Given a surface $S$ in $\mathbb{R}^3$, we can calculate the area of its projection $S'$ onto a given plane $P$ using the formula $[S'] = \iint_{S} \cos\beta \space dA$, where $\beta$ is the angle between the surface and $P$ at a given point on $S$.

Is there a corresponding formula for the perimeter of a surface projected onto a plane? It seems like it would be messier, since not all lengths are scaled equally under projection.

  • $\begingroup$ What about $\int_{\partial S}\cos\beta\,ds$? $\endgroup$ – Aretino Jul 18 at 17:07
  • $\begingroup$ I don't think the above formula works. In the case that $S$ is a polygon that lies on a single plane $Q$, then if you have a side $AB$ of $S$ that lies parallel to the plane $P$, the projected length of $AB$ will equal its original length, even if $\beta \neq 0$. $\endgroup$ – yojan_sushi Jul 18 at 18:13
  • 2
    $\begingroup$ Of course in this case $\beta$ should represent the angle between between the line and $P$ at a given point. $\endgroup$ – Aretino Jul 18 at 19:46

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