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WORLD OF CURVES VS FLAT WORLD

     


      A convex surface (facing us) has positive curvature and a concave surface has a negative curvature.  A flat surface has zero curvature.  For example, a banana peel has positive curvature outside and negative curvature inside.
     Take a paper and roll it as cylinder.  Length wise, you can draw a straight line.  Hence, lengthwise its curvature is zero.  But, across you can only draw circle or curve.  In this direction it has positive curvature.  If you multiply '+' and '0', you get 0.  Hence the curvature of cylinder at any point is 'zero'.  This is called "Gaussian curvature" of cylinder. Because of 'zero' curvature, you can roll a flat paper into cylinder and back.

      Next take a sphere, all sides, it is curved convexly.  Hence at any point, it has positive  Gaussian curvature.  Hence spherical surface cannot be converted into flat one.  Take an orange.  Peel its skin.  Put on the floor.  stomp on it.  It will not be made into smooth circle.  It will only tear and break.

      Similarly, our globe surface cannot made into smooth flat map.  It will be somewhat distorted.  The areas of countries cannot be brought into uniform scale easily.
     You cannot wrap a foot ball in a gift paper smoothly.  The wrinkles will always be there.
     When flat clothes are made into dress.  It has to cover curved body gracefully.  That is the challenge of fashion designer.

      The geometry, we studied is only applicable to flat surface and called Euclidean geometry.  Convex and concave surfaces have different geometries.  For example, a triangle on a sphere has a total of more than 180 degrees.

      How to measure the curvature?  Take the curve to be measured.  Draw a closely touching and smoothly fitting circle to it.  Measure its radius.  The reciprocal of radius gives the curvature. For a straight line, the radius is infinity and the curvature is zero.  In physics, the amount of electric charges present on a metal surface is proportional to the curvature.

      We live in a spherical world.  But we can easily understand the flat world only.    
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