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计算机代写|计算机图形学代考Computer Graphics代考|3D Affine Transformations

Three dimensional transformations are used for many different purposes, such as coordinate transforms, shape modeling, animation, and camera modeling.

An affine transform in 3D looks the same as in 2D: $F(\bar{p})=A \bar{p}+\vec{t}$ for $A \in \mathbb{R}^{3 \times 3}, \bar{p}, \vec{t} \in \mathbb{R}^3$. A homogeneous affine transformation is
$$\hat{F}(\hat{p})=\hat{M} \hat{p} \text {, where } \hat{p}=\left[\begin{array}{l} \bar{p} \ 1 \end{array}\right], \hat{M}=\left[\begin{array}{cc} A & \vec{t} \ \overrightarrow{0}^T & 1 \end{array}\right] .$$
Translation: $A=I, \vec{t}=\left(t_x, t_y, t_z\right)$.
Scaling: $A=\operatorname{diag}\left(s_x, s_y, s_z\right), \vec{t}=\overrightarrow{0}$.
Rotation: $A=R, \vec{t}=\overrightarrow{0}$, and $\operatorname{det}(R)=1$.
3D rotations are much more complex than 2D rotations, so we will consider only elementary rotations about the $x, y$, and $z$ axes.

For a rotation about the $z$-axis, the $z$ coordinate remains unchanged, and the rotation occurs in the $x-y$ plane. So if $\bar{q}=R \bar{p}$, then $q_z=p_z$. That is,
$$\left[\begin{array}{l} q_x \ q_y \end{array}\right]=\left[\begin{array}{cc} \cos (\theta) & -\sin (\theta) \ \sin (\theta) & \cos (\theta) \end{array}\right]\left[\begin{array}{l} p_x \ p_y \end{array}\right] .$$
Including the $z$ coordinate, this becomes
$$R_z(\theta)=\left[\begin{array}{ccc} \cos (\theta) & -\sin (\theta) & 0 \ \sin (\theta) & \cos (\theta) & 0 \ 0 & 0 & 1 \end{array}\right] .$$

计算机代写|计算机图形学代考Computer Graphics代考|Spherical Coordinates

Any three dimensional vector $\vec{u}=\left(u_x, u_y, u_z\right)$ may be represented in spherical coordinates. By computing a polar angle $\phi$ counterclockwise about the $y$-axis from the $z$-axis and an azimuthal angle $\theta$ counterclockwise about the $z$-axis from the $x$-axis, we can define a vector in the appropriate direction. Then it is only a matter of scaling this vector to the correct length $\left(u_x^2+u_y^2+u_z^2\right)^{-1 / 2}$ to match $\vec{u}$.

Given angles $\phi$ and $\theta$, we can find a unit vector as $\vec{u}=(\cos (\theta) \sin (\phi), \sin (\theta) \sin (\phi), \cos (\phi))$.
Given a vector $\vec{u}$, its azimuthal angle is given by $\theta=\arctan \left(\frac{u_y}{u_x}\right)$ and its polar angle is $\phi=$ $\arctan \left(\frac{\left(u_x^2+u_y^2\right)^{1 / 2}}{u_z}\right)$. This formula does not require that $\vec{u}$ be a unit vector.

计算机代写|计算机图形学代考Computer Graphics代考|3D Affine Transformations

3D 中的仿射音换看起来与 $2 \mathrm{D}$ 中的相同: $F(\bar{p})=A \bar{p}+\vec{t}$ 为了 $A \in \mathbb{R}^{3 \times 3}, \bar{p}, \vec{t} \in \mathbb{R}^3$. 齐次仿射变换是
$$\hat{F}(\hat{p})=\hat{M} \hat{p} \text {, where } \hat{p}=[\bar{p} 1], \hat{M}=\left[\begin{array}{lll} A \vec{t} \hat{0}^T & 1 \end{array}\right]$$

$3 \mathrm{D}$ 旋转比 $2 \mathrm{D}$ 旋转复杂得多，所以我们只考虑基本旋转 $x, y$ ，和 $z$ 轴。

$$\left[\begin{array}{ll} q_x q_y \end{array}\right]=\left[\begin{array}{lll} \cos (\theta) & -\sin (\theta) \sin (\theta) & \cos (\theta) \end{array}\right]\left[p_x p_y\right] .$$

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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计算机代写|计算机图形学代考Computer Graphics代考|3D Objects

As with $2 \mathrm{D}$ objects, we can represent 3D objects in parametric and implicit forms. (There are also explicit forms for 3D surfaces – sometimes called “height fields” – but we will not cover them here).

Implicit: $\left(\bar{p}-\bar{p}_0\right) \cdot \vec{n}=0$, where $\bar{p}_0$ is a point in $\mathbb{R}^3$ on the plane, and $\vec{n}$ is a normal vector perpendicular to the plane.
$\underset{\vec{b}}{\text { A plane can be defined uniquely by three non-colinear points }} \bar{p}_1, \bar{p}_2, \bar{p}_3$. Let $\vec{a}=\bar{p}_2-\bar{p}_1$ and $\vec{b}=\bar{p}_3-\bar{p}_1$, so $\vec{a}$ and $\vec{b}$ are vectors in the plane. Then $\vec{n}=\vec{a} \times \vec{b}$. Since the points are not colinear, $|\vec{n}| \neq 0$

Parametric: $\bar{s}(\alpha, \beta)=\bar{p}_0+\alpha \vec{a}+\beta \vec{b}$, for $\alpha, \beta \in \mathbb{R}$.
Note:
This is similar to the parametric form of a line: $\bar{l}(\alpha)=\bar{p}_0+\alpha \vec{a}$.
A planar patch is a parallelogram defined by bounds on $\alpha$ and $\beta$.

计算机代写|计算机图形学代考Computer Graphics代考|Surface Tangents and Normals

The tangent to a curve at $\bar{p}$ is the instantaneous direction of the curve at $\bar{p}$.
The tangent plane to a surface at $\bar{p}$ is analogous. It is defined as the plane containing tangent vectors to all curves on the surface that go through $\bar{p}$.
A surface normal at a point $\bar{p}$ is a vector perpendicular to a tangent plane.

Curves on Surfaces
The parametric form $\bar{p}(\alpha, \beta)$ of a surface defines a mapping from 2D points to 3D points: every $2 \mathrm{D}$ point $(\alpha, \beta)$ in $\mathbb{R}^2$ corresponds to a $3 \mathrm{D}$ point $\bar{p}$ in $\mathbb{R}^3$. Moreover, consider a curve $\bar{l}(\lambda)=$ $(\alpha(\lambda), \beta(\lambda))$ in 2D – there is a corresponding curve in 3D contained within the surface: $\bar{l}^*(\lambda)=$ $\bar{p}(\bar{l}(\lambda))$
Parametric Form
For a curve $\bar{c}(\lambda)=(x(\lambda), y(\lambda), z(\lambda))^T$ in 3D, the tangent is
$$\frac{d \bar{c}(\lambda)}{d \lambda}=\left(\frac{d x(\lambda)}{d \lambda}, \frac{d y(\lambda)}{d \lambda}, \frac{d z(\lambda)}{d \lambda}\right) .$$
For a surface point $\bar{s}(\alpha, \beta)$, two tangent vectors can be computed:
$$\frac{\partial \bar{s}}{\partial \alpha} \text { and } \frac{\partial \bar{s}}{\partial \beta}$$

计算机代写|计算机图形学代考Computer Graphics代考|3D Objects

A plane can be defined uniquely by three non-colinear points $\bar{p}_1, \bar{p}_2, \bar{p}_3$. 让 $\vec{a}=\bar{p}_2-\bar{p}_1$ 和 $\vec{b}=\bar{p}_3-\bar{p}_1$ ，所以 $\vec{a}$ 和 $\vec{b}$ 是平面上的向量。然后 $\vec{n}=\vec{a} \times \vec{b}$. 由于点不共线， $|\vec{n}| \neq 0$

计算机代写|计算机图形学代考Computer Graphics代考|Surface Tangents and Normals

$$\frac{d \bar{c}(\lambda)}{d \lambda}=\left(\frac{d x(\lambda)}{d \lambda}, \frac{d y(\lambda)}{d \lambda}, \frac{d z(\lambda)}{d \lambda}\right)$$

$$\frac{\partial \bar{s}}{\partial \alpha} \text { and } \frac{\partial \bar{s}}{\partial \beta}$$

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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计算机代写|计算机图形学代考Computer Graphics代考|2D Transformations

Given a point cloud, polygon, or sampled parametric curve, we can use transformations for several purposes:

1. Change coordinate frames (world, window, viewport, device, etc).
2. Compose objects of simple parts with local scale/position/orientation of one part defined with regard to other parts. For example, for articulated objects.
3. Use deformation to create new shapes.
4. Useful for animation.
There are three basic classes of transformations:
5. Rigid body – Preserves distance and angles.
• Examples: translation and rotation.
1. Conformal – Preserves angles.
• Examples: translation, rotation, and uniform scaling.
1. Affine – Preserves parallelism. Lines remain lines.
• Examples: translation, rotation, scaling, shear, and reflection.

计算机代写|计算机图形学代考Computer Graphics代考|Affine Transformations

An affine transformation takes a point $\bar{p}$ to $\bar{q}$ according to $\bar{q}=F(\bar{p})=A \bar{p}+\vec{t}$, a linear transformation followed by a translation. You should understand the following proofs.

The inverse of an affine transformation is also affine, assuming it exists.
Proof:
Let $\bar{q}=A \bar{p}+\vec{t}$ and assume $A^{-1}$ exists, i.e. $\operatorname{det}(A) \neq 0$.
Then $A \bar{p}=\bar{q}-\vec{t}$, so $\bar{p}=A^{-1} \bar{q}-A^{-1} \vec{t}$. This can be rewritten as $\bar{p}=B \bar{q}+\vec{d}$, where $B=A^{-1}$ and $\vec{d}=-A^{-1} \vec{t}$.
Note:
The inverse of a $2 \mathrm{D}$ linear transformation is
$$A^{-1}=\left[\begin{array}{ll} a & b \ c & d \end{array}\right]^{-1}=\frac{1}{a d-b c}\left[\begin{array}{rr} d & -b \ -c & a \end{array}\right] .$$

Lines and parallelism are preserved under affine transformations.
Proof:
To prove lines are preserved, we must show that $\bar{q}(\lambda)=F(\bar{l}(\lambda))$ is a line, where $F(\bar{p})=A \bar{p}+\vec{t}$ and $\bar{l}(\lambda)=\bar{p}_0+\lambda \vec{d}$.
\begin{aligned} \bar{q}(\lambda) & =A \bar{l}(\lambda)+\vec{t} \ & =A\left(\bar{p}_0+\lambda \vec{d}\right)+\vec{t} \ & =\left(A \bar{p}_0+\vec{t}\right)+\lambda A \vec{d} \end{aligned}
This is a parametric form of a line through $A \bar{p}_0+\vec{t}$ with direction $A \vec{d}$.

计算机代写|计算机图形学代考Computer Graphics代考|2D Transformations

• 示例: 平移和旋转。
1. 共形 $-$ 保留角度。
• 示例: 平移、旋转和均匀缩放。
1. 仿射 – 保留并行性。线仍然是线。

计算机代写|计算机图形学代考Computer Graphics代考|Affine Transformations

$\mathrm{a}$ 的倒数 $2 \mathrm{D}$ 线性音换是
$$A^{-1}=\left[\begin{array}{lll} a & b c & d \end{array}\right]^{-1}=\frac{1}{a d-b c}\left[\begin{array}{lll} d & -b-c & a \end{array}\right] .$$

$$\bar{q}(\lambda)=A \bar{l}(\lambda)+\vec{t} \quad=A\left(\bar{p}_0+\lambda \vec{d}\right)+\vec{t}=\left(A \bar{p}_0+\vec{t}\right)+\lambda A \vec{d}$$

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:CS代写, 计算机代写, 计算机图形

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计算机代写|计算机图形学代考Computer Graphics代考|Background

Modern algebraic notation has evolved over thousands of years where different civilisations developed ways of annotating mathematical and logical problems. The word ‘algebra’ comes from the Arabic ‘al-jabr w’al-muqabal’ meaning ‘restoration and reduction’. In retrospect, it does seem strange that centuries passed before the ‘equals’ sign $(=)$ was invented, and concepts such as ‘zero’ (CE 876) were introduced, especially as they now seem so important. But we are not at the end of this evolution, because new forms of annotation and manipulation will continue to emerge as new mathematical objects are invented.

One fundamental concept of algebra is the idea of giving a name to an unknown quantity. For example, $m$ is often used to represent the slope of a $2 \mathrm{D}$ line, and $c$ is the line’s $y$-coordinate where it intersects the $y$-axis. René Descartes formalised the idea of using letters from the beginning of the alphabet $(a, b, c, \ldots)$ to represent arbitrary quantities, and letters at the end of the alphabet $(p, q, r, s, t, \ldots, x, y, z)$ to represent quantities such as pressure $(p)$, time $(t)$ and coordinates $(x, y, z)$.

With the aid of the basic arithmetic operators: $+,-, \times$, / we can develop expressions that describe the behaviour of a physical process or a logical computation. For example, the expression $a x+b y-d$ equals zero for a straight line. The variables $x$ and $y$ are the coordinates of any point on the line and the values of $a, b$ and $d$ determine the position and orientation of the line. The $=$ sign permits the line equation to be expressed as a self-evident statement:
$$0=a x+b y-d .$$
Such a statement implies that the expressions on the left- and right-hand sides of the $=$ sign are ‘equal’ or ‘balanced’, and in order to maintain equality or balance,whatever is done to one side, must also be done to the other. For example, adding $d$ to both sides, the straight-line equation becomes
$$d=a x+b y .$$
Similarly, we could double or treble both expressions, divide them by 4 , or add 6 , without disturbing the underlying relationship. When we are first taught algebra, we are often given the task of rearranging a statement to make different variables the subject. For example, (3.1) can be rearranged such that $x$ is the subject:
\begin{aligned} y &=\frac{x+4}{2-\frac{1}{z}} \ y\left(2-\frac{1}{z}\right) &=x+4 \ x &=y\left(2-\frac{1}{z}\right)-4 \end{aligned}

计算机代写|计算机图形学代考Computer Graphics代考|Solving the Roots of a Quadratic Equation

Problem solving is greatly simplified if one has solved it before, and having a good memory is always an advantage. In mathematics, we keep coming across problems that have been encountered before, apart from different numbers. For example, $(a+b)(a-b)$ always equals $a^2-b^2$, therefore factorising the following is a trivial exercise:
\begin{aligned} &a^2-16=(a+4)(a-4) \ &x^2-49=(x+7)(x-7) \ &x^2-2=(x+\sqrt{2})(x-\sqrt{2}) . \end{aligned}
A perfect square has the form:
$$a^2+2 a b+b^2=(a+b)^2 .$$
Consequently, factorising the following is also a trivial exercise:
\begin{aligned} a^2+4 a b+4 b^2 &=(a+2 b)^2 \ x^2+14 x+49 &=(x+7)^2 \ x^2-20 x+100 &=(x-10)^2 \end{aligned}

计算机代写|计算机图形学代考Computer Graphics代考|Background

$$0=a x+b y-d .$$

$$d=a x+b y .$$

$$y=\frac{x+4}{2-\frac{1}{z}} y\left(2-\frac{1}{z}\right) \quad=x+4 x=y\left(2-\frac{1}{z}\right)-4$$

计算机代写|计算机图形学代考Computer Graphics代考|Solving the Roots of a Quadratic Equation

$$a^2-16=(a+4)(a-4) \quad x^2-49=(x+7)(x-7) x^2-2=(x+\sqrt{2})(x-\sqrt{2}) .$$

$$a^2+2 a b+b^2=(a+b)^2 .$$

$$a^2+4 a b+4 b^2=(a+2 b)^2 x^2+14 x+49 \quad=(x+7)^2 x^2-20 x+100=(x-10)^2$$

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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计算机代写|计算机图形学代考Computer Graphics代考|Binary Numbers

The binary number system has $B=2$, and $a$ to $h$ are 0 or 1 :
$$\ldots a 2^3+b 2^2+c 2^1+d 2^0+e 2^{-1}+f 2^{-2}+g 2^{-3}+h 2^{-4} \ldots$$
and the first 13 binary numbers are:
$$1_2, 10_2, 11_2, 100_2, 101_2, 110_2, 111_2, 1000_2, 1001_2, 1010_2, 1011_2, 1100_2, 1101_2 .$$
Thus $11011.11_2$ is converted to decimal as follows:
$$\begin{gathered} \left(1 \times 2^4\right)+\left(1 \times 2^3\right)+\left(0 \times 2^2\right)+\left(1 \times 2^1\right)+\left(1 \times 2^0\right)+\left(1 \times 2^{-1}\right)+\left(1 \times 2^{-2}\right) \ (1 \times 16)+(1 \times 8)+(0 \times 4)+(1 \times 2)+(1 \times 0.5)+(1 \times 0.25) \ (16+8+2)+(0.5+0.25) \ 26.75 \end{gathered}$$
The reason why computers work with binary numbers-rather than decimal-is due to the difficulty of designing electrical circuits that can store decimal numbers in a stable fashion. A switch, where the open state represents 0 , and the closed state represents 1 , is the simplest electrical component to emulate. No matter how often it is used, or how old it becomes, it will always behave like a switch. The main advantage of electrical circuits is that they can be switched on and off trillions of times a second, and the only disadvantage is that the encoded binary numbers and characters contain a large number of bits, and humans are not familiar with binary.

The hexadecimal number system has $B=16$, and $a$ to $h$ can be 0 to 15 , which presents a slight problem, as we don’t have 15 different numerical characters. Consequently, we use 0 to 9 , and the letters $A, B, C, D, E, F$ to represent $10,11,12,13,14,15$ respectively:

$\ldots a 16^3+b 16^2+c 16^1+d 16^0+e 16^{-1}+f 16^{-2}+g 16^{-3}+h 16^{-4} \ldots$
and the first 17 hexadecimal numbers are:
$$1_{16}, 2_{16}, 3_{16}, 4_{16}, 5_{16}, 6_{16}, 7_{16}, 8_{16}, 9_{16}, A_{16}, B_{16}, C_{16}, D_{16}, E_{16}, F_{16}, 10_{16}, 11_{16} \text {. }$$
Thus $1 E .8_{16}$ is converted to decimal as follows:
$$\begin{gathered} (1 \times 16)+(E \times 1)+\left(8 \times 16^{-1}\right) \ (16+14)+(8 / 16) \ 30.5 \end{gathered}$$
Although it is not obvious, binary, octal and hexadecimal numbers are closely related, which is why they are part of a programmer’s toolkit. Even though computers work with binary, it’s the last thing a programmer wants to use. So to simplify the manmachine interface, binary is converted into octal or hexadecimal. To illustrate this, let’s convert the 16-bit binary code 1101011000110001 into octal.
Using the following general binary integer
$$a 2^8+b 2^7+c 2^6+d 2^5+e 2^4+f 2^3+g 2^2+h 2^1+i 2^0$$
we group the terms into threes, starting from the right, because $2^3=8$ :
$$\left(a 2^8+b 2^7+c 2^6\right)+\left(d 2^5+e 2^4+f 2^3\right)+\left(g 2^2+h 2^1+i 2^0\right) .$$

计算机代写|计算机图形学代考Computer Graphics代考|Binary Numbers

$$\ldots a 2^3+b 2^2+c 2^1+d 2^0+e 2^{-1}+f 2^{-2}+g 2^{-3}+h 2^{-4} \ldots$$

$$1_2, 10_2, 11_2, 100_2, 101_2, 110_2, 111_2, 1000_2, 1001_2, 1010_2, 1011_2, 1100_2, 1101_2 .$$

$$\left(1 \times 2^4\right)+\left(1 \times 2^3\right)+\left(0 \times 2^2\right)+\left(1 \times 2^1\right)+\left(1 \times 2^0\right)+\left(1 \times 2^{-1}\right)+\left(1 \times 2^{-2}\right)(1 \times 16)+(1 \times 8)+(0 \times 4)+(1 \times 2)+(1 \times 0.5)+(1 \times 0.25)(16+8$$

0 ，闭合状态代表 1 ，是最简单的电子组件来模拟。无论使用多久，或使用多久，它的行为始終像一个开关。电路的主要优点是每 秒可以开关数万亿次，唯一的缺点是编的的二进制数和字符包含大量的比特，而人类对二进制并不孰悉。

$\ldots a 16^3+b 16^2+c 16^1+d 16^0+e 16^{-1}+f 16^{-2}+g 16^{-3}+h 16^{-4} \ldots$

$$1_{16}, 2_{16}, 3_{16}, 4_{16}, 5_{16}, 6_{16}, 7_{16}, 8_{16}, 9_{16}, A_{16}, B_{16}, C_{16}, D_{16}, E_{16}, F_{16}, 10_{16}, 11_{16} .$$

$$(1 \times 16)+(E \times 1)+\left(8 \times 16^{-1}\right)(16+14)+(8 / 16) 30.5$$

$$a 2^8+b 2^7+c 2^6+d 2^5+e 2^4+f 2^3+g 2^2+h 2^1+i 2^0$$

$$\left(a 2^8+b 2^7+c 2^6\right)+\left(d 2^5+e 2^4+f 2^3\right)+\left(g 2^2+h 2^1+i 2^0\right) .$$

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:CS代写, 计算机代写, 计算机图形

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计算机代写|计算机图形学代考Computer Graphics代考|Counting

Our brain’s visual cortex possesses some incredible image processing features. For example, children know instinctively when they are given less sweets than another child, and adults know instinctively when they are short-changed by a Parisian taxi driver, or driven around the Arc de Triumph several times, on the way to the airport! Intuitively, we can assess how many donkeys are in a field without counting them, and generally, we seem to know within a second or two, whether there are just a few, dozens, or hundreds of something. But when accuracy is required, one can’t beat counting. But what is counting?

Well normally, we are taught to count by our parents by memorising first, the counting words ‘one, two, three, four, five, six, seven, eight, nine, ten, ..’ and second, associating them with our fingers, so that when asked to count the number of donkeys in a picture book, each donkey is associated with a counting word. When each donkey has been identified, the number of donkeys equals the last word mentioned. However, this still assumes that we know the meaning of ‘one, two, three, four, ..’ etc. Memorising these counting words is only part of the problem-getting them in the correct sequence is the real challenge. The incorrect sequence ‘one, two, five, three, nine, four, ..’ etc., introduces an element of randomness into any calculation, but practice makes perfect, and it’s useful to master the correct sequence before going to university!

计算机代写|计算机图形学代考Computer Graphics代考|Sets of Numbers

A set is a collection of arbitrary objects called its elements or members. For example, each system of number belongs to a set with given a name, such as $\mathbb{N}$ for the natural numbers, $\mathbb{R}$ for real numbers, and $\mathbb{Q}$ for rational numbers. When we want to indicate that something is whole, real or rational, etc., we use the notation:
$$n \in \mathbb{N}$$
which reads ‘ $n$ is a member of $(\epsilon)$ the set $\mathbb{N}$ ‘, i.e. $n$ is a whole number. Similarly:
$$x \in \mathbb{R}$$
stands for ‘ $x$ is a real number.’
A well-ordered set possesses a unique order, such as the natural numbers $\mathbb{N}$. Therefore, if $P$ is the well-ordered set of prime numbers and $\mathbb{N}$ is the well-ordered set of natural numbers, we can write:
\begin{aligned} &P={2,3,5,7,11,13,17,19,23,29,31,37,41,43,47, \ldots} \ &\mathbb{N}={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17, \ldots} \end{aligned}
By pairing the prime numbers in $P$ with the numbers in $\mathbb{N}$, we have:
$${{2,1},{3,2},{5,3},{7,4},{11,5},{13,6},{17,7},{19,8},{23,9}, \ldots}$$
and we can reason that 2 is the 1 st prime, and 3 is the 2 nd prime, etc. However, we still have to declare what we mean by $1,2,3,4,5, \ldots$ etc., and without getting too philosophical, I like the idea of defining them as follows. The word ‘one’, represented by 1 , stands for ‘oneness’ of anything: one finger, one house, one tree, one donkey, etc. The word ‘two’, represented by 2, is ‘one more than one’. The word ‘three’, represented by 3 , is ‘one more than two’, and so on.

We are now in a position to associate some mathematical notation with our numbers by introducing the $+$ and $=$ signs. We know that $+$ means add, but it also can stand for ‘more’. We also know that $=$ means equal, and it can also stand for ‘is the same as’. Thus the statement:
$$2=1+1$$
is read as ‘two is the same as one more than one.’
We can also write:
$$3=1+2$$
which is read as ‘three is the same as one more than two.’ But as we already have a definition for 2 , we can write
\begin{aligned} 3 &=1+2 \ &=1+1+1 . \end{aligned}

计算机代写|计算机图形学代考Computer Graphics代考|Sets of Numbers

$$n \in \mathbb{N}$$

$$x \in \mathbb{R}$$

$$P=2,3,5,7,11,13,17,19,23,29,31,37,41,43,47, \ldots \quad \mathbb{N}=1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17, \ldots$$

$$2,1,3,2,5,3,7,4,11,5,13,6,17,7,19,8,23,9, \ldots$$

$$2=1+1$$

$$3=1+2$$

$$3=1+2 \quad=1+1+1 .$$

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。