Solution

Past exam of the mathematics course of the University of Cambridge / 2003 / ib / Paper 3 / 2D / Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 26F / b / Solution

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 $dR^2+R^2d\Theta^2$.

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 Incoming links: Local isometry from a circular cone to the plane