1 ，y1 _） 。A 、 B和C分别描述过这两点_的直线吗？因为这些点位于直线上，所以它们都必须满足公式（2.15 _） ：

Unfortunately, we have two equations and three unknowns: A, B, and C. This problem arises because of the arbitrary multiplier we can have with an implicit equation. We could set C = 1 for convenience:
不幸的是，我们有两个方程和三个未知数： A、B和C。这个问题的出现是因为我们可以用隐式方程得到任意乘数。为了方便起见，我们可以设置C = 1：

but we have a similar problem to the infinite slope case in slope-intercept form: lines through the origin would need to have A(0) + B(0) + 1 = 0, which is a contradiction. For example, the equation for a 45–° line through the origin can be written x – y = 0, or equally well y – x = 0, or even 17y – 17x = 0, but it cannot be written in the form Ax + By +1 = 0.
但在斜率截距形式中，我们有一个与无限斜率情况类似的问题：通过原点的直线需要有A (0) + B (0) + 1 = 0，这是一个矛盾。例如，通过原点的 45 °直线方程可以写成x – y = 0，或者同样可以写成y – x = 0，甚至可以写成 17 y – 17 x = 0，但不能写成Ax + By +1 = 0 的形式。

Whenever we have such pesky algebraic problems, we try to solve the problems using geometric intuition as a guide. One tool we have, as discussed in Section 2.7.2, is the gradient. For the line Ax + By + C = 0, the gradient vector is (A, B) . This vector is perpendicular to the line (Figure 2.31), and points to the side of the line where Ax + By + C is positive. Given two points on the line (x₀,y₀) and (x₁,y₁) , we know that the vector between them points in the same direction as the line. This vector is just (x₁ – x₀,y₁ – y₀), and because it is parallel to the line, it must also be perpendicular to the gradient vector (A, B) . Recall that there are an infinite number of (A, B, C) that describe the line because of the arbitrary scaling property of implicits. We want any one of the valid (A, B, C) .
每当我们遇到这类棘手的代数问题时，我们都会尝试以几何直觉为指导来解决这些问题。如第 2.7.2 节所述，我们拥有的一个工具是梯度。对于直线Ax + By + C ₌ 0，梯度向量为（ A，B ）。该向量垂直于直线（图 2.31 ），并指向直线上Ax + By + C为正的一侧。给定直线上_的两点（ x0 ，y0 _）和_（ x1 ，y1 _） ，我们知道它们之间的向量指向与_直线相同的方向。这个向量就是（ x1 - x0 ，y1 _- y0 ） ，因为_它平行于直线，所以它也必须垂直于梯度向量（ A，B ）。回想一下，由于隐式函数的任意缩放属性，描述直线的（ A，B，C ）有无数个。我们想要有效的（ A，B，C ）中的任意一个。

We can start with any (A, B) perpendicular to (x₁–x₀,y₁–y₀). Such a vector is just (A, B) = (y₀ –y₁, x₁ – x₀) by the same reasoning as in Section 2.7.2. This means that the equation of the line through (x₀,y₀) and (x₁,y₁) is
我们可以从任何垂直于 ( x ₁ – x ₀ , y ₁ – y ₀ ) 的 ( A, B ) 开始。根据与第 2.7.2 节相同的推理，这样的向量就是 ( A, B ) = ( y ₀ –y ₁ , x ₁ – x ₀ )。这意味着通过 ( x ₀ ,y ₀ ) 和 ( x ₁ ,y ₁ ) 的直线方程为

Now we just need to find C. Because (x₀,y₀) and (x₁,y₁) are on the line, they must satisfy Equation (2.16). We can plug either value in and solve for C. Doing this for (x₀,y₀) yields C = x₀y₁ – x₁y₀, and thus, the full equation for the line is
现在我们只需要找到C 。因为（ x ₀ ,y ₀ ）和（ x ₁ ,y ₁ ）在直线上，所以它们必须满足公式 (2.16)。我们可以代入任意一个值并求解C 。对（ x ₀ ,y ₀ ）执行此操作可得出C = x ₀ y ₁ – x ₁ y ₀ ，因此，该直线的完整方程为

Again, this is one of infinitely many valid implicit equations for the line through two points, but this form has no division operation and thus no numerically degenerate cases for points with finite Cartesian coordinates. A nice thing about Equation (2.17) is that we can always convert to the slope-intercept form (when it exists) by moving the non-y terms to the right-hand side of the equation and dividing by the multiplier of the y term:
同样，这是通过两点的直线的无数有效隐式方程之一，但这种形式没有除法运算，因此对于具有有限笛卡尔坐标的点，没有数值退化的情况。方程 (2.17) 的一个好处是，我们总是可以通过将非y项移到等式的右侧并除以y项的乘数来转换为斜率截距形式（当它存在时）：

An interesting property of the implicit line equation is that it can be used to find the signed distance from a point to the line. The value of Ax + By + C is proportional to the distance from the line (Figure 2.32). As shown in Figure 2.33, the distance from a point to the line is the length of the vector k(A, B) ,which is
隐式直线方程的一个有趣特性是，它可以用来求出从点到直线的有符号距离。Ax + By + C的值与到直线的距离成正比（图 2.32 ）。如图 2.33所示，从点到直线的距离是矢量k ( A, B ) 的长度，即

For the point (x, y)+ k(A, B) ,thevalueof f (x, y) = Ax + By + C is
对于点 ( x, y )+ k ( A, B ) ， f ( x, y ) = Ax + By + C 的值为

