# 线性代数网课代修|计算机图形学代写Computer Graphics代考|COMP7204

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## 线性代数作业代写linear algebra代考|Central Projections and Perspectivities

Definition. Let $\mathbf{O}$ be a fixed point of $\mathbf{R}^{\mathrm{n}}$. For every point $\mathbf{p}$ of $\mathbf{R}^{\mathrm{n}}$ distinct from $\mathbf{0}$, let $\mathbf{L}{\mathbf{p}}$ denote the line through $\mathbf{O}$ and $\mathbf{p}$. If $\mathbf{Y}$ is a hyperplane in $\mathbf{R}^{\mathrm{n}}$ not containing $\mathbf{O}$, then define a map $$\pi{\mathrm{o}}: \mathbf{R}^{\mathrm{n}} \rightarrow \mathbf{Y}$$
by
\begin{aligned} \pi_{\mathbf{o}}(\mathbf{p}) &=\mathbf{L}{\mathbf{p}} \cap \mathbf{Y}, \text { if } \mathbf{L}{\mathbf{p}} \text { intersects } \mathbf{Y} \text { in a single point, } \ &=\text { undefined, otherwise. } \end{aligned}
The map $\pi_{\mathbf{o}}$ is called the central projection with center $\boldsymbol{O}$ of $\boldsymbol{R}^{n}$ to the plane $\boldsymbol{Y}$. If $\mathbf{X}$ is another hyperplane in $\mathbf{R}^{\mathrm{n}}$, then the restriction of $\pi_{\mathbf{O}}$ to $\mathbf{X}, \pi_{\mathbf{0}} \mid \mathbf{X}: \mathbf{X} \rightarrow \mathbf{Y}$, is called the perspective transformation or perspectivity from $\boldsymbol{X}$ to $\boldsymbol{Y}$ with center $\boldsymbol{O}$.

Note that our terminology makes a slight distinction between central projections and perspectivities. Both send points to a plane, but the former is defined on all of Euclidean space, whereas the latter is only defined on a plane; however, they clearly are closely related.

Clearly, from the point of view of formulas, one would not expect our new maps to be complicated because they simply involve finding the intersection of a line with a hyperplane. Let us look at some simple examples to get a feel for what geometric propertics these maps posscss. First, consider perspectivitics betwecn lincs in $\mathbf{R}^{2}$. Figure $3.1$ shows the case where the two lines parallel. In this case, the ratio of the distance between points and the distance between their images is constant. The perspectivity is one-to-one and onto. It preserves parallelism, concurrence, ratio of division, and betweenness.

What happens when the two lines are not parallel? See Figure 3.2. The point $\mathbf{V}$ on $\mathbf{L}$ has no image and the point $\mathbf{W}$ on $\mathbf{L}^{\prime}$ has no preimage. These points are called vanishing points. Betweenness is not preserved as is demonstrated by the points $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}$ in Figure 3.2. Furthermore, the fact that betweenness is not preserved leads to other properties not being preserved. In particular, segments, rays, and ratios of division are not preserved, and distances are distorted by different constants.

Next, consider perspectivities between planes. When the planes are parallel, things behave pretty well just like for parallel lines. The interesting case is when the planes are not parallel. Consider a perspectivity with center $\mathbf{O}$ from a plane $\mathbf{X}$, which we shall call the object plane, to another plane $\mathbf{Y}$, which we shall call the view plane. The following facts are noteworthy.

## 线性代数作业代写linear algebra代考|Homogeneous Coordinates

One of the key idcas in the study of analytic projective geometry is that of homogeneous coordinates. The standard Cartesian coordinates are sometimes referred to as “nonhomogeneous” coordinates and are simply one of many ways to specify points in space with real numbers. Other ways are polar coordinates in the plane and cylindrical and spherical coordinates in 3-space. Barycentric coordinates are a type of “homogeneous” coordinates. They specify points relative to a fixed set of points.
Out of the many ways that one can coordinatize points, which is the most convenient depends completely on the type of problem we are trying to solve. Homogeneous coordinates are just another way of coordinatizing points. Historically they find their roots in Moebius’ work on barycentric coordinates (Der Barycentrische Calcul, 1827) and the fact that they are useful with central projections. Here we motivate their definition by looking at the relationship between points and solutions to linear equations.

We shall start with the real line $\mathbf{R}$. What we are about to do may seem a little silly at first, but if the reader will bear with us, it should make more sense in the end. Linear equations in $\mathbf{R}$ have the form
$$a x+b=0, \quad \text { with } \quad a \neq 0 .$$
We can think of $\mathbf{R}$ as the set of solutions to all equations of the form (3.12). Equation (3.12) is homogeneous in a and $\mathrm{b}$, but not in $\mathrm{x}$. We can achieve more symmetry by introducing another variable $\mathrm{Y}$ and consider the equation
$$a X+b Y=0, \text { with }(a, b) \neq(0,0)$$
The trivial solution $(\mathrm{X}, \mathrm{Y})=(0,0)$ is uninteresting and will be excluded from consideration. Note that if we have a solution $x$ to equation (3.12), then we have a solution $(\mathrm{x}, 1)$ to equation (3.13). In fact, $(\mathrm{kx}, \mathrm{k})$ will also be a solution to (3.13) for all $\mathrm{k} \neq 0$. Conversely, if $(\mathrm{X}, \mathrm{Y})$ is a solution to (3.13), then $\mathrm{X} / \mathrm{Y}$ is a solution to (3.12) if $\mathrm{Y} \neq 0$. In short, each solution $x$ to (3.12) gives rise to a class of solutions $(\mathrm{kx}, \mathrm{k}), \mathrm{k} \neq 0$, to (3.13) and each class of solutions ( $\mathrm{kX}, \mathrm{kY})$ to (3.13) with $\mathrm{k} \neq 0$ and $\mathrm{Y} \neq 0$ gives rise to a unique solution $\mathrm{X} / \mathrm{Y}$ to (3.12).

## 线性代数作业代写linear algebra代考|Central Projections and Perspectivities

$\pi \mathrm{o}: \mathbf{R}^{\mathrm{n}} \rightarrow \mathbf{Y}$

$\pi_{\mathrm{o}}(\mathbf{p})=\mathbf{L} \mathbf{p} \cap \mathbf{Y}$, if $\mathbf{L} \mathbf{p}$ intersects $\mathbf{Y}$ in a single point, $\quad=$ undefined, otherwise.

## 线性代数作业代写linear algebra代考|Homogeneous Coordinates

$$a x+b=0, \quad \text { with } \quad a \neq 0 .$$

$$a X+b Y=0, \text { with }(a, b) \neq(0,0)$$

# 计量经济学代写

## 在这种情况下，如何学好线性代数？如何保证线性代数能获得高分呢？

1.1 mark on book

【重点的误解】划重点不是书上粗体，更不是每个定义，线代概念这么多，很多朋友强迫症似的把每个定义整整齐齐用荧光笔标出来，然后整本书都是重点，那期末怎么复习呀。我认为需要标出的重点为

A. 不懂，或是生涩，或是不熟悉的部分。这点很重要，有的定义浅显，但证明方法很奇怪。我会将晦涩的定义，证明方法标出。在看书时，所有例题将答案遮住，自己做，卡住了就说明不熟悉这个例题的方法，也标出。

B. 老师课上总结或强调的部分。这个没啥好讲的，跟着老师走就对了

C. 你自己做题过程中，发现模糊的知识点

1.2 take note

1.3 understand the relation between definitions