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Q) Looking at the surface of a smooth $3$-dimensional object from the outside, which one of the following options is TRUE?

$A)$ The surface of the object must be concave everywhere.

$B)$ The surface of the object must be convex everywhere.

$C)$ The surface of the object may be concave in some places and convex in other places.

$D)$ The object can have edges, but no corners.

This question was asked here in aptitude section of the exam. I have provided the answer there also.

According to me, both (C) and (D) are the correct options here because option $(D)$ seems to be ambiguous for me. It is not specified what an "edge" means. It could be straight line segment or it could be the curved edges, for example, like we can take curved edges in graph theory.

And in this way, for $(D)$ we can take an example of "Cylinder" which has $2$ curved edges, one curved surface and $2$ faces and no corners which makes option $(D)$ correct.

So, according to me, both (C) and (D) are correct options here. Can anyone please verify it whether I am correct or not. Any help would be appreciated.

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  • $\begingroup$ Doesn't a curved edge contradict smoothness? $\endgroup$
    – Ivan Neretin
    Commented Feb 22, 2023 at 7:54
  • $\begingroup$ @IvanNeretin how ? $\endgroup$
    – ankit
    Commented Feb 22, 2023 at 7:56
  • $\begingroup$ Please look at the animation here: math.stackexchange.com/questions/4386764/…. This has concave and convex areas and an edge, to boot. Is this a smooth object by your definition? $\endgroup$
    – Cye Waldman
    Commented Feb 24, 2023 at 16:35
  • $\begingroup$ @CyeWaldman smoothness is defined for surfaces, right ? I am considering, smooth 3D object means it has all the surfaces smooth. Since Cylindrical surfaces are smooth and so, I thing we can consider the cylinder as smooth object. Please find the proof at the end of page 8 here pdfhost.io/v/MB8EhcBCO_Overleaf_Example and please correct me if I am wrong ? $\endgroup$
    – ankit
    Commented Feb 24, 2023 at 16:54
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    $\begingroup$ @tbhaxor, B includes the word "must" $\endgroup$
    – ankit
    Commented Sep 25, 2023 at 9:52

1 Answer 1

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I assume we're talking about a 3D surface that doesn't have a boundary.

In both of your references you state "A smooth surface has a well-defined tangent plane at every point of the surface". This precludes surfaces with edges, because a point on an edge (straight or curved) does not have a well-defined tangent plane.

So option (D) is ruled out .. because smooth surfaces cannot have either edges or corners (or conical points for that matter).

That leaves (C) as the only true statement.

Discussion

OP argues (see comments, and materials referenced therein) that a surface (e.g. a cylinder, see picture below adapted from Wikipedia) that is composed of smooth patches connected by smooth edges is itself everywhere smooth, including at the edges, presumably because points on the edges inherit normals from the surrounding surfaces.

A surface is smooth at a point if it has a well-defined normal (equivalently a well-defined tangent plane). The figure below shows two points on the top and side of the cylinder, along with their normals (in red). It also shows a point on the top edge of the cylinder, along with the normals inherited from the top and side surfaces. Both are plausible, but their existence contradicts the claim that there is a single well defined normal.

This counter argument admittedly appeals to visual intuition, but I don't think that a more formal mathematical argument would help matters.

enter image description here

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  • $\begingroup$ I am considering here "edges" as "curved edges" . And for example, if we take example of cylinder then two curved edges can't meet and we can't get a corner and so we can't get a well-defined tangent plane in case of cylinder.. It is proved that cylindrical surfaces are smooth..Can you please look at this link pdfhost.io/v/MB8EhcBCO_Overleaf_Example and can you please me is there any wrong with the proof at page no. 8 in the pdf ? $\endgroup$
    – ankit
    Commented Feb 27, 2023 at 20:38
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    $\begingroup$ I don't see a problem with it. And the conclusion that cylinder $x^2+y^2=1$ is smooth is fine. $\endgroup$
    – brainjam
    Commented Feb 27, 2023 at 22:28
  • $\begingroup$ Thank you for the verification. so both the options C and D are correct here, right ? I am assuming that a 3D mathematical object is smooth if all of its surfaces are smooth and in option (D), it is given as "can have" so it is a possibility that edges can be curves edges. Please verify once if I am right here ? $\endgroup$
    – ankit
    Commented Feb 28, 2023 at 2:46
  • $\begingroup$ And as you have confirmed that cylinder is smooth and cylinder is an example which I have taken for option (D), so should not it be a correct option along with option (C) ? $\endgroup$
    – ankit
    Commented Feb 28, 2023 at 3:46
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    $\begingroup$ @ankit - The infinite cylinder $x^2+y^2=1$ is smooth. The finite cylinder $\max\{x^2+y^2,|z|\}=1$ is not smooth. $\endgroup$
    – mr_e_man
    Commented Mar 3, 2023 at 4:23

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