GATE CSE 2023: Question Number GA-5 1 Master Paper: Question GA-5 Figure 1: Question Description Screenshot from the master question paper GATE Answer Key: (C) My Opinion: Possible answers should be both ( C ) and ( D ) for this ques- tion as question is based on certain assumptions for option ( D ) which are not given in the question. Explanation: We can combine the convex lens and the concave lens and the combined lens is called a convexo-concave or concavo-convex lens for which one side is convex and other side is concave. Since, convex and concave lenses are 3 − dimensional object because each one is formed from two spheres (a three- dimensional object) and so the combined object is also a 3 − dimensional object. 1 More Details: ( 1 ) difference-between-convexo-concave-and-concavo-convex-lenses ( 2 ) Formation of Convex and concave Lenses Hence, options ( A ) and ( B ) are eliminated and ( C ) is correct. Also, the texture of this 3 − dimensional object is smooth because it is hav- ing continuous even surface and not having bumps, holes, ridges or any rough surface. A smooth surface has a well-defined tangent plane at every point of the surface. (We will see what is the meaning of ”smoothness”) mathematically later. Now, option ( D ) may be correct depends on what ”edge” implies here. If we consider the curved edge (like a self loop in graphs) as an edge then option ( C ) is also correct because consider a cylinder which has 2 curved edges, 1 curved surface, 2 flat faces and no cor- ners but if we consider the edge as an straight line then for a finite three-dimensional object, option ( D ) is wrong because where at least two lines or straight edges meet, it creates a corner and according to the definition of smoothness, it should not have a sudden rise or fall (a sharp corner) and so it will not be a smooth object and so if edge means straight edges then ( D ) is wrong but at some places authors consider edges as curved edges and at some places edges are considered as straight line segment. So, we can consider both choices and that’s why option ( D ) should also be correct. The shape of the cylinder is as following: 2 Figure 2: Image Source: Wikipedia Since question implies only one option is correct, so both ( C ) and ( D ) are possible answers because if we assume an edge as ”curved edge” then it makes option ( D ) as correct. For example, Cylinder which is smooth three-dimentional object (we will prove that each surface of the cylinder is smooth later.) 2 Evidences in the support of option (D) As part of the evidences, we will show two things: first that edges can be curved and second all the surfaces of cylinder are smooth. So, first, we will provide evidences that edges can be curved also. ( 1 ) Purdue university Computer Science Course Course Link: Representation of Curved Edges and Faces Here, in Section 5.7: Edge Identification , it is mentioned that: If curved edges are not specified carefully, the boundary description of objects could contain ambiguities and to remove the ambiguities, 3 it should be defined as the ”the intersection of surfaces”. So, If we consider the shape of the cylinder and if we take the intersection of flat and the curved surfaces, we get the curved edges according to the defi- nition of intersection of two surfaces. So, here, edges are considered as curved edges.g ( 2 ) Introduction to Graph Theory, Second Edition, by Douglas B. West We can represent a three-dimensional mathematical object on a plane by ver- tices and the edges. According to the definition 1.1.2, A graph G is a triple consisting of a vertex set V ( G ) , an edge set E ( G ) , and a relation that associates with each edge, two vertices (not necessarily distinct) called its endpoints. We draw a graph on paper by placing each vertex at a point and representing each edge by a curve joining the locations of its end- points. Please find the attachment for a graph with curved edges from the chap- ter: Embeddings and Euler’s Formula Figure 3: Image Source: Introduction to Graph Theory by Douglas West Note: Various Graph Theory Books like Diestel, Bondy and Murthy, Harary,Narsingh Deo follows the definition of graphs in which curved edges are allowed and a graph can be non-planar and can be a three-dimensional object, Hence, it shows that curved edges are allowed for a three-dimensional object. 