Curvature and its types

this ppt is detailed with curvature and its types , normal,geodesic,Gaussian,principal,mean curvatures.

1. COMPLEX ANALYSIS PRESENTED TO : MA’AM HUNZA PRESENTED BY : AMENAH GONDAL CLASS: BS.ED (VI)

2. NORMAL , PRINCIPAL , MEAN , GUASSIAN , GEODESIC CURVATURES

3. CURVATURE: In  differential geometry ,  curvature  is the rate of change of direction of a curve at a point on that curve, or the rate of change of inclination of the tangent to a certain curve relative to the length of arc.

4. NORMAL CURVATURE: “The   normal curvature  at a point is the amount of the curve's  curvature  in the direction of the surface  normal . The curve on the surface passes through a point , with tangent,   curvature  and  normal .”

5. EXAMPLES:

6. FORMULA: Kn is the function of the surface parameters L, M, N, E, F, G and of the direction du/ dt.

7. EXAMPLE:

8. PRINCIPAL CURVATURE: “The maximum and minimum of the normal  curvature k1  and k2 at a given point on a surface are called the  principal curvatures . The  principal curvatures  measure the maximum and minimum bending of a regular surface at each point.” FORMULA: A number k is a principal curvature iff k is a solution of the equation

9. MEAN CURVATURE: (1) Let  K1  and K2 be the  principal curvatures , then their  mean is called the mean curvature. Let   and   be the radii corresponding to the  principal curvatures , then the  multiplicative inverse  of the mean curvature   is given by the  multiplicative inverse  of the  harmonic mean , =

10. GUASSIAN CURVATURE: “The Gaussian curvature or Gauss curvature Κ of a surface at a point is the product of the principal curvatures, κ₁ and κ₂, at the given point.” For example, a sphere of radius r has Gaussian curvature everywhere, and a flat plane and a cylinder have Gaussian curvature zero everywhere.

11. FORMULA:

12. GEODESIC CURVATURE: GEODESIC: In differential geometry, a geodesic is a curve representing in some sense the shortest path between two points in a surface. GEODESCI CURVATURE: The geodesic curvature of a curve measures how far the curve is from being a geodesic.

13. The curvature vector at P of the projection of the curve C onto the tangent at P is called the Geodesic Curvature Vector of C at P is denoted by Kg.

14. FORMULA:     k g  = k g U   The scalar k g  is called the  geodesic curvature  of C at P.