= Geodesics of the product-weighted diagonal plane metric
{title2=$ds^2=y^2dx^2+x^2dy^2$}
On a quadrant $xy\ne0$, the <geodesic Lagrangian> gives $y^2\ddot x+2y\dot y\dot x-x\dot y^2=0$ and $x^2\ddot y+2x\dot x\dot y-y\dot x^2=0$. Rays $y=mx$ reduce these to $(x\dot x)'=0$ and hence $x^2=a\lambda+b$. The coordinate change $\rho=|xy|/\sqrt2$, $\vartheta=\log|y/x|$ gives the locally flat <metric tensor> $d\rho^2+\rho^2d\vartheta^2$. Here $\vartheta$ ranges over the real line, so the polar-plane representation is local, rather than a globally one-to-one map onto the punctured plane. The axes are excluded by the <degenerate metric tensor>; these rays do not establish continuation through the origin as a regular <geodesic>.
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