Two links

If there are two links, ${\cal A}_1$ and ${\cal A}_2$, then the C-space can be nicely visualized as a square with opposite faces identified. Each coordinate, $\theta_1$ and $\theta_2$, ranges from 0 to $2 \pi$, for which $0 \sim 2 \pi$. Suppose that each link has length $1$. This yields $a_1 = 1$. A point $(x,y) \in {\cal A}_2$ is transformed as

$$\begin{pmatrix}\cos\theta_1 & -\sin\theta_1 & 0 \sin\theta_1 &... ...& 0 0 & 0 & 1 \end{pmatrix} \begin{pmatrix}x y 1 \end{pmatrix} .$$ (4.58)

To obtain polynomials, the technique from Section 4.2.2 is applied to replace the trigonometric functions using $a_i = \cos\theta_i$ and $b_i=\sin\theta_i$, subject to the constraint $a_i^2 + b_i^2 = 1$. This results in

$$\begin{pmatrix}a_1 & -b_1 & 0 b_1 & a_1 & 0 0 & 0 & 1 \e... ...& 0 0 & 0 & 1 \end{pmatrix} \begin{pmatrix}x y 1 \end{pmatrix} ,$$ (4.59)

for which the constraints $a_i^2 + b_i^2 = 1$ for $i = 1, 2$ must be satisfied. This preserves the torus topology of ${\cal C}$, but now the C-space is embedded in ${\mathbb{R}}^4$. The coordinates of each point are $(a_1,b_1,a_2,b_2) \in {\mathbb{R}}^4$; however, there are only two degrees of freedom because each $a_i,b_i$ pair must lie on a unit circle.

Multiplying the matrices in (4.59) yields the polynomials, $f_1, f_2 \in {\mathbb{R}}[a_1,b_1,a_2,b_2]$,

$$f_1 = x a_1 a_2 - y a_1 b_2 - x b_1 b_2 + y a_2 b_1 + a_1$$ (4.60)

and

$$f_2 = -y a_1 a_2 + x a_1 b_2 + x a_2 b_1 - y b_1 b_2 + b_1 ,$$ (4.61)

for the $x$ and $y$ coordinates, respectively. Note that the polynomial variables are configuration parameters; $x$ and $y$ are not polynomial variables. For a given point $(x,y) \in {\cal A}_2$, all coefficients are determined.