We define a system of two equations with two variables and six parameters: $\begin{cases} a_1x+b_1y=c_1\\ a_2x+b_2y=c_2 \end{cases}$

We obtain a function $f:\mathbb{R^3\times\mathbb{R}^2\rightarrow\mathbb{R}^2}$,

$$\pmatrix{a_1 & a_2 \\ b_1 & b_2 \\ c_1 & c_2}\mapsto \pmatrix{x \\ y} $$

If we fix the five parameters $a_1$, $a_2$, $b_1$, $b_2$, $c_1$ and we vary the value of $c_2$. We obtain a function $f_{c_2}: \mathbb{R}\rightarrow\mathbb{R}^2$, $c_2\mapsto\left(x,y\right)$. In the same manner we obtain the five other functions $f_{a_1}$, $f_{a_2}$, $f_{b_1}$, $f_{b_2}$, $f_{c_1}$.

My question is how are theses functions called and what are their properties?


1 Answer 1


The functions in OP are defined by the system: $$\begin{cases} a_1x+b_1y=c_1\\ a_2x+b_2y=c_2 \end{cases}$$ so that: $$ x=\dfrac {\Delta_x}{\Delta} \qquad y=\dfrac {\Delta_y}{\Delta} $$ where: $$ \Delta=a_1b_2-a_2b_1 \quad \Delta_x=b_1c_2-b_2c_1 \quad \Delta_y=a_1c_2-a_2c_1 $$ so that the functions $f_{c_1}:\mathbb{R}\to\mathbb{R}^2$ and $f_{c_2}:\mathbb{R}\to\mathbb{R}^2$ are: $$ f_{c_1}(t)= \begin{bmatrix} \frac{-b_2}{\Delta}t+\frac{b_1c_2}{\Delta}\\ \frac{-a_2}{\Delta}t+\frac{a_1c_2}{\Delta}\\ \end{bmatrix} $$ and $$ f_{c_2}(t)= \begin{bmatrix} \frac{b_1}{\Delta}t-\frac{b_2c_1}{\Delta}\\ \frac{a_1}{\Delta}t-\frac{a_2c_1}{\Delta}\\ \end{bmatrix} $$ so these functions represent straight lines on $\mathbb{R}^2$.

The functions $f_{a_i}$ and $f_{b_i}$ are different. As example the function $f_{a_1}:\mathbb{R} \to \mathbb{R}^2$ is: $$ f_{a_1}(t)= \begin{bmatrix} \dfrac{b_1c_2-b_2c_1}{b_2t-a_2b_1}\\ \dfrac{c_2t-a_2c_1}{b_2t-a_2b_1} \end{bmatrix} $$ that represent an homographic function in $\mathbb{R}^2$ whose graph is an hyperbola.

I dont know if all these function have some collective name (I never see it).



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