Local isometry from a circular cone to the plane
= Local isometry from a circular cone to the plane
For the cone
$$
X(r,\theta)=(r\cos\theta,r\sin\theta,\sqrt a\,r),
\qquad r>0,
$$
the <first fundamental form> is
$$
(1+a)\,dr^2+r^2\,d\theta^2.
$$
The local change of <polar coordinates>
$$
R=\sqrt{1+a}\,r,
\qquad
\Theta=\frac{\theta}{\sqrt{1+a}}
$$
turns this into the Euclidean metric $dR^2+R^2d\Theta^2$.