Hyperbola
Parametric representation

An affine transformation of the Euclidean plane has the form x → → f → 0 + A x → {\displaystyle {\vec {x}}\to {\vec {f}}_{0}+A{\vec {x}}}, where A {\displaystyle A} is a regular matrix (its determinant is not 0) and f → 0 {\displaystyle {\vec {f}}_{0}} is an arbitrary vector. If f → 1, f → 2 {\displaystyle {\vec {f}}_{1},{\vec {f}}_{2}} are the column vectors of the matrix A {\displaystyle A}, the unit hyperbola (± cosh (t), sinh (t)), t ∈ R, {\displaystyle (\pm \cosh(t),\sinh(t)),t\in \mathbb {R},} is mapped onto the hyperbola

x → = p → (t) = f → 0 ± f → 1 cosh t + f → 2 sinh t. {\displaystyle {\vec {x}}={\vec {p}}(t)={\vec {f}}_{0}\pm {\vec {f}}_{1}\cosh t+{\vec {f}}_{2}\sinh t\.}

f → 0 {\displaystyle {\vec {f}}_{0}} is the center, f → 0 + f → 1 {\displaystyle {\vec {f}}_{0}+{\vec {f}}_{1}} a point of the hyperbola and f → 2 {\displaystyle {\vec {f}}_{2}} a tangent vector at this point.
length_constraints:number_paragraphs[2], combination:two_responses