Projecting a surface segment of a cone onto a 2D plane?

Question

Firstly, I'd like to apologise - I do not know the correct terms for what I am asking.

Diagram showing a section of a cone highlighted

Assume that the top/bottom of the highlighted portion there is actually aligned with the base.

To help explain: I need to wrap that section of the cone using a piece of paper. What shape (exactly) do I need to cut out from said paper so that it will wrap flawlessly?

Maazul · Accepted Answer · 2013-05-15T04:42:46.550

3

If I've understood your problem correctly, I think this should help. This one's a right circular cone and its opened up paper version. Here we've taken an arbitrary curved surface of the form shown on the cone and visualized. enter image description here

edited May 15 '13 at 04:42

answered May 14 '13 at 23:39

Maazul

2,498

What about r? Would that affect the space seen on the paper cutout? – Steffan Donal May 15 '13 at 06:12
It depends. If $a$ and $b$ are given to you, then $r$ is not needed. If the perpendicular heights $h_a$ and $h_b$ are what's given, you will need $r$ to find $a$ and $b$, respectively. – genepeer May 15 '13 at 08:37
@genepeer Agreed... and obviously the outer circular perimeter of the paper cutout is $2\pi r$. – Maazul May 15 '13 at 09:37
Don't mind me, just forgetting very basic geometry here. – Steffan Donal May 15 '13 at 10:20

score 1 · Answer 2 · answered May 14 '13 at 23:41

Since the distance from the apex to the base is constant. Assuming the cone is made of paper, cutting it open along your yellow line will make a circular sector whose radius is the former distance. Your cross-sections will correspond to bands concentric with the sector's circumference.

Projecting a surface segment of a cone onto a 2D plane?

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