Computer Graphics Linear Algebra Review

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CG-Math-Review

Vectors

Vectors are mathematical objects that have both a magnitude (size) and a direction. They are widely used in computer graphics for representing points, directions, velocities, and more in 2D, 3D, or higher-dimensional spaces.

Vector Normalization

Normalization is the process of converting a vector to a unit vector (a vector with a magnitude of 1) while maintaining its direction. This is often useful in shading calculations or directional movement.

To normalize a vector v:

$$\hat{\mathbf{v}}=\frac{\mathbf{v}}{\parallel \mathbf{v}\parallel }$$

Where:

• $\mathbf{v}$ is the original vector.
• $\parallel \mathbf{v}\parallel$ is the magnitude (or length) of the vector, calculated as:$$\parallel \mathbf{v}\parallel =\sqrt{v_x^2+v_y^2+v_z^2}\$\$(in3D)$$

Cartesian coordinates are used to describe points in a 2D or 3D space as tuples of numbers (x, y) or (x, y, z). In vector representation, they are used to describe the components of a vector.

For example:

• A point in 2D space: $\mathbf{p}=(x,y)$
• A point in 3D space: $\mathbf{p}=(x,y,z)$

Vector Multiplication

Vector multiplication comes in three major forms:

1. Scalar (dot) product: Produces a scalar value, useful for projections.
2. Vector (cross) product: Produces a new vector orthogonal to the two input vectors.
3. Scalar multiplication: A vector is multiplied by a scalar value.

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Dot (Scalar) Product

The dot product of two vectors produces a single scalar value and is a measure of how much one vector extends in the direction of another. It is calculated using:

$$\text{mathbf{a} cdot mathbf{b} = a\_x b\_x + a\_y b\_y + a\_z b\_z\$\$ (in 3D) \#\#\# Dot Product in Cartesian Coordinates For two vectors \$mathbf{a} = (a\_x, a\_y, a\_z)\$ and \$mathbf{b} = (b\_x, b\_y, b\_z)\$, the dot product is: \$\$mathbf{a} cdot mathbf{b} = a\_x b\_x + a\_y b\_y + a\_z b\_z}$$

Dot Product for Projection

The dot product can also be used to compute the projection of one vector onto another:

$$\text{Projection of }\mathbf{a}\text{ onto }\mathbf{b}=\frac{\mathbf{a}\cdot \mathbf{b}}{\parallel \mathbf{b}\parallel }$$

In vector form, the projected vector is:

$${\text{Proj}}_{\mathbf{b}}(\mathbf{a})=\frac{\mathbf{a}\cdot \mathbf{b}}{\parallel \mathbf{b}{\parallel }^2}\cdot \mathbf{b}$$

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Cross (Vector) Product

In 3D, the cross product of two vectors produces a new vector that is perpendicular to both of the input vectors. The magnitude of the result is equal to the area of the parallelogram formed by the two vectors.

$$\mathbf{a}\times \mathbf{b}=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ a_x& a_y& a_z\\ b_x& b_y& b_z\end{array}\right|$$

Where $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ are the unit vectors of the x, y, and z axes.

Cross Product: Properties

1. $\mathbf{a}\times \mathbf{b}=-(\mathbf{b}\times \mathbf{a})$
2. $\mathbf{a}\times \mathbf{a}=\mathbf{0}$ (zero vector)
3. The magnitude of the cross product is:$$\parallel \mathbf{a}\times \mathbf{b}\parallel =\parallel \mathbf{a}\parallel \parallel \mathbf{b}\parallel \sin \theta$$Where $\theta$ is the angle between the vectors.
4. The direction of the resultant vector follows the right-hand rule.

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Orthonormal Coordinate Frames

An orthonormal coordinate frame is a set of basis vectors that are:

1. Orthogonal: Each pair of vectors is perpendicular.
2. Normalized: Each basis vector has a magnitude of 1.

In 3D computer graphics, orthonormal frames (like the basis vectors $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$) are often used for camera orientation, transformation matrices, and object alignment.

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Matrix

A matrix is a 2D array of numbers that can represent a variety of transformations or systems of equations. In computer graphics, matrices are widely used to perform linear transformations like scaling, rotation, and translation.

Matrix-Matrix Multiplication

The product of two matrices $A$ (of size $m\times n$) and $B$ (of size $n\times p$) is a new matrix $C=AB$ (of size $m\times p$):

$$C_{ij}=\sum _{k=1}^n{A}_{ik}{B}_{kj}$$

Matrix-Vector Multiplication

A matrix can transform a vector by the operation $\mathbf{y}=A\mathbf{x}$, where $A$ is a matrix and $\mathbf{x}$ is a vector.

Example: For $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ and $\mathbf{x}=\left[\begin{array}{c}x\\ y\end{array}\right]$,

$$A\mathbf{x}=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}ax+by\\ cx+dy\end{array}\right]$$

Transpose of a Matrix

The transpose of a matrix $A$ is obtained by flipping its rows and columns:

$$A_{ij}^T=A_{ji}$$

Identity Matrix and Inverses

1. Identity Matrix: An identity matrix $I$ is a square matrix with 1s on the diagonal and 0s elsewhere:

   $$I=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$$

2. Inverse Matrix: For a square matrix $A$, the inverse $A^{-1}$ satisfies:

   $$A{A}^{-1}=A^{-1}A=I$$

   Not all matrices have inverses; a matrix must be non-singular (determinant $\ne 0$).

Vector Multiplication in Matrix Form

The dot product of two vectors can also be represented as a matrix multiplication:

$$\mathbf{a}\cdot \mathbf{b}={\mathbf{a}}^T\mathbf{b}$$

For example:

$$\mathbf{a}=\left[\begin{array}{c}{a}_x\\ a_y\end{array}\right],\mathbf{b}=\left[\begin{array}{c}{b}_x\\ b_y\end{array}\right]$$$${\mathbf{a}}^T\mathbf{b}=\left[\begin{array}{cc}{a}_x& a_y\end{array}\right]\left[\begin{array}{c}{b}_x\\ b_y\end{array}\right]=a_x{b}_x+a_y{b}_y$$