diff --git a/Tutorials/Appendices/01-MathBasics/vectors.html b/Tutorials/Appendices/01-MathBasics/vectors.html index 09c9a2ce..f5b1468c 100755 --- a/Tutorials/Appendices/01-MathBasics/vectors.html +++ b/Tutorials/Appendices/01-MathBasics/vectors.html @@ -772,10 +772,8 @@
Gram-Schmidt Orthogonalization

So, we have that

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

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: \(\mathbf{w_0}\ \bot\ \mathbf{w_1}\).

In the 3D case, we introduce a third vector \(\mathbf{v_2}\) that we need to modify in order to make it orthogonal to both \(\mathbf{w_0}\) and \(\mathbf{w_1}\). As before, we start by setting \(\mathbf{w_0}=\mathbf{v_0}\), and we can still use the method of subtracting the projection of \(\mathbf{v_1}\) onto \(\mathbf{w_0}\) to compute \(\mathbf{w_1}\). This is possible because we can consider \(\mathbf{v_0}\) and \(\mathbf{v_1}\) as lying in the same plane, thereby reducing the problem to the 2D case. To calculate \(\mathbf{w_2}\), we proceed similarly. We subtract the projection of \(\mathbf{v_2}\) onto \(\mathbf{w_0}\) from \(\mathbf{v_2}\), obtaining an intermediate vector. Then, we can subtract the projection of \(\mathbf{v_2}\) onto \(\mathbf{w_1}\) from this intermediate vector. By doing this, we ensure that the resultant vector \(\mathbf{w_2}\) is orthogonal to both \(\mathbf{w_0}\) and \(\mathbf{w_1}\).

@@ -916,7 +914,6 @@

HLSL#

The color set: \(\ r,g,b,a\)

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

-
vector<int, 1> iVector = 1;                             // int iVector = 1;
 vector<float, 4> dVector = { 0.2, 0.3, 0.4, 0.5 };      // float4 dVector = { 0.2, 0.3, 0.4, 0.5 };  
   
@@ -961,8 +958,6 @@ 
C++#

 ../../../_images/SIMD.jpg
 
 
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.

-

-
Listing 52 DirectXMath.h#

 
// 32-bit signed floating-point components
 struct XMFLOAT2
 {
@@ -988,8 +983,6 @@ 
C++#

 };
 

 

-

-

 
// 32-bit signed integer components
 struct XMINTT2
 {
@@ -1071,7 +1064,6 @@ 
C++#
-

[WIP]


Support this project

diff --git a/_sources/Tutorials/Appendices/01-MathBasics/vectors.md b/_sources/Tutorials/Appendices/01-MathBasics/vectors.md index 804ad295..b056ee9c 100755 --- a/_sources/Tutorials/Appendices/01-MathBasics/vectors.md +++ b/_sources/Tutorials/Appendices/01-MathBasics/vectors.md @@ -352,12 +352,8 @@ $$\mathbf{v_1}=\mathbf{w_1}+\text{proj}_{\mathbf{w_0}}(\mathbf{v_1})$$ So, we have that -
- $$\mathbf{w_1}=\mathbf{v_1}-\text{proj}_{\mathbf{w_0}}(\mathbf{v_1})$$ -
- 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: $\mathbf{w_0}\ \bot\ \mathbf{w_1}$. @@ -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: -
- ```{code-block} hlsl vector iVector = 1; // int iVector = 1; @@ -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 @@ -609,9 +601,8 @@ struct XMFLOAT4 float w; }; ``` -
-```cpp +```{code-block} cpp // 32-bit signed integer components struct XMINTT2 { @@ -701,8 +692,6 @@ __declspec(align(16)) struct XMVECTORF32 static const XMVECTORF32 vZero = { 0.0f, 0.0f, 0.0f, 0.0f }; ``` -[WIP] -
````{admonition} Support this project diff --git a/objects.inv b/objects.inv index 29af0d6e..499d0bcd 100755 Binary files a/objects.inv and b/objects.inv differ diff --git a/searchindex.js b/searchindex.js index 6a876f37..e65f897b 100755 --- a/searchindex.js +++ b/searchindex.js @@ -1 +1 @@ -Search.setIndex({"docnames": ["Tutorials/00-Intro/intro-and-prerequisites", "Tutorials/01-HelloWorld/hello-bundles", "Tutorials/01-HelloWorld/hello-cbs", "Tutorials/01-HelloWorld/hello-frame-buffering", "Tutorials/01-HelloWorld/hello-texture", "Tutorials/01-HelloWorld/hello-triangle", "Tutorials/01-HelloWorld/hello-window", "Tutorials/01-HelloWorld/hello-world", "Tutorials/Appendices/01-MathBasics/math-basics", "Tutorials/Appendices/01-MathBasics/vectors", "presentation"], "filenames": ["Tutorials/00-Intro/intro-and-prerequisites.md", "Tutorials/01-HelloWorld/hello-bundles.md", "Tutorials/01-HelloWorld/hello-cbs.md", "Tutorials/01-HelloWorld/hello-frame-buffering.md", 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