4 ( 3 ) Geometry For Dummies (2nd Edition) The text of this book is available on Amazon and it can be also downloaded in the pdf format from here In Chapter 1 , it is mentioned that ”You may be familiar with some shapes that have curved sides , such as circles, ellipses, and parabolas. The circle is the only curved 2-D shape.” For three-dimensional objects, they have given the example of ”Cone” as: Figure 4: Image Source: Geometry For Dummies And on Page No. 295, they have written that: ”The lateral area of a cone is one “triangle” that’s been rolled into a cone shape like a snow-cone cup (it’s only kind of a triangle because when flattened out, it’s actually a sector of a circle with a curved bottom side. ” ”side” is nothing but an edge. So, this book also confirms the possibility of curved edges. Also, on Page no. 288, they have written that: A cylinder is a solid figure with two congruent, parallel bases that have rounded sides (in other words, the bases are not straight-sided polygons); these bases are connected by a rounded surface. A right circular cylinder is a cylinder with circular bases that are directly above and below each other. (And the circular bases are at a right angle with the curving sides.) All cylinders in this book, and almost all cylinders you find in other geometry books, are right circular cylin- 5 ders. When I say cylinder, I mean a right circular cylinder ( 4 ) Plane and Solid Geometry The text of this book is available on Springer Link and it can be also down- loaded in the pdf format from here Here, curved edge or line is defined as A curved line is a line, no por- tion of which is straight and A curved surface is a surface no portion of which is plane According to this definition, curved edge is also possible for a three-dimensional object. Note that Cylinder: C = { ( x, y, z ) ∈ R 3 | x 2 + y 2 = 1 } is also curved, but the map ψ : R 2 → C defined by ψ ( s, t ) = (cos( t ) , sin( t ) , s ) preserves the length of the curves. ( 5 ) Various Blogs about Curved edges of a 3D- Mathematical ob- ject: ( A ) 2D and 3D shapes ( B ) Does a cylinder have edges? ( C ) a Cylinder has one curved surface, two curved edges (always circles) and no vertices. ( D ) Third Space Blog Note: The two-dimensional version of a mathematical object which has edge(s) but no corner is M ̈ obius strip and Stadium. Since, we have shown that curved edges are possible for the three-dimensional object. So, now, we comes to ”smoothness” of a three-dimensional mathemati- cal object. We will prove that Cylinder has smooth surfaces mathematically as: 6 3 Proof for the Smoothness of Cylindrical Sur- faces Before going to the actual proof, we will go through the following definitions: ( 1 ) Surface: If S ⊆ R 3 and ∀ p ∈ S where p is any point in surface S, we have ∃ open set w ∈ R 3 and ∃ open set u ∈ R 2 such that p ∈ S ∩ W and S ∩ W is homeomor- phic to U i.e. S ∩ W ≈ U which means we get a function σ : U → S ∩ W and this function σ should be bijective, continuous and σ − 1 should also be continuous. We call this function as parametrization. ( 2 ) Curve: If we have a function γ : ( α, β ) → R n where R n is an Euclidean space, then we say, γ is curve or parametrization of the curve. Now, if every order of derivatives of γ i.e. γ ′ ( t ) , γ ′′ ( t ) , γ ′′′ ( t ) , ..., γ n ( t ) exists then we say curve is smooth. The basic meaning of smoothness of a curve is that there is no corner or sudden jump for every point of the curve. Like curve is made from the function, the surface S is also made from the function but it is not necessary that it is made from only one function. For surfaces, we have surface patches or parameterization i.e. ( σ, u ), And with the help of derivatives of these patches, we define the smoothness of the surfaces. Here, also, the basic meaning of smoothness of a surface is that there is no corner or sudden jump for every point of the curve. For a three-dimensional object, every patches of a surface, we have a func- tion σ : R 2 → R 3 with σ ( u, v ) = ( σ 1 ( u, v ) , σ 2 ( u, v ) , σ 3 ( u, v )) where σ 1 , σ 2 , σ 3 are defined for R 2 → R We say a multivariable function f smooth if all the partial derivatives of f exists and are continuous. So, if functions σ 1 , σ 2 , σ 3 are smooth then function σ is smooth and so, sur- face patch ( σ, u ) is a smooth surface patch. 7 A curve is regular if its derivative σ ′ ( t ) ̸ = 0 For any surface, if we take a point and draw various curves passing through this point then we get the various corresponding tangent vectors and with these tangent vectors, we get a plane which is called the tangent plane. If the derivatives of the curves get zero at a point on the surface from which many curves are passes, then we can’t get the tangent vectors and in this situa- tion, we can’t get the tangent plane and if we can’t get the tangent plane then we say, that surface is not regular. Now, if we want to make the tangent plane then we need a normal vector and it can be obtained by the cross product of two tangent vectors, say, γ ′ 1 ( t ) × γ ′ 2 ( t ) and so, if we get a normal vector or a tangent plane at a point of the surface then we can say, surface is regular at that point. For any mathematical object, we have a patch corresponding to each point of the surface and when this patch σ operates on u and v i.e. σ ( u, v ) , we get that point. From this patch σ ( u, v ) , if we fix one parameter, say, u is constant and v varies and in this way, we have a curve on that surface. Similarly, we also get a curve if we do u constant and v varies. Hence, we get two tangent vectors in this case i.e. σ v and