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Local isometry from a circular cone to the plane

For the cone

X (r, θ) = (r cos θ, r sin θ, a r), r > 0,

the first fundamental form is

(1 + a) d r^{2} + r^{2} d θ^{2} .

The local change of polar coordinates

R = 1 + a r, Θ = \frac{θ}{1 + a}

turns this into the Euclidean metric

d R^{2} + R^{2} d Θ^{2}

.

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