The simplification in that equation is a result of the fact that we know (x, y) is on the line, so Ax + By + C = 0. From Equations (2.18) and (2.19), we can see that the signed distance from line Ax + By + C = 0 to a point (a, b) is
该方程的简化是因为我们知道 ( x, y ) 在线上，所以Ax + By + C = 0。从方程 (2.18) 和 (2.19)，我们可以看出从直线Ax + By + C = 0 到点 ( a, b ) 的有符号距离为

Here, “signed distance” means that its magnitude (absolute value) is the geometric distance, but on one side of the line, distances are positive and on the other, they are negative. You can choose between the equally valid representations f (x, y) = 0 and –f (x, y) = 0 if your problem has some reason to prefer a particular side being positive. Note that if (A, B) is a unit vector, then f (a, b) is the signed distance. We can multiply Equation (2.17) by a constant that ensures that (A, B) is a unit vector:
此处，“有符号距离”表示其量级（绝对值）是几何距离，但在线的一侧，距离为正，而另一侧则为负。如果您的问题有某种原因需要某一侧为正，则可以在同样有效的表示f ( x, y ) = 0 和–f ( x, y ) = 0 之间进行选择。请注意，如果 ( A, B ) 是单位向量，则f ( a, b ) 是有符号距离。我们可以将公式 (2.17) 乘以一个常数，以确保 ( A, B ) 是单位向量：

Note that evaluating f (x, y) in Equation (2.20) directly gives the signed distance, but it does require a square root to set up the equation. Implicit lines will turn out to be very useful for triangle rasterization (Section 9.1.2). Other forms for 2D lines are discussed in Chapter 13.
请注意，在公式 (2.20) 中直接求f ( x, y ) 可得出有符号距离，但需要平方根才能建立公式。隐式线对于三角形光栅化非常有用（第 9.1.2 节）。第 13 章将讨论二维线的其他形式。

Implicit Quadric Curves
隐式二次曲线

In the previous section, we saw that a linear function f (x, y) gives rise to an implicit line f (x, y) = 0. If f is instead a quadratic function of x and y, with the general form
在上一节中，我们看到线性函数f ( x, y ) 会产生一条隐式直线f ( x, y ) = 0。如果f是x和y的二次函数，则一般形式为

the resulting implicit curve is called a quadric. Two-dimensional quadric curves include ellipses and hyperbolas, as well as the special cases of parabolas, circles, and lines.
由此产生的隐式曲线称为二次曲线。二维二次曲线包括椭圆和双曲线，以及特殊情况的抛物线、圆和直线。

Examples of quadric curves include the circle with center (x_c, y_c) and radius r,
二次曲线的例子包括以 ( xc _, yc ) 为中心、半径为r 的_圆，

and axis-aligned ellipses of the form
以及轴对齐椭圆的形式

where (x_c,y_c) is the center of the ellipse, and a and b are the minor and major semi-axes (Figure 2.34).
其中（ _xc ， _yc ）是椭圆的中心， a和b分别是短半轴和长半轴（图 2.34 ）。

2.7.3 3D Implicit Surfaces
2.7.3 3D隐式曲面

Just as implicit equations can be used to define curves in 2D, they can be used to define surfaces in 3D. As in 2D, implicit equations implicitly define a set of points that are on the surface:
隐式方程可用于定义二维曲线，同样，它也可用于定义三维曲面。与二维一样，隐式方程隐式定义了曲面上的一组点：

Any point (x, y, z) that is on the surface results in zero when given as an argument to f . Any point not on the surface results in some number other than zero. You can check whether a point is on the surface by evaluating f , or you can check which side of the surface the point lies on by looking at the sign of f , but you cannot always explicitly construct points on the surface. Using vector notation, we will write such functions of p = (x, y, z) as
任何位于表面上的点 ( x, y, z ) 作为f的参数给出时都会返回零。任何不在表面上的点都会返回除零以外的某个数字。您可以通过评估f来检查某个点是否位于表面上，或者您可以通过查看f的符号来检查该点位于表面的哪一侧，但您无法始终明确构造表面上的点。使用向量符号，我们将p = ( x, y, z ) 的函数写为

2.7.4 Surface Normal to an Implicit Surface
2.7.4 隐式曲面的法向曲面

A surface normal (which is needed for lighting computations, among other things) is a vector perpendicular to the surface. Each point on the surface may have a different normal vector. In the same way that the gradient provides a normal to an implicit curve in 2D, the surface normal at a point p on an implicit surface is given by the gradient of the implicit function
表面法线（除其他外，还用于照明计算）是垂直于表面的向量。表面上的每个点可能具有不同的法线向量。与梯度为二维中的隐式曲线提供法线的方式相同，隐式表面上点p处的表面法线由隐式函数的梯度给出

The reasoning is the same as for the 2D case: the gradient points in the direction of fastest increase in f , which is perpendicular to all directions tangent to the surface, in which f remains constant. The gradient vector points toward the side of the surface where f (p) > 0, which we may think of as “into” the surface or “out from” the surface in a given context. If the particular form of f creates inward-facing gradients, and outward-facing gradients are desired, the surface –f (p) = 0 is the same as surface f (p) = 0 but has directionally reversed gradients, i.e., –∇f (p) = ∇(–f (p)) .
其原因与二维情况相同：梯度指向f增长最快的方向，该方向垂直于与表面相切的所有方向，其中f保持不变。梯度向量指向f ( p ) > 0 所在的表面一侧，在特定情况下，我们可以将其视为“进入”表面或“离开”表面。如果f的特定形式会产生向内的梯度，而需要向外的梯度，则表面–f ( p ) = 0 与表面f ( p ) = 0 相同，但梯度方向相反，即 –∇ f ( p ) = ∇ (–f ( p ))。