σ u and if we take cross product of σ v and σ u , we get the normal vector and if σ v × σ u ̸ = 0 then we say, we are getting the tangent plane with the help of normal vector and so, Surface is regular So, ( σ, u ) is a surface patch of surface S and it is regular surface patch(Allowable Surface Patch) if this surface patch is smooth and σ u and σ v are linearly independent. Definition of Smooth Surface: A surface S ⊆ R 3 is called smooth if for any p ∈ S there exists a regu- lar(allowable) surface patch ( σ, u ) such that p ∈ σ ( u ) Note: Plane, Cylinder and Spheres are smooth surfaces. Proof: Unit Cylinder is a smooth surface S = { ( x, y, z ) ∈ R 3 | x 2 + y 2 = 1 } Now, take two open sets as: A = { ( u, v ) ∈ R 2 | 0 < u < 2 π } and B = { ( u, v ) ∈ R 2 | − π < v < π } σ : A → S ∩ R 3 8 σ : B → S ∩ R 3 σ ( u, v ) = (cos u, sin u, v ) So, we get the atlas as { ( σ, A ) , ( σ, B ) } ( σ, A ) gives cylinder without one vertical line which is visible on a plane and same for ( σ, B ) and after merging we get the whole cylinder and now we have to prove this surface is smooth. So, we need to prove every patch of this atlas is regular. First we prove σ is smooth. Here, partial derivatives σ v = ( − sin u, cos u, 0) and σ u = (0 , 0 , 1) and we can also calculate next derivates and all partial derivates exist and these are also continuous. Hence, σ is smooth. And, now we need to prove, σ u and σ v are linearly independent. Here, σ u ̸ = cσ v ∀ c ∈ R − { 0 } Hence, σ u and σ v are linearly independent. Therefore, patches ( σ, A ) and ( σ, B ) are regular surface patches. Hence, it is proved that unit cylinder has all smooth surfaces. 4 Evidences in the support of option (C) As we have already given one example of a three-dimensional mathematical ob- ject which has one side convex and one side concave but still we still try to understand the meaning of convex and concave surfaces. For a 3 D mathematical object which might not be smooth, can have convex and concave surface and these surfaces can be behave like a convex and concave function. So, what a convex function means ? We start with two-dimensional function: Here, Let, x 3 = θx 1 + (1 − θ ) x 2 ; 0 ≤ θ ≤ 1 9 Figure 5: Convex Function That is, x 3 is a linear combination of x 1 and x 2 As it is cleared from the figure 5 that z ≥ f ( x 3 ) And equation of line AB is y − f ( x 1 ) = f ( x 2 ) − f ( x 1 ) x 2 − x 1 ( x − x 1 ) If we put x = x 3 = θx 1 + (1 − θ ) x 2 z = y = f ( x 1 ) + f ( x 2 ) − f ( x 1 ) x 2 − x 1 ( θx 1 − x 1 + (1 − θ ) x 2 ) After simplification and using z ≥ f ( x 3 ) , we get the condition for the con- vex function: θf ( x 1 ) + (1 − θ ) f ( x 2 ) ≥ f ( θx 1 + (1 − θ ) x 2 ) ∀ θ ∈ [0 , 1] and x 1 , x 2 ∈ dom ( f ) 1) For multidimensional case, we can write, f ( y ) ≥ f ( x ) + ( y − x ) T ∇ f ( x ) 2) 2 nd order characterization of convex function is: ∇ 2 f ( x ) ≥ 0 i.e. Hessian Matrix is positive definite. 3) Convex set contains line segment between any two points in the set i.e. 10 x 1 , x 2 ∈ C and 0 ≤ θ ≤ 1 then θx 1 + (1 − θ ) x 2 ∈ C 4) If f is convex then − f is concave and vice versa. 5) Some examples of convex function are e x , − log x, x log x etc. 6) Intersection of two convex sets is also convex. These are some properties of convex function and based on this function, we can take convex surface and similarly concave surface and it is possible to create a three-dimensional object for which one side is convex and one side is concave as mentioned in the beginning. Note: ( 1 ) A 3 − dimensional object can both convex and concave. Visualization: f ( x, y ) = x + 2 y (Check here) ( 2 ) A 3 − dimensional object can neither be convex nor concave. For example: Hyperbolic Paraboloid because the graph contains the saddle points everywhere. This is quite famous function and some facts about it are written here. Visualization of Hyperbolic Paraboloid: Check Here 11 Figure 6: Convex Function 5 Final Remark As we have already seen many evidences that for a smooth three-dimensional object, edge(s) can be curved and if we take an example of ”Cylinder” then it is already proved that it has all smooth surfaces and since it contains 2 curved edges, 1 curved surface, 2 faces and no corners and Hence, option ( D ) is also correct along with option ( C ). So, both options (C) and (D) are logically correct. Hence, I am kindly requesting you, to please change answer key from ( C ) to ( C ) , ( D ) because both the are the possible answers for this ques- tion and they are also logically correct based on the above evidences. 12 6 References 1. Purdue university Computer Science Course 2. GRAPH THEORY WITH APPLICATIONS by J. A. Bondy and U. S. R. Murty, Departmnent of Combinatorics and Optimization, University of Waterloo, Ontario, Canada. 3. Geometry For Dummies (2nd Edition) by Mark Ryan 4. Plane and Solid Geometry,Springer-Verlag New York Inc 5. https://physics.stackexchange.com/questions/217757/difference-between- convexo-concave-and-concavo-convex-lenses 13