commit 32a7d479960560cfe42a53b7a5269de2779908bf
parent 198f489ad65d3fc4cff8eaa94674d7abb7bb39d5
Author: Kenji Matsuda <ftvda283@gmail.com>
Date: Thu, 30 Sep 2021 20:38:29 +0900
start Transformations.html
Diffstat:
14 files changed, 49 insertions(+), 246 deletions(-)
diff --git a/translation/Getting-started/Transformations.html b/translation/Getting-started/Transformations.html
@@ -1,5 +1,3 @@
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-
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<h1 id="content-title">Transformations</h1>
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<h1 id="content-url" style='display:none;'>Getting-started/Transformations</h1>
<p>
We now know how to create objects, color them and/or give them a detailed appearance using textures, but they're still not that interesting since they're all static objects. We could try and make them move by changing their vertices and re-configuring their buffers each frame, but that's cumbersome and costs quite some processing power. There are much better ways to <def>transform</def> an object and that's by using (multiple) <def>matrix</def> objects. This doesn't mean we're going to talk about Kung Fu and a large digital artificial world.
+ これまでオブジェクトを作成し、色を着け、あるいはテクスチャにより精細な見た目を与えることを学びました。しかしここまで作成した物体は静止したままで、あまり面白くありません。フレーム毎に頂点の座標を動かしバッファを再設定することで物体を動かすこともできましょうが、これは面倒であるうえに処理にコストがかかります。よりよい方法として、いくつかの行列を用いて物体を<def>変換</def>するというものがあります。ただし美味しいラーメン屋の話をするわけではありません。
</p>
<p>
Matrices are very powerful mathematical constructs that seem scary at first, but once you'll grow accustomed to them they'll prove extremely useful. When discussing matrices, we'll have to make a small dive into some mathematics and for the more mathematically inclined readers I'll post additional resources for further reading.
+ 行列とはとても便利な数学の道具です。はじめは得体の知れないものかもしれませんが、一度慣れてしまえばその便利さは手放せません。行列の話をする上で、少しばかり数学に足を踏み入れなければなりません。また数学が好きな人の為にさらに発展的な内容の文献も挙げておきます。
</p>
<p>
However, to fully understand transformations we first have to delve a bit deeper into vectors before discussing matrices. The focus of this chapter is to give you a basic mathematical background in topics we will require later on. If the subjects are difficult, try to understand them as much as you can and come back to this chapter later to review the concepts whenever you need them.
+ しかし変換について完全に理解するには行列の話をする前にもう少しベクトルについて深掘りする必要があります。この章では後程必要となる基礎的な数学的の知識を身に付けていただきます。もし難しく感じた場合、可能な限り理解しようとしたうえで、後から必要な概念についてこの章を振り返るようにしてください。
</p>
<h1>Vectors</h1>
+<h1>ベクトル</h1>
<p>
In its most basic definition, vectors are directions and nothing more. A vector has a <def>direction</def> and a <def>magnitude</def> (also known as its strength or length). You can think of vectors like directions on a treasure map: 'go left 10 steps, now go north 3 steps and go right 5 steps'; here 'left' is the direction and '10 steps' is the magnitude of the vector. The directions for the treasure map thus contains 3 vectors. Vectors can have any dimension, but we usually work with dimensions of 2 to 4. If a vector has 2 dimensions it represents a direction on a plane (think of 2D graphs) and when it has 3 dimensions it can represent any direction in a 3D world.
+ 最も基本的な定義において、ベクトルとは矢印以外のなにものでもありません。ベクトルは<def>方向</def>と<def>大きさ</def>（あるいは強さや長さとも）を持ちます。ベクトルは宝の地図に書かれた指示のように捉えることができます: 「左に10歩進み、北に3歩、その後右に5歩進む」この場合「左」がベクトルの向きで「10歩」がベクトルの大きさです。つまりこの宝の地図には3つのベクトルがあります。ベクトルは任意の次元を持つことができますが、この本で利用するのはだいたい2次元から4次元までです。ベクトルが2次元であればそれは平面上の方向を表し（2次元の座標空間を思いうかべて下さい）、3次元であれば3次元空間での任意の方向を表わします。
</p>
<p>
Below you'll see 3 vectors where each vector is represented with <code>(x,y)</code> as arrows in a 2D graph. Because it is more intuitive to display vectors in 2D (rather than 3D) you can think of the 2D vectors as 3D vectors with a <code>z</code> coordinate of <code>0</code>. Since vectors represent directions, the origin of the vector does not change its value. In the graph below we can see that the vectors \(\color{red}{\bar{v}}\) and \(\color{blue}{\bar{w}}\) are equal even though their origin is different:
+ 下の図は2次元の座標空間上に<code>(x, y)</code>と共に矢印で図示した3つのベクトルです。ベクトルは3次元より2次元で図示する方が直感的なので、2次元ベクトルは<code>z</code>座標が<code>0</code>である3次元ベクトルだと見做せます。ベクトルは方向を表わしたものなので、始点が変化してもベクトル自身は同じものです。下の図において2つのベクトル\(\color{red}{\bar{v}}\)と\(\color{blue}{\bar{w}}\)は始点は違いますがベクトルとしては同じものです:
</p>
<img src="/img/getting-started/vectors.png" class="clean" />
<p>
When describing vectors mathematicians generally prefer to describe vectors as character symbols with a little bar over their head like \(\bar{v}\). Also, when displaying vectors in formulas they are generally displayed as follows:
+ 数学者はよく\(\bar{v}\)のように文字の上に小さな棒を書いてベクトルを表わす書き方をします。そして数式の中にベクトルを書く際は一般に以下のような形になります:
\[\bar{v} = \begin{pmatrix} \color{red}x \\ \color{green}y \\ \color{blue}z \end{pmatrix} \]
</p>
<p>
Because vectors are specified as directions it is sometimes hard to visualize them as positions. If we want to visualize vectors as positions we can imagine the origin of the direction vector to be <code>(0,0,0)</code> and then point towards a certain direction that specifies the point, making it a <def>position vector</def> (we could also specify a different origin and then say: 'this vector points to that point in space from this origin'). The position vector <code>(3,5)</code> would then point to <code>(3,5)</code> on the graph with an origin of <code>(0,0)</code>. Using vectors we can thus describe directions <strong>and</strong> positions in 2D and 3D space.
+ ベクトルは方向を表わすので、位置として可視化するのは困難です。ベクトルを位置として捉えるには<vode>(0, 0, 0)</code>をベクトルの始点としたときにそのベクトルが差す点をそのベクトルの位置とします。この見方でのベクトルを<def>位置ベクトル</def>といいます（他のある点を始点として見た場合、「このベクトルは空間上でその点を始点として見たときにあそこの点を差す」というような言い方をします）。<code>(3, 5)</code>という位置ベクトルは<code>(0, 0)</code>を始点と見たときに<code>(3, 5)</code>を指し示すようなベクトルです。ベクトルを利用することで、2次元空間や3次元空間での方向<strong>及び</strong>位置を表わせるようになりました。
</p>
<p>
Just like with normal numbers we can also define several operations on vectors (some of which you've already seen).
+ 通常の数字と同様にベクトルに関してもいくつかの演算を定義できます（既に少しでてきました）。
</p>
<h2>Scalar vector operations</h2>
+<h2>スカラとベクトルの演算</h2>
<p>
A <def>scalar</def> is a single digit. When adding/subtracting/multiplying or dividing a vector with a scalar we simply add/subtract/multiply or divide each element of the vector by the scalar. For addition it would look like this:
+ <def>スカラ</def>というのは数字です。ベクトルに対してスカラを足す／引く／掛ける演算は単にベクトルの各要素に対してスカラを足す／引く／掛けるだけです。足し算は以下のようになります:
\[ \begin{pmatrix} \color{red}1 \\ \color{green}2 \\ \color{blue}3 \end{pmatrix} + x \rightarrow \begin{pmatrix} \color{red}1 \\ \color{green}2 \\ \color{blue}3 \end{pmatrix} + \begin{pmatrix} x \\ x \\ x \end{pmatrix} = \begin{pmatrix} \color{red}1 + x \\ \color{green}2 + x \\ \color{blue}3 + x \end{pmatrix} \]
Where \(+\) can be \(+\),\(-\),\(\cdot\) or \(\div\) where \(\cdot\) is the multiplication operator.
+ \(+\)になっているところは\(+\)、\(-\)、\(\cdot\)あるいは\(\div\)が使えます。ただし\(\cdot\)は掛け算です。
</p>
<h2>Vector negation</h2>
+<h2>ベクトルの反転</h2>
<p>
Negating a vector results in a vector in the reversed direction. A vector pointing north-east would point south-west after negation. To negate a vector we add a minus-sign to each component (you can also represent it as a scalar-vector multiplication with a scalar value of <code>-1</code>):
+ ベクトルを反転させるというのはベクトルの向きを逆にするということです。北東を向いていたベクトルは反転により南西を向くようになります。ベクトルを反転させるには各要素にマイナスの記号を付ければよいです（あるいはベクトルに対して<code>-1</code>を掛けることでも反転できます）:
\[-\bar{v} = -\begin{pmatrix} \color{red}{v_x} \\ \color{blue}{v_y} \\ \color{green}{v_z} \end{pmatrix} = \begin{pmatrix} -\color{red}{v_x} \\ -\color{blue}{v_y} \\ -\color{green}{v_z} \end{pmatrix} \]
</p>
<h2>Addition and subtraction</h2>
+<h2>加法、減法</h2>
<p>
Addition of two vectors is defined as <def>component-wise</def> addition, that is each component of one vector is added to the same component of the other vector like so:
+ 2つのベクトルの足し算は<def>要素毎</def>の足し算として定義されます。つまり以下のように、一方のベクトルの各要素が他方のベクトルの同じ要素と足し合せられるということです:
\[\bar{v} = \begin{pmatrix} \color{red}1 \\ \color{green}2 \\ \color{blue}3 \end{pmatrix}, \bar{k} = \begin{pmatrix} \color{red}4 \\ \color{green}5 \\ \color{blue}6 \end{pmatrix} \rightarrow \bar{v} + \bar{k} = \begin{pmatrix} \color{red}1 + \color{red}4 \\ \color{green}2 + \color{green}5 \\ \color{blue}3 + \color{blue}6 \end{pmatrix} = \begin{pmatrix} \color{red}5 \\ \color{green}7 \\ \color{blue}9 \end{pmatrix} \]
Visually, it looks like this on vectors <code>v=(4,2)</code> and <code>k=(1,2)</code>, where the second vector is added on top of the first vector's end to find the end point of the resulting vector (head-to-tail method):
+ 視覚的には以下の図のように、2つのベクトル<code>v=(4, 2)</code>と<code>k=(1, 2)</code>を足す場合、2つ目のベクトルが1つ目のベクトルの終点から始まり、結果として足し合せられたベクトルは1つ目のベクトルの始点から2つ目のベクトルの終点を差すようになります:
</p>
<img src="/img/getting-started/vectors_addition.png" class="clean"/>
<p>
Just like normal addition and subtraction, vector subtraction is the same as addition with a negated second vector:
+ 通常の加法、減法と同様に、ベクトルの引き算は2つ目のベクトルを反転させたものを足し合せるだけです:
\[\bar{v} = \begin{pmatrix} \color{red}{1} \\ \color{green}{2} \\ \color{blue}{3} \end{pmatrix}, \bar{k} = \begin{pmatrix} \color{red}{4} \\ \color{green}{5} \\ \color{blue}{6} \end{pmatrix} \rightarrow \bar{v} + -\bar{k} = \begin{pmatrix} \color{red}{1} + (-\color{red}{4}) \\ \color{green}{2} + (-\color{green}{5}) \\ \color{blue}{3} + (-\color{blue}{6}) \end{pmatrix} = \begin{pmatrix} -\color{red}{3} \\ -\color{green}{3} \\ -\color{blue}{3} \end{pmatrix} \]
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<p>
Subtracting two vectors from each other results in a vector that's the difference of the positions both vectors are pointing at. This proves useful in certain cases where we need to retrieve a vector that's the difference between two points.
+ 2つのベクトルの引き算の結果は、双方が差す位置の差分となります。この性質は2つのベクトルの指し示す位置の違いを得る必要がある場合に便利です。
</p>
<img src="/img/getting-started/vectors_subtraction.png" class="clean"/>
<h2>Length</h2>
+<h2>長さ</h2>
<p>
To retrieve the length/magnitude of a vector we use the <def>Pythagoras theorem</def> that you may remember from your math classes. A vector forms a triangle when you visualize its individual <code>x</code> and <code>y</code> component as two sides of a triangle:
+ ベクトルの長さを得るには数学の授業で習った<def>ピタゴラスの定理</def>を利用します。ベクトルの<code>x</code>と<code>y</code>の要素をそれぞれ三角形の辺と見ることで、ベクトルが三角形を形成します:
</p>
<img src="/img/getting-started/vectors_triangle.png" class="clean"/>
<p>
Since the length of the two sides <code>(x, y)</code> are known and we want to know the length of the tilted side \(\color{red}{\bar{v}}\) we can calculate it using the Pythagoras theorem as:
+ 2辺の長さ<code>(x, y)</code>は既知なので、斜辺の長さはピタゴラスの定理より:
\[||\color{red}{\bar{v}}|| = \sqrt{\color{green}x^2 + \color{blue}y^2} \]
Where \(||\color{red}{\bar{v}}||\) is denoted as <em>the length of vector \(\color{red}{\bar{v}}\)</em>. This is easily extended to 3D by adding \(z^2\) to the equation.
+ となります。ここで\(||\color{red}{\bar{v}}||\)は<em>ベクトル\(\color{red}{\bar{v}}\)の長さ</em>を表わします。ここに\(z^2\)を加えることで簡単に3次元に拡張できます。
</p>
<p>
In this case the length of vector <code>(4, 2)</code> equals:
+ これを用いると、ベクトル<code>(4, 2)</code>の長さは:
\[||\color{red}{\bar{v}}|| = \sqrt{\color{green}4^2 + \color{blue}2^2} = \sqrt{\color{green}16 + \color{blue}4} = \sqrt{20} = 4.47 \]
Which is <code>4.47</code>.
+ となり、これは<code>4.47</code>です。
</p>
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