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I want to prove that the shapes are isometric. How to prove?

There is no info except for the picture. First of all I need to write surface patches. Please can someone help me? enter image description here


The definition of isometry: let $S_1$ and $S_2$ be two regular surfaces. Let $f: S_1 \to S_2$ be diffeomorphism. $f$ is isometry if the length of $\gamma$ in $S_1$ must be equal to the length of $f\circ \gamma$ in $S_2$ for any curve $\gamma$ in $S_1$. i.e $L_{S_1}(\gamma )= L_{S_2}(f \circ \gamma )$

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  • $\begingroup$ Daniel Rust's comments apply to this question as well. Even if by "isometry" you mean a "local isometry" (same first fundamental form), your surfaces are not "isometric", because two sides of the triangle (or of the rectangle, in your other question) are being identified in your picture. $\endgroup$ – Andrew D. Hwang Nov 27 '13 at 17:14
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    $\begingroup$ From context, your instructor clearly wants you to verify that the given pairs of surfaces have the same first fundamental form in the sets of points corresponding to the interior of the triangle/rectangle. To show this, just calculate the first fundamental form of the cone/cylinder using the "obvious" parametrizations. $\endgroup$ – Andrew D. Hwang Nov 27 '13 at 17:17
  • $\begingroup$ I need to calculate firs fundamental form for cone and I need to calculate the triangle's first fundamental form. Right? Their first fundamental forms are the same. So they are isometric. Well, what is the parametrization of triangle Dear @user86418 $\endgroup$ – user315 Nov 27 '13 at 17:25
  • $\begingroup$ Use polar coordinates, putting the origin at the triangle vertex that maps to the cone vertex. (Incidentally, it is clear that "rolling up" an angular sector made of paper gives a piece of cone, right? The subtended angle at the origin must equal the incident angle at the cone's vertex. If your instructor really didn't say/write more than you have in your notes, they presumably expected you to work out this relationship using elementary geometry.) $\endgroup$ – Andrew D. Hwang Nov 27 '13 at 18:44

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