2.7.5 Implicit Planes
2.7.5 隐式平面

As an example, consider the infinite plane through point a with surface normal n. The implicit equation to describe this plane is given by
例如，考虑通过点a且表面法线为n的无限平面。描述该平面的隐式方程为

Note that a and n are known quantities. The point p is any unknown point that satisfies the equation. In geometric terms this equation says “the vector from a to p is perpendicular to the plane normal.” If p were not in the plane, then (p – a) would not make a right angle with n (Figure 2.35).
请注意， a和n是已知量。点p是满足该方程的任意未知点。从几何角度看，该方程表示“从a到p的矢量垂直于平面法线”。如果p不在平面上，则 ( p - a ) 不会与n成直角（图 2.35 ）。

Sometimes, we want the implicit equation for a plane through points a, b, and c. The normal to this plane can be found by taking the cross product of any two vectors in the plane. One such cross product is
有时，我们想要通过点a 、 b和c 的平面的隐式方程。可以通过对平面中任意两个向量进行叉积来找到该平面的法线。其中一个叉积是

This allows us to write the implicit plane equation:
这使得我们可以写出隐式平面方程：

A geometric way to read this equation is that the volume of the parallelepiped defined by p – a, b – a, and c – a is zero; i.e., they are coplanar. This can only be true if p is in the same plane as a, b, and c. The full-blown Cartesian representation for this is given by the determinant (this is discussed in more detail in Section 6.3):
用几何学的方法来理解这个方程，就是p – a 、 b – a和c – a定义的平行六面体的体积为零；也就是说，它们是共面的。只有当p与a 、 b和c位于同一平面时，这才是正确的。完整的笛卡尔表示由行列式给出（第 6.3 节将对此进行更详细的讨论）：

The determinant can be expanded (see Section 6.3 for the mechanics of expanding determinants) to the bloated form with many terms.
行列式可以展开（有关展开行列式的机制见第 6.3 节）为具有多项式的膨胀形式。

Equations (2.22) and (2.23) are equivalent, and comparing them is instructive. Equation (2.22) is easy to interpret geometrically and will yield efficient code. In addition, it is relatively easy to avoid a typographic error that compiles into incorrect code if it takes advantage of debugged cross and dot product code. Equation (2.23) is also easy to interpret geometrically and will be efficient provided an efficient 3 × 3 determinant function is implemented. It is also easy to implement without a typo if a function determinant (a, b, c) is available. It will be especially easy for others to read your code if you rename the determinant function volume. So both Equations (2.22) and (2.23) map well into code. The full expansion of either equation into x-, y-, and z-components is likely to generate typos. Such typos are likely to compile and, thus, to be especially pesky. This is an excellent example of clean math generating clean code and bloated math generating bloated code.
方程 (2.22) 和 (2.23) 是等价的，对它们进行比较很有启发。方程 (2.22) 很容易从几何角度解释，并且可以产生高效的代码。此外，如果利用已调试的交叉积和点积代码，则可以相对容易地避免编译成错误代码的印刷错误。方程 (2.23) 也很容易从几何角度解释，并且只要实现了高效的 3 × 3 行列式函数，它就会很高效。如果有函数行列式( a , b , c )，它也很容易实现而没有打字错误。如果将行列式函数重命名为体积，其他人将特别容易阅读您的代码。因此，方程 (2.22) 和 (2.23) 都可以很好地映射到代码中。将任何一个方程完全展开为x -、 y - 和z -分量都可能产生打字错误。这种打字错误很可能通过编译，因此特别麻烦。这是干净的数学生成干净的代码和臃肿的数学生成臃肿的代码的绝佳例子。

2.7.6 2D Parametric Curves
2.7.6 2D 参数曲线

A parametric curve is controlled by a single parameter that can be considered a sort of index that moves continuously along the curve. Such curves have the form
参数曲线由单个参数控制，该参数可以看作是沿曲线连续移动的一种指数。此类曲线的形式为

Here, (x, y) is a point on the curve, and t is the parameter that influences the curve. For a given t, there will be some point determined by the functions g and h. For continuous g and h, a small change in t will yield a small change in x and y. Thus, as t continuously changes, points are swept out in a continuous curve. This is a nice feature because we can use the parameter t to explicitly construct points on the curve. Often, we can write a parametric curve in vector form,
这里，（ x，y ）是曲线上的一个点， t是影响曲线的参数。对于给定的t ，将存在由函数g和h确定的某个点。对于连续的g和h ， t的微小变化将导致x和y的微小变化。因此，随着t的连续变化，点会以连续曲线的形式扫过。这是一个很好的特性，因为我们可以使用参数t来明确构造曲线上的点。通常，我们可以以矢量形式写出参数曲线，

where f is a vector-valued function, . Such vector functions can generate very clean code, so they should be used when possible.
其中f是矢量值函数， f ： R → R 2 。此类向量函数可以生成非常干净的代码，因此应尽可能使用它们。

We can think of the curve with a position as a function of time. The curve can go anywhere and could loop and cross itself. We can also think of the curve as having a velocity at any point. For example, the point p(t) is traveling slowly near t = –2 and quickly between t = 2 and t = 3. This type of “moving point” vocabulary is often used when discussing parametric curves even when the curve is not describing a moving point.
我们可以将具有位置的曲线视为时间函数。曲线可以到达任何地方，可以循环并交叉自身。我们还可以认为曲线在任何一点都有速度。例如，点p ( t ) 在t = -2附近缓慢移动，在t = 2 和t = 3 之间快速移动。即使曲线不是在描述移动点，在讨论参数曲线时也经常使用这种“移动点”词汇。

