0

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.

ankit
  • 343
  • 2
  • 14
  • Doesn't a curved edge contradict smoothness? – Ivan Neretin Feb 22 '23 at 07:54
  • @IvanNeretin how ? – ankit Feb 22 '23 at 07:56
  • Please look at the animation here: https://math.stackexchange.com/questions/4386764/how-do-i-map-a-sphere-to-a-sphube-a-special-superellipsoid/4387047#4387047. This has concave and convex areas and an edge, to boot. Is this a smooth object by your definition? – Cye Waldman Feb 24 '23 at 16:35
  • @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 https://pdfhost.io/v/MB8EhcBCO_Overleaf_Example and please correct me if I am wrong ? – ankit Feb 24 '23 at 16:54
  • Why not $B$ as the answer? For example check this. I need to know how you come to conclusion on $C$ and $D$ options. – tbhaxor Sep 25 '23 at 06:49
  • 1
    @tbhaxor, B includes the word "must" – ankit Sep 25 '23 at 09:52
  • @ankit, if we have a solid sphere (no hollow) and looking outside? – tbhaxor Sep 25 '23 at 10:14
  • Then it would be convex. Sphere is convex set. – ankit Sep 25 '23 at 12:03

1 Answers1

2

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

brainjam
  • 8,626
  • 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 https://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 ? – ankit Feb 27 '23 at 20:38
  • 2
    I don't see a problem with it. And the conclusion that cylinder $x^2+y^2=1$ is smooth is fine. – brainjam Feb 27 '23 at 22:28
  • 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 ? – ankit Feb 28 '23 at 02:46
  • 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) ? – ankit Feb 28 '23 at 03:46
  • Why are you assuming that a 3D mathematical object is smooth if all of its surfaces are smooth? The assumption is incorrect. Point on the edges do not have well defined normals or tangent planes. – brainjam Feb 28 '23 at 14:41
  • Then what is the definition of smoothness of 3D mathematical object ? I have found definitions of smooth surfaces and smooth functions but didn't find definition of smooth mathematical object. – ankit Feb 28 '23 at 16:05
  • And, why you are saying edges don not have well-defined normals or tangent planes. It is proved and also you have confirmed that "cylinder is smooth" and from definition $S \subseteq R^3$ and we take $\forall P \in S$ and so according to this, for every point on cylinder, whether it is on curved edges or not, has well-defined tangent plane. You are saying cylinder is smooth and I am also taking an example of cylinder and so it should also be smooth and should have tangent plane on every point of the cylinder. Could you please tell me where I am wrong ? – ankit Feb 28 '23 at 16:05
  • I've added a diagram and discussion to the answer that hopefully clarifies matters. – brainjam Feb 28 '23 at 21:11
  • 2
    @ankit - The infinite cylinder $x^2+y^2=1$ is smooth. The finite cylinder $\max{x^2+y^2,|z|}=1$ is not smooth. – mr_e_man Mar 03 '23 at 04:23