Give an example of a bijective, continuous mapping φ : R 2 → R 2 {\displaystyle {}\varphi \colon \mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2}} , with φ ( S 1 ) = S 1 {\displaystyle {}\varphi (S^{1})=S^{1}} and with φ ( s v ) = s φ ( v ) {\displaystyle {}\varphi (sv)=s\varphi (v)} (for all s ∈ R {\displaystyle {}s\in \mathbb {R} } and v ∈ R 2 {\displaystyle {}v\in \mathbb {R} ^{2}} ) that is not an isometry.
