Parameter transformation

In analysis, a parameter transformation is a continuous and strictly monotonic mapping that changes the parameter of a path .

Formal definition

Are and two ways and is a continuous and strictly monotonic function with ${\displaystyle \gamma _{1}\colon [a,b]\rightarrow \mathbb {R} ^{n}}$ ${\displaystyle \gamma _{2}\colon [\alpha ,\beta ]\rightarrow \mathbb {R} ^{n}}$ ${\displaystyle f\colon [a,b]\rightarrow [\alpha ,\beta ]}$

${\displaystyle \gamma _{1}(t)=\gamma _{2}(f(t))}$for all , therefore , ${\displaystyle t\in [a,b]}$ ${\displaystyle \gamma _{1}=\gamma _{2}\circ f}$

this is what is called a parameter transformation. One then also calls a reparameterization of means . ${\displaystyle f}$ ${\displaystyle \gamma _{1}}$ ${\displaystyle \gamma _{2}}$ ${\displaystyle f}$

If it increases strictly monotonically, the parameter transformation is called orientation- true. If the parameter transformation is strictly monotonically decreasing, it is called orientation reversal. ${\displaystyle f}$ ${\displaystyle f}$

If and the inverse function are continuously differentiable , then one calls a -parameter transformation. ${\displaystyle f}$ ${\displaystyle f^{-1}}$ ${\displaystyle f}$ ${\displaystyle C^{1}}$

properties

• The parameter transformation changes the path, but not the associated curve .
• The path is rectifiable exactly when it is rectifiable. In this case the path lengths from and are equal. ${\displaystyle \gamma _{1}}$ ${\displaystyle \gamma _{2}}$ ${\displaystyle \gamma _{1}}$ ${\displaystyle \gamma _{2}}$

literature

• Otto Forster : Analysis 2 . Differential calculus in R ⁿ , ordinary differential equations. 8th edition. Vieweg + Teubner Verlag, Wiesbaden 2008, ISBN 978-3-8348-0575-1 , p. 44-46 .

Individual evidence

1. ^ Otto Forster : Analysis 2 . Differential calculus in R ⁿ , ordinary differential equations. 8th edition. Vieweg + Teubner Verlag, Wiesbaden 2008, ISBN 978-3-8348-0575-1 , p. 44 .
2. ↑ Florian Modler, Martin Kreh: Tutorial Analysis 2 and Lineare Algebra 2 . Mathematics explained and commented on by students for students. 2011, ISBN 978-3-8274-2895-0 , pp. 139 .
3. ^ Otto Forster : Analysis 2 . Differential calculus in IRn, ordinary differential equations. 9th edition. Vieweg + Teubner Verlag, 2011, ISBN 978-3-8348-1231-5 , pp. 48 .
4. ↑ Harro Heuser : Textbook of Analysis . Part 2. 13th edition. Teubner Verlag, 2004, ISBN 3-519-62232-7 , pp. 360 .