2D Parametric Lines
二维参数线

A parametric line in 2D that passes through points p₀ = (x₀, y₀) and p₁ = (x₁,y₁) can be written as
经过点p ₀ = ( x ₀ , y ₀ ) 和p ₁ = ( x ₁ , y ₁ ) 的二维参数线可以写成

Because the formulas for x and y have such similar structure, we can use the vector form for p = (x, y) (Figure 2.36):
由于x和y的公式具有非常相似的结构，我们可以使用p = ( x, y ) 的向量形式（图 2.36 ）：

You can read this in geometric form as “start at point p₀ and go some distance toward p₁ determined by the parameter t.” A nice feature of this form is that p(0) = p₀ and p(1) = p₁. Since the point changes linearly with t, the value of t between p₀ and p₁ measures the fractional distance between the points. Points with t < 0 are to the “far” side of p₀, and points with t > 1 are to the “far” side of p₁.
您可以用几何形式将其理解为“从点p ₀开始，向p ₁移动一段距离，该距离由参数t决定”。此形式的一个很好的特点是p (0) = p ₀和p (1) = p ₁ 。由于点随t线性变化，因此p ₀和p ₁之间的t值测量了点之间的分数距离。t < 0 的点位于p ₀的“远”侧， t > 1 的点位于p ₁的“远”侧。

Parametric lines can also be described as just a point o and a vector d:
参数线也可以描述为一个点o和一个向量d ：

When the vector d has unit length, the line is arc-length parameterized. This means t is an exact measure of distance along the line. Any parametric curve can be arc-length parameterized, which is obviously a very convenient form, but not all can be converted analytically.
当向量d具有单位长度时，直线是弧长参数化的。这意味着t是沿直线距离的精确测量值。任何参数曲线都可以是弧长参数化的，这显然是一种非常方便的形式，但并非所有曲线都可以进行解析转换。

2D Parametric Circles
2D 参数圆

A circle with center (x_c,y_c) and radius r has a parametric form:
以 ( _xc , _yc ) 为圆心，以r为半径的圆具有参数形式：

To ensure that there is a unique parameter ϕ for every point on the curve, we can restrict its domain: ϕ ∈ [0, 2π) or ϕ ∈ (–π, π] or any other half-open interval of length 2π.
为了确保曲线上每个点都有一个唯一的参数 ϕ，我们可以限制它的定义域： ϕ ∈ [0, 2 π ) 或 ϕ ∈ (-π, π] 或任何其他长度为 2 π的半开区间。

An axis-aligned ellipse can be constructed by scaling the x and y parametric equations separately:
可以通过分别缩放x和y参数方程来构建轴对齐椭圆：

2.7.7 3D Parametric Curves
2.7.7 3D 参数曲线

A 3D parametric curve operates much like a 2D parametric curve:
3D 参数曲线的操作与 2D 参数曲线非常相似：

For example, a spiral around the z-axis is written as
例如，绕z轴的螺旋线写为

As with 2D curves, the functions f , g, and h are defined on a domain D ⊂ R if we want to control where the curve starts and ends. In vector form, we can write
与二维曲线一样，如果我们想控制曲线的起点和终点，函数f 、 g和h定义在域D ⊂ R 上。以矢量形式，我们可以写成

In this chapter, we only discuss 3D parametric lines in detail. General 3D parametric curves are discussed more extensively in Chapter 15.
在本章中，我们仅详细讨论 3D 参数线。第 15 章将更广泛地讨论一般 3D 参数曲线。

2.7.8 3D Parametric Surfaces
2.7.8 3D 参数曲面

The parametric approach can be used to define surfaces in 3D space in much the same way we define curves, except that there are two parameters to address the two-dimensional area of the surface. These surfaces have the form
参数化方法可用于定义三维空间中的曲面，其方式与定义曲线的方式非常相似，只是有两个参数用于处理曲面的二维区域。这些曲面具有以下形式

or, in vector form,
或者以矢量形式，

With implicit surfaces, the derivative of the function f gave us the surface normal. With parametric surfaces, the derivatives of p also give information about the surface geometry.
对于隐式曲面，函数f的导数给出了曲面法线。对于参数曲面， p的导数还给出了曲面几何的信息。

Consider the function q(t) = p(t, v₀) . This function defines a parametric curve obtained by varying u while holding v fixed at the value v₀. This curve, called an isoparametric curve (or sometimes “isoparm” for short), lies in the surface. The derivative of q gives a vector tangent to the curve, and since the curve lies in the surface, the vector q also lies in the surface. Since it was obtained by varying one argument of p, the vector q is the partial derivative of p with respect to u, which we’ll denote p_u. A similar argument shows that the partial derivative p_v gives the tangent to the isoparametric curves for constant u, which is a second tangent vector to the surface.
考虑函数q ( t ) = p ( t, v ₀ ) 。该函数定义了一条参数曲线，该曲线通过改变u获得，同时将v保持在v _{0 的}值不变。这条曲线称为等参曲线（有时简称为“等参线”）位于曲面上。q的导数给出与曲线相切的向量，由于曲线位于曲面上，向量q也位于曲面上。由于它是通过改变p的一个参数获得的，因此向量q是p对u的偏导数，我们将其表示为p _u 。类似的论证表明，偏导数p _v给出常数u 时等参曲线的切线，它是曲面的第二个切向量。

