Points, Vectors, Coordinate Systems

Most of this is review but the Visual SLAM uses some fancy notation so I took notes.

• See Vector Review

A point is a basic element in space with no length or volume. Connecting two points forms a vector.

A vector $\mathbf{a}$ in 3D space can be described by ${\mathbb{R}}^3$ with a set of base $({\mathbf{e}}_1,{\mathbf{e}}_2,{\mathbf{e}}_3)$:

$$\mathbf{a}=[{\mathbf{e}}_1,{\mathbf{e}}_2,{\mathbf{e}}_3]\left[\begin{array}{c}{a}_1\\ a_2\\ a_3\end{array}\right]=a_1{\mathbf{e}}_1+a_2{\mathbf{e}}_2+a_3{\mathbf{e}}_3$$

where $(a_1,a_2,a_3{)}^T$ are $\mathbf{a}$‘s coordinates.

Inner Product

Also called a dot product.

$$a\cdot b=a^{T}b=\sum _{i=1}^3{a}_i{b}_i=\mid a\mid \mid b\mid \cos ⟨a,b⟩$$

where $\cos ⟨a,b⟩$ refers to the cosine of the angle between vector $a$ and $b$. This is used to describe the projection between vectors

Outer Product

Also called a cross product.

$$\begin{array}{rl}a\times b& =\left|\begin{array}{ccc}{\mathbf{e}}_1& {\mathbf{e}}_2& {\mathbf{e}}_3\\ a_1& a_2& a_3\\ b_1& b_2& b_3\end{array}\right|=\left[\begin{array}{c}{a}_2{b}_3-a_3{b}_2\\ a_3{b}_1-a_1{b}_3\\ a_1{b}_2-a_2{b}_1\end{array}\right]=\left[\begin{array}{ccc}0& -a_3& a_2\\ a_3& 0& -a_1\\ -a_2& a_1& 0\end{array}\right]b\\ & \\ & \triangleq a^{\wedge }b\end{array}$$

The result of this is a vector whose direction is perpendicular to the two vectors, and the length is $\mid \mathbf{a}\mid \mid \mathbf{b}\mid \sin ⟨\mathbf{a},\mathbf{b}⟩$. The $\triangleq$ symbol means “is equivalent to”.

The $\wedge$ operator is a skew-symmetric symbol, which means writing $\mathbf{a}$ as a skew-symmetric matrix, where:

$${\mathbf{a}}^{\wedge }=\left[\begin{array}{ccc}0& -a_3& a_2\\ a_3& 0& -a_1\\ -a_2& a_1& 0\end{array}\right]$$

The turns the outer product $a\times b$ into the multiplication of the matrix $a^{\wedge }$ and the matrix $b$, which is a linear operation.

Skew-Symmetric

Skew-symmetric matrix means that $\mathbf{A}$ satisfies ${\mathbf{A}}^T=-\mathbf{A}$.