3
$\begingroup$

A flexible rectangle sheet size $(a,b),a>b $ is folded half along side $a$ and glued to make a circular cone cut segment of vertex angle $60^{\circ}$ as shown with three edges $(b,a,b).$

( $60^{\circ}$ choice for cone apex angle deformation arises due to maximum volume created by internal pressure at $90^{\circ}$ corner obtained by maintaining second order continuity along a line perpendicular to glue line.)

After bending distorted edges $(a,b)$ are curved/mapped as conical helices with Clairaut minimal radii nearer to cone vertex as $ (r_a,r_b)= (a/4,b).$ The cone surface is a single boat shaped nappe.

Calculate bent area to verify $A= ab $ conserved due to isometry.

Calculate volume enclosed by parallel displacement of edge $AB$ (skew perpendicular to cone axis) along the helices.

It refers to Jack D'Aurizio A4 paper sheet bent volume problem with two nappes.

Isometric Bending of Sheet to Cone

$\endgroup$
  • $\begingroup$ Are you asking for the volume of the convex hull of this surface? $\endgroup$ – David K Jul 22 '18 at 21:39
  • $\begingroup$ Yes, if $AB$ is a rubber band then it is the concept of convex hull. (Jack D'Aurizio's problem is different, it asks for holding maximum water at an inclination of cone/ object that may need to be defined). in this case no edges are in a horizontal plane. $\endgroup$ – Narasimham Jul 22 '18 at 22:56
  • $\begingroup$ I don't think the resulting solid is convex, because the segment joining the midpoint of $AB$ with the midpoint $C$ or $C'$ of either curve lays outside the solid. The convex hull is simply the solid having for boundaries the cone and planes $ABC$ and $ABC'$. $\endgroup$ – Aretino Jul 23 '18 at 7:07
  • $\begingroup$ I was imagining lines parallel to $AB$ contacting and sliding along both helices,so that surface has an area $=ab$ $\endgroup$ – Narasimham Jul 23 '18 at 7:15
  • $\begingroup$ Why should lines parallel to $AB$ sweep an area $ab$? $\endgroup$ – Aretino Jul 23 '18 at 8:17
0
$\begingroup$

You can see below a diagram of the solid, made with GeoGebra. Green surface $ABC$ is described by $AB$ sliding along the border: it is apparent from the diagram that segment $CM$ lies over the surface, hence the solid is not convex.

enter image description here

The equation for the lateral surface (orange) can be obtained by noticing that a generic point $P$ on the sheet of paper, having distance $r$ from the midpoint $V$ of the folded side and forming an angle $\angle CVP=\theta$, gets mapped to point $$ P'=\left({1\over2}r\cos2\theta,\ {1\over2}r\sin2\theta,\ {\sqrt3\over2}r \right) $$

of the cone, if we take its axis along $z$ and $V$ as origin.

enter image description here

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.