The derivative of p, then, gives two tangent vectors at any point on the surface. The normal to the surface may be found by taking the cross product of these vectors: since both are tangent to the surface, their cross product, which is perpendicular to both tangents, is normal to the surface. The right-hand rule for cross products provides a way to decide which side is the front, or outside, of the surface; we will use the convention that the vector
然后， p的导数给出曲面上任意一点的两个切向量。曲面的法线可以通过取这些向量的叉积来找到：由于两者都与曲面相切，因此它们的叉积（垂直于两个切线）垂直于曲面。叉积的右手定则提供了一种确定哪一侧是曲面的前端或外侧的方法；我们将使用以下惯例：向量

points toward the outside of the surface.
指向表面的外侧。

2.7.9 Summary of Curves and Surfaces
2.7.9 曲线和曲面总结

Implicit curves in 2D or surfaces in 3D are defined by scalar-valued functions of two or three variables, f : ℝ² → ℝ or f : ℝ³ → ℝ, and the surface consists of all points where the function is zero:
二维中的隐式曲线或三维中的曲面由两个或三个变量的标量值函数定义， f : ℝ ² → ℝ 或f : ℝ ³ → ℝ，​​曲面由函数为零的所有点组成：

Parametric curves in 2D or 3D are defined by vector-valued functions of one variable, p : D ⊂ ℝ → ℝ² or p : D ⊂ ℝ → ℝ³, and the curve is swept out as t varies over all of D:
二维或三维中的参数曲线由一个变量的向量值函数定义， p : D ⊂ ℝ → ℝ ²或p : D ⊂ ℝ → ℝ ³ ，并且曲线随着t在整个D上的变化而变化：

Parametric surfaces in 3D are defined by vector-valued functions of two variables, p : D ⊂ ℝ² → ℝ³, and the surface consists of the images of all points (u, v) in the domain:
三维中的参数曲面由^两个变量的向量值函数定义， p : D⊂ℝ2 → ^ℝ3 ，曲面由域内所有点（ u，v ）的图像组成：

For implicit curves and surfaces, the normal vector is given by the derivative of f (the gradient), and the tangent vector (for a curve) or vectors (for a surface) can be derived from the normal by constructing a basis.
对于隐式曲线和曲面，法向量由f的导数（梯度）给出，并且可以通过构建基从法向量中导出切向量（对于曲线）或向量（对于曲面）。

For parametric curves and surfaces, the derivative of p gives the tangent vector (for a curve) or vectors (for a surface), and the normal vector can be derived from the tangents by constructing a basis.
对于参数曲线和曲面， p的导数给出切向量（对于曲线）或向量（对于曲面），而法向量可以通过构建基从切线推导出来。

2.9.1 2D Triangles
2.9.1 二维三角形

If we have a 2D triangle defined by 2D points a, b, and c, we can first find its area:
如果我们有一个由二维点a 、 b和c定义的二维三角形，我们首先可以找到它的面积：

The derivation of this formula can be found in Section 6.3. This area will have a positive sign if the points a, b,and c are in counterclockwise order and a negative sign, otherwise.
这个公式的推导可以在6.3 节中找到。如果点a 、 b和c是按逆时针顺序排列的，则该区域为正号，否则为负号。

Often in graphics, we wish to assign a property, such as color, at each triangle vertex and smoothly interpolate the value of that property across the triangle. There are a variety of ways to do this, but the simplest is to use barycentric coordinates. One way to think of barycentric coordinates is as a nonorthogonal coordinate system as was discussed briefly in Section 2.4.2. Such a coordinate system is shown in Figure 2.38, where the coordinate origin is a and the vectors from a to b and c are the basis vectors. With that origin and those basis vectors, any point p can be written as
在图形学中，我们经常希望为每个三角形顶点分配一个属性（例如颜色），并在整个三角形中平滑地插入该属性的值。有多种方法可以做到这一点，但最简单的方法是使用重心坐标。一种思考重心坐标的方法是将其视为非正交坐标系，如第 2.4.2 节中简要讨论的那样。这种坐标系如图 2.38所示，其中坐标原点为a ，从a到b和c的向量为基向量。有了该原点和这些基向量，任何点p都可以写成

Note that we can reorder the terms in Equation (2.28) to get
请注意，我们可以重新排序公式 (2.28) 中的项，得到

Often people define a new variable α to improve the symmetry of the equations:
人们常常定义一个新变量α来改善方程的对称性：

which yields the equation
得出以下等式

with the constraint that
约束条件是

Barycentric coordinates seem like an abstract and unintuitive construct at first, but they turn out to be powerful and convenient. You may find it useful to think of how street addresses would work in a city where there are two sets of parallel streets, but where those sets are not at right angles. The natural system would essentially be barycentric coordinates, and you would quickly get used to them. Barycentric coordinates are defined for all points on the plane. A particularly nice feature of barycentric coordinates is that a point p is inside the triangle formed by a, b,and c if and only if
重心坐标乍一看似乎是一种抽象且不直观的结构，但事实证明它功能强大且方便。您可能会发现，想象一下在有两组平行街道但两组街道不成直角的城市中街道地址的工作原理很有用。自然系统本质上是重心坐标，您很快就会习惯它们。重心坐标适用于平面上的所有点。重心坐标的一个特别好的特征是，当且仅当点p位于由a 、 b和c形成的三角形内时

If one of the coordinates is zero and the other two are between zero and one, then you are on an edge. If two of the coordinates are zero, then the other is one, and you are at a vertex. Another nice property of barycentric coordinates is that Equation (2.29) in effect mixes the coordinates of the three vertices in a smooth way. The same mixing coefficients (α, β, γ) can be used to mix other properties, such as color, as we will see in the next chapter.
如果其中一个坐标为零，而其他两个坐标介于零和一之间，则您位于边缘。如果两个坐标为零，则另一个坐标为一，则您位于顶点。重心坐标的另一个好特性是，方程 (2.29) 实际上以平滑的方式混合了三个顶点的坐标。相同的混合系数 ( α、β、γ ) 可用于混合其他属性，例如颜色，我们将在下一章中看到。

