Curvature and Torsion

# Curvature and Torsion - ME 210 Applied Mathematics for...

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ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010 Curvature and Torsion of a Curve, Normal, Binormal, TNB Frame The velocity, , of a point P moving along a space curve C is always tangent to this curve traced by the tip of the position vector, , therefore is in the direction of the unit tangent vector defined as ) t ( v ) t ( r ds ) s ( r d ) s ( r ) s ( T As our representