Skip to content

Commit

Permalink
Update documentation
Browse files Browse the repository at this point in the history
  • Loading branch information
PAMinerva committed Feb 18, 2024
1 parent 68289c8 commit dde4318
Show file tree
Hide file tree
Showing 4 changed files with 2 additions and 21 deletions.
8 changes: 0 additions & 8 deletions Tutorials/Appendices/01-MathBasics/vectors.html
View file
Original file line number Diff line number Diff line change
Expand Up @@ -772,10 +772,8 @@ <h5>Gram-Schmidt Orthogonalization<a class="headerlink" href="#gram-schmidt-orth
<img alt="../../../_images/2D-Gram-Schmidt.png" src="../../../_images/2D-Gram-Schmidt.png" />
</figure>
<p>So, we have that</p>
<br>
<div class="math notranslate nohighlight">
\[\mathbf{w_1}=\mathbf{v_1}-\text{proj}_{\mathbf{w_0}}(\mathbf{v_1})\]</div>
<br>
<p>where <span class="math notranslate nohighlight">\(\ \text{proj}_{\mathbf{w_0}}(\mathbf{v_1})=\displaystyle\frac{\mathbf{v_1}\cdot\mathbf{w_0}}{|\mathbf{w_0}|^2}\mathbf{w_0}\)</span></p>
<p>To prove that <span class="math notranslate nohighlight">\(\mathbf{w_0}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{w_1}\)</span> are orthogonal to each other, we can first observe that a projection is orthogonal if the direction of projection forms a right angle <span class="math notranslate nohighlight">\((90°)\)</span> with the vector we project onto (see the dashed line in the illustration above). Also, we know that the sum of two vectors is the diagonal of the parallelogram with sides the two vectors. In this case we obtain a rectangle since we just established that an angle of the parallelogram with diagonal <span class="math notranslate nohighlight">\(v_1\)</span> is <span class="math notranslate nohighlight">\(90°\)</span>. So, we verified that <span class="math notranslate nohighlight">\(\mathbf{w_0}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{w_1}\)</span> are orthogonal: <span class="math notranslate nohighlight">\(\mathbf{w_0}\ \bot\ \mathbf{w_1}\)</span>.</p>
<p>In the 3D case, we introduce a third vector <span class="math notranslate nohighlight">\(\mathbf{v_2}\)</span> that we need to modify in order to make it orthogonal to both <span class="math notranslate nohighlight">\(\mathbf{w_0}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{w_1}\)</span>. As before, we start by setting <span class="math notranslate nohighlight">\(\mathbf{w_0}=\mathbf{v_0}\)</span>, and we can still use the method of subtracting the projection of <span class="math notranslate nohighlight">\(\mathbf{v_1}\)</span> onto <span class="math notranslate nohighlight">\(\mathbf{w_0}\)</span> to compute <span class="math notranslate nohighlight">\(\mathbf{w_1}\)</span>. This is possible because we can consider <span class="math notranslate nohighlight">\(\mathbf{v_0}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{v_1}\)</span> as lying in the same plane, thereby reducing the problem to the 2D case. To calculate <span class="math notranslate nohighlight">\(\mathbf{w_2}\)</span>, we proceed similarly. We subtract the projection of <span class="math notranslate nohighlight">\(\mathbf{v_2}\)</span> onto <span class="math notranslate nohighlight">\(\mathbf{w_0}\)</span> from <span class="math notranslate nohighlight">\(\mathbf{v_2}\)</span>, obtaining an intermediate vector. Then, we can subtract the projection of <span class="math notranslate nohighlight">\(\mathbf{v_2}\)</span> onto <span class="math notranslate nohighlight">\(\mathbf{w_1}\)</span> from this intermediate vector. By doing this, we ensure that the resultant vector <span class="math notranslate nohighlight">\(\mathbf{w_2}\)</span> is orthogonal to both <span class="math notranslate nohighlight">\(\mathbf{w_0}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{w_1}\)</span>.</p>
Expand Down Expand Up @@ -916,7 +914,6 @@ <h3>HLSL<a class="headerlink" href="#hlsl" title="Link to this heading">#</a></h
<li><p>The color set: <span class="math notranslate nohighlight">\(\ r,g,b,a\)</span></p></li>
</ul>
<p>Specifying one or more vector components is called swizzling. For example:</p>
<br>
<div class="highlight-hlsl notranslate"><div class="highlight"><pre><span></span><span class="kt">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="mi">1</span><span class="o">&gt;</span><span class="w"> </span><span class="n">iVector</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// int iVector = 1;</span>
<span class="kt">vector</span><span class="o">&lt;</span><span class="kt">float</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="o">&gt;</span><span class="w"> </span><span class="n">dVector</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="mf">0.2</span><span class="p">,</span><span class="w"> </span><span class="mf">0.3</span><span class="p">,</span><span class="w"> </span><span class="mf">0.4</span><span class="p">,</span><span class="w"> </span><span class="mf">0.5</span><span class="w"> </span><span class="p">};</span><span class="w"> </span><span class="c1">// float4 dVector = { 0.2, 0.3, 0.4, 0.5 }; </span>
<span class="w"> </span>
Expand Down Expand Up @@ -961,8 +958,6 @@ <h3>C++<a class="headerlink" href="#c" title="Link to this heading">#</a></h3>
<img alt="../../../_images/SIMD.jpg" src="../../../_images/SIMD.jpg" />
</figure>
<p>However, <strong>__m128</strong> variables require 16-byte alignment in memory. This isn’t an issue for global or local variables of this type, as the compiler automatically aligns them. Problems arise when using an <strong>XMVECTOR</strong> (which is an alias for <strong>__m128</strong>) as a member of a structure or class, where C++ packing rules might cause misalignment. To address this, DirectXMath provides specific types for including integer or floating-point vectors as class members without alignment concerns.</p>
<div class="literal-block-wrapper docutils container" id="vectors-xmfloats-code">
<div class="code-block-caption"><span class="caption-number">Listing 52 </span><span class="caption-text">DirectXMath.h</span><a class="headerlink" href="#vectors-xmfloats-code" title="Link to this code">#</a></div>
<div class="highlight-cpp notranslate"><div class="highlight"><pre><span></span><span class="c1">// 32-bit signed floating-point components</span>
<span class="k">struct</span><span class="w"> </span><span class="nc">XMFLOAT2</span>
<span class="p">{</span>
Expand All @@ -988,8 +983,6 @@ <h3>C++<a class="headerlink" href="#c" title="Link to this heading">#</a></h3>
<span class="p">};</span>
</pre></div>
</div>
</div>
<br>
<div class="highlight-cpp notranslate"><div class="highlight"><pre><span></span><span class="c1">// 32-bit signed integer components</span>
<span class="k">struct</span><span class="w"> </span><span class="nc">XMINTT2</span>
<span class="p">{</span>
Expand Down Expand Up @@ -1071,7 +1064,6 @@ <h3>C++<a class="headerlink" href="#c" title="Link to this heading">#</a></h3>
</pre></div>
</div>
</div>
<p>[WIP]</p>
<br>
<div class="admonition-support-this-project admonition">
<p class="admonition-title">Support this project</p>
Expand Down
13 changes: 1 addition & 12 deletions _sources/Tutorials/Appendices/01-MathBasics/vectors.md
View file
Original file line number Diff line number Diff line change
Expand Up @@ -352,12 +352,8 @@ $$\mathbf{v_1}=\mathbf{w_1}+\text{proj}_{\mathbf{w_0}}(\mathbf{v_1})$$

So, we have that

<br>

$$\mathbf{w_1}=\mathbf{v_1}-\text{proj}_{\mathbf{w_0}}(\mathbf{v_1})$$

<br>

where $\ \text{proj}_{\mathbf{w_0}}(\mathbf{v_1})=\displaystyle\frac{\mathbf{v_1}\cdot\mathbf{w_0}}{|\mathbf{w_0}|^2}\mathbf{w_0}$

To prove that $\mathbf{w_0}$ and $\mathbf{w_1}$ are orthogonal to each other, we can first observe that a projection is orthogonal if the direction of projection forms a right angle $(90°)$ with the vector we project onto (see the dashed line in the illustration above). Also, we know that the sum of two vectors is the diagonal of the parallelogram with sides the two vectors. In this case we obtain a rectangle since we just established that an angle of the parallelogram with diagonal $v_1$ is $90°$. So, we verified that $\mathbf{w_0}$ and $\mathbf{w_1}$ are orthogonal: $\mathbf{w_0}\ \bot\ \mathbf{w_1}$.
Expand Down Expand Up @@ -526,8 +522,6 @@ The components of a vector can be accessed using the subscript operator, [ ], to

Specifying one or more vector components is called swizzling. For example:

<br>

```{code-block} hlsl
vector<int, 1> iVector = 1; // int iVector = 1;
Expand Down Expand Up @@ -582,8 +576,6 @@ SIMD allows the CPU to execute four operations (denoted as OP in the image below
However, **__m128** variables require 16-byte alignment in memory. This isn't an issue for global or local variables of this type, as the compiler automatically aligns them. Problems arise when using an **XMVECTOR** (which is an alias for **__m128**) as a member of a structure or class, where C++ packing rules might cause misalignment. To address this, DirectXMath provides specific types for including integer or floating-point vectors as class members without alignment concerns.

```{code-block} cpp
:caption: DirectXMath.h
:name: vectors-xmfloats-code
// 32-bit signed floating-point components
struct XMFLOAT2
Expand All @@ -609,9 +601,8 @@ struct XMFLOAT4
float w;
};
```
<br>

```cpp
```{code-block} cpp
// 32-bit signed integer components
struct XMINTT2
{
Expand Down Expand Up @@ -701,8 +692,6 @@ __declspec(align(16)) struct XMVECTORF32
static const XMVECTORF32 vZero = { 0.0f, 0.0f, 0.0f, 0.0f };
```

[WIP]

<br>

````{admonition} Support this project
Expand Down
Binary file modified objects.inv
View file
Binary file not shown.
2 changes: 1 addition & 1 deletion searchindex.js
View file

Large diffs are not rendered by default.

0 comments on commit dde4318

Please sign in to comment.