Given a point p, how do we compute its barycentric coordinates? One way is to write Equation (2.28) as a linear system with unknowns β and γ,solve,andset α = 1 – β – γ . That linear system is
给定一个点p ，我们如何计算它的重心坐标？一种方法是将方程 (2.28) 写成具有未知数β和γ的线性系统，求解并设置 α = 1 – β – γ 。该线性系统是

Although it is straightforward to solve Equation (2.31) algebraically, it is often fruitful to compute a direct geometric solution.
尽管用代数方法求解方程 (2.31) 很简单，但计算直接几何解通常会很有成效。

One geometric property of barycentric coordinates is that they are the signed scaled distance from the lines through the triangle sides, as is shown for β in Figure 2.39. Recall from Section 2.7.2 that evaluating the equation f (x, y) for the line f (x, y) = 0 returns the scaled signed distance from (x, y) to the line. Also recall that if f (x, y) = 0 is the equation for a particular line, so is kf (x, y) = 0 for any nonzero k. Changing k scales the distance and controls which side of the line has positive signed distance, and which negative. We would like to choose k such that, for example, kf (x, y) = β. Since k is only one unknown, we can force this with one constraint, namely, that at point b, we know β = 1. So if the line f_ac(x, y) = 0 goes through both a and c, then we can compute β for a point (x, y) as follows:
重心坐标的一个几何性质是，它们是从通过三角形边的直线到原点的带符号的缩放距离，如图 2.39中β所示。回想一下2.7.2 节，对直线f ( x, y ) = 0 求方程f ( x, y ) 的值将返回从 ( x, y ) 到该直线的带符号的缩放距离。还记得吗，如果f ( x, y ) = 0 是某条直线的方程，那么对于任何非零k ， kf ( x, y ) = 0 也是这样的。改变k会缩放距离并控制直线的哪一侧具有正符号距离，哪一侧具有负符号距离。我们希望选择k使得，例如， kf ( x, y ) = β 。由于k只有一个未知数，我们可以用一个约束来强制实现这一点，即在点b处，我们知道β = 1。因此，如果直线f _ac ( x, y ) = 0 同时经过a和c ，则我们可以按如下方式计算点 ( x, y ) 的β ：

and we can compute γ and α in a similar fashion. For efficiency, it is usually wise to compute only two of the barycentric coordinates directly and to compute the third using Equation (2.30).
我们可以用类似的方式计算γ和 α。为了提高效率，通常明智的做法是直接计算两个重心坐标，然后使用公式 (2.30) 计算第三个坐标。

To find this “ideal” form for the line through p₀ and p₁, we can first use the technique of Section 2.7.2 to find some valid implicit lines through the vertices. Equation (2.17) gives us
为了找到通过p ₀和p _{1 的}直线的“理想”形式，我们可以首先使用2.7.2 节中的技巧来找到一些通过顶点的有效隐式直线。公式 (2.17) 给出

Note that f_ab(x_c,y_c) probably does not equal one, so it is probably not the ideal form we seek. By dividing through by f_ab(x_c,y_c) ,weget
请注意， f _ab ( x _c , y _c ) 可能不等于 1，因此它可能不是我们寻求的理想形式。通过除以f _ab ( x _c , y _c )，我们得到

The presence of the division might worry us because it introduces the possibility of divide-by-zero, but this cannot occur for triangles with areas that are not near zero. There are analogous formulas for α and β, but typically only one is needed:
除法的存在可能会让我们担心，因为它引入了除以零的可能性，但对于面积不接近零的三角形，这种情况不会发生。α 和β有类似的公式，但通常只需要一个：

Another way to compute barycentric coordinates is to compute the areas A_a, A_b, and A_c, of subtriangles as shown in Figure 2.40. Barycentric coordinates obey
计算重心坐标的另一种方法是计算子三角形的面积A _a 、 A _b和A _c ，如图 2.40所示。重心坐标遵循

the rule
规则

where A is the area of the triangle. Note that A = A_a + A_b + A_c, so it can be computed with two additions rather than a full area formula. This rule still holds for points outside the triangle if the areas are allowed to be signed. The reason for this is shown in Figure 2.41. Note that these are signed areas and will be computed correctly as long as the same signed area computation is used for both A and the subtriangles A_a, A_b,and A_c.
其中A是三角形的面积。注意A = A _a + A _b + A _c ，因此可以用两次加法来计算，而不必使用完整的面积公式。如果允许对面积进行符号表示，则此规则对于三角形外部的点仍然适用。原因如图 2.41所示。注意，这些是有符号的面积，只要对A以及子三角形A _a 、 A _b和A _c使用相同的有符号面积计算，就可以正确计算。

2.9.2 3D Triangles
2.9.2 3D三角形

One wonderful thing about barycentric coordinates is that they extend almost transparently to 3D. If we assume the points a, b, and c are 3D, then we can still use the representation
重心坐标的奇妙之处在于，它们几乎可以透明地扩展到三维空间。如果我们假设点a 、 b和c是三维的，那么我们仍然可以使用以下表示

Now, as we vary β and γ, we sweep out a plane.
现在，随着我们改变β和γ ，我们扫出一个平面。

The normal vector to a triangle can be found by taking the cross product of any two vectors in the plane of the triangle (Figure 2.42). It is easiest to use two of the three edges as these vectors, for example,
三角形的法向量可以通过对三角形平面中的任意两个向量进行叉积来找到（图 2.42 ）。最简单的方法是使用三条边中的两条作为这些向量，例如，

Note that this normal vector is not necessarily of unit length, and it obeys the right-hand rule of cross products.
注意，这个法向量不一定是单位长度，而且它遵循叉积的右手定则。

The area of the triangle can be found by taking the length of the cross product:
三角形的面积可以通过取叉积的长度来求得：

Note that this is not a signed area, so it cannot be used directly to evaluate barycentric coordinates. However, we can observe that a triangle with a “clockwise” vertex order will have a normal vector that points in the opposite direction to the normal of a triangle in the same plane with a “counterclockwise” vertex order. Recall that
请注意，这不是一个有符号区域，因此不能直接用于评估重心坐标。但是，我们可以观察到，具有“顺时针”顶点顺序的三角形将具有指向与同一平面中具有“逆时针”顶点顺序的三角形的法线相反方向的法向量。回想一下

where ϕ is the angle between the vectors. If a and b are parallel, then cos ϕ = ±1, and this gives a test of whether the vectors point in the same or opposite directions. This, along with Equations (2.33)–(2.35), suggest the formulas
其中 φ 是向量之间的角度。如果a和b平行，则 cos φ = ±1，这可以测试向量指向相同还是相反的方向。这与方程 (2.33)–(2.35) 一起，提出了以下公式

where n is Equation (2.34) evaluated with vertices a, b, and c; n_a is Equation (2.34) evaluated with vertices b, c,and p, and so on, i.e.,
其中n是通过顶点a 、 b和c求得的方程（2.34）； na是通过顶点b 、 c和p求得_的方程（2.34），依此类推，即

2.12.1 Importance Sampling
2.12.1 重要性抽样

When a function we want to take a random average of has a wide variation in its high and low values, it can be to our advantage to concentrate samples in some areas and then correct for the nonuniformity with weights. The probability density functions give us the right tool for that: if we know the PDF of a sample, that is a direct measure of how “oversampled” that region is. If we use nonuniform samples, then we can get thus
当我们想要随机取平均值的函数的高值和低值变化很大时，将样本集中在某些区域，然后用权重校正不均匀性对我们有利。概率密度函数为我们提供了正确的工具：如果我们知道样本的 PDF，那么它就是该区域“过采样”程度的直接度量。如果我们使用非均匀样本，那么我们可以得到

integrate = average_of_nonuniform_samples(f()/p(),
            domain).

A neat thing about this formula is it also works for uniform random samples. In that case, the PDF p() = 1/ integrate(1, domain) so the “size” of the domain is encoded in the PDF.
这个公式的一个巧妙之处在于它也适用于均匀随机样本。在这种情况下，PDF p() = 1/integrate(1, domain) ，因此域的“大小”被编码在 PDF 中。

For any given Monte Carlo importance sampling problem, there is a pretty formulaic approach we follow, at least to get started:
对于任何给定的蒙特卡洛重要性采样问题，我们遵循一个非常公式化的方法，至少在开始时是这样：

1. Identify what is the function f () and the domain of integration (e.g., points on the unit sphere or points on a triangle).
   确定函数f () 是什么以及积分域（例如，单位球面上的点或三角形上的点）。

2. Pick a method for generating random samples x_i on that domain, and make sure there is a way to evaluate the PDF p(x_i) for each sample.
   选择一种在该域上生成随机样本 x 的方法，并确保有一种方法可以评估每个样本的 PDF p (x)。

3. Average the ratio f (x_i)/p(x_i) for many x_i. This is our estimate of the integral.
   对多个 x 计算f (x) /p (x) 比率的平均值。这是我们对积分的估计。

A neat thing is that any p() can be used and you will converge to the right answer (with the caveat that where f () is nonzero p() must be nonzero). Which p() you use merely influences how fast your estimate converges. So we usually start with a constant p() for debugging our code.
巧妙之处在于，可以使用任何p ()，并且您将收敛到正确答案（但要注意，当f () 非零时， p () 必须非零）。您使用哪个p () 仅影响您的估计收敛速度。因此，我们通常从常数p () 开始调试代码。

3.1.1 Displays
3.1.1 显示

Current displays, including televisions and digital cinematic projectors as well as displays and projectors for computers, are nearly universally based on fixed arrays of pixels. They can be separated into emissive displays, which use pixels that directly emit controllable amounts of light, and transmissive displays, in which the pixels themselves don’t emit light but instead vary the amount of light that they allow to pass through them. Transmissive displays require a light source to illuminate them: in a direct-viewed display, this is a backlight behind the array; in a projector, it is a lamp that emits light that is projected onto the screen after passing through the array. An emissive display is its own light source.
当前的显示器，包括电视和数字电影放映机以及计算机显示器和投影仪，几乎普遍基于固定的像素阵列。它们可以分为发射显示器，使用直接发射可控光量的像素，以及透射显示器，其中像素本身不发光，而是改变允许穿过它们的光量。透射显示器需要光源来照亮它们：在直视显示器中，这是阵列后面的背光；在投影仪中，它是一盏灯，发出的光穿过阵列后投射到屏幕上。发射显示器有自己的光源。

Light-emitting diode (LED) displays are an example of the emissive type. Each pixel is composed of one or more LEDs, which are semiconductor devices (based on inorganic or organic semiconductors) that emit light with intensity depending on the electrical current passing through them (see Figure 3.1).
发光二极管 (LED) 显示器是自发光类型的一个例子。每个像素由一个或多个 LED 组成，LED 是一种半导体器件（基于无机或有机半导体），其发光强度取决于通过它们的电流（见图3.1 ）。

The pixels in a color display are divided into three independently controlled subpixels—one red, one green, and one blue—each with its own LED made using different materials so that they emit light of different colors (Figure 3.2). When the display is viewed from a distance, the eye can’t separate the individual subpixels, and the perceived color is a mixture of red, green, and blue.
彩色显示器中的像素分为三个独立控制的子像素（一个红色、一个绿色和一个蓝色）每个都有自己的 LED，这些 LED 使用不同的材料制成，因此它们会发出不同颜色的光（图 3.2 ）。从远处观看显示器时，眼睛无法区分各个子像素，感知到的颜色是红色、绿色和蓝色的混合。

Liquid crystal displays (LCDs) are an example of the transmissive type. A liquid crystal is a material whose molecular structure enables it to rotate the polarization of light that passes through it, and the degree of rotation can be adjusted by an applied voltage. An LCD pixel (Figure 3.3) has a layer of polarizing film behind it, so that it is illuminated by polarized light—let’s assume it is polarized horizontally.
液晶显示器 (LCD) 是透射型显示器的一个例子。液晶是一种材料，其分子结构使其能够旋转穿过它的光的偏振，并且可以通过施加电压来调整旋转的程度。LCD 像素（图 3.3 ）后面有一层偏振膜，因此它被偏振光照射——我们假设它是水平偏振的。

A second layer of polarizing film in front of the pixel is oriented to transmit only vertically polarized light. If the applied voltage is set so that the liquid crystal layer in between does not change the polarization, all light is blocked and the pixel is in the “off” (minimum intensity) state. If the voltage is set so that the liquid crystal rotates the polarization by 90°, then all the light that entered through the back of the pixel will escape through the front, and the pixel is fully “on”—it has its maximum intensity. Intermediate voltages will partly rotate the polarization so that the front polarizer partly blocks the light, resulting in intensities between the minimum and maximum (Figure 3.4). Like color LED displays, color LCDs have red, green, and blue subpixels within each pixel, which are three independent pixels with red, green, and blue color filters over them.
像素前面的第二层偏光膜仅透射垂直偏振光。如果设置施加的电压使得中间的液晶层不改变偏振，则所有光都将被阻挡，像素处于“关闭”（最小强度）状态。如果设置电压使液晶将偏振旋转 90°，则所有从像素后面进入的光都将从前面逸出，像素完全“开启” - 具有最大强度。中间电压将部分旋转偏振，使得前偏振器部分阻挡光，从而产生最小值和最大值之间的强度（图 3.4 ）。与彩色 LED 显示器一样，​​彩色 LCD 每个像素内都有红、绿、蓝子像素，它们是三个独立的像素，上面有红、绿、蓝滤光片。

Any type of display with a fixed pixel grid, including these and other technologies, has a fundamentally fixed resolution determined by the size of the grid. For displays and images, resolution simply means the dimensions of the pixel grid: if a desktop monitor has a resolution of 1920 × 1200 pixels, this means that it has 2,304,000 pixels arranged in 1920 columns and 1200 rows.
任何具有固定像素网格的显示器（包括这些技术和其他技术）都具有由网格大小决定的基本固定分辨率。对于显示器和图像而言，分辨率只是指像素网格的尺寸：如果台式机显示器的分辨率为 1920 × 1200 像素，则意味着它有 2,304,000 个像素，排列在 1920 列和 1200 行中。

An image of a different resolution, to fill the screen, must be converted into a 1920 × 1200 image using the methods of Chapter 10.
为了填满屏幕，必须使用第 10 章的方法将不同分辨率的图像转换为 1920 × 1200 的图像。

3.1.2 Hardcopy Devices
3.1.2 硬拷贝设备

The process of recording images permanently on paper has very different constraints from showing images transiently on a display. In printing, pigments are distributed on paper or another medium so that when light reflects from the paper it forms the desired image. Printers are raster devices like displays, but many printers can only print binary images—pigment is either deposited or not at each grid position, with no intermediate amounts possible.
在纸上永久记录图像的过程与在显示器上短暂显示图像的过程有着非常不同的限制。在印刷过程中，颜料分布在纸张或其他介质上，这样当光线从纸张上反射时，就会形成所需的图像。打印机是像显示器一样的光栅设备，但许多打印机只能打印二元图像——颜料在每个网格位置上要么沉积，要么不沉积，不存在中间的量。

An ink-jet printer (Figure 3.5) is an example of a device that forms a raster image by scanning. An ink-jet print head contains liquid ink carrying pigment, which can be sprayed in very small drops under electronic control. The head moves across the paper, and drops are emitted as it passes grid positions that should receive ink; no ink is emitted in areas intended to remain blank. After each sweep, the paper is advanced slightly, and then, the next row of the grid is laid down. Color prints are made by using several print heads, each spraying ink with a different pigment, so that each grid position can receive any combination of different colored drops. Because all drops are the same, an ink-jet printer prints binary images: at each grid point, there is a drop or no drop; there are no intermediate shades.
喷墨打印机（图 3.5 ）是通过扫描形成光栅图像的设备的一个示例。喷墨打印头含有载有颜料的液态墨水，可以在电子控制下以非常小的液滴形式喷射。打印头在纸张上移动，当打印头经过应接收墨水的网格位置时，就会喷射液滴；在应保持空白的区域则不会喷射墨水。每次扫描之后，纸张都会稍微前进，然后放下下一行网格。彩色打印是使用多个打印头进行的，每个打印头喷射的墨水都带有不同的颜料，这样每个网格位置都可以接收不同颜色液滴的任意组合。由于所有液滴都是相同的，因此喷墨打印机打印的是二元图像：在每个网格点，要么有一滴液滴，要么没有液滴；没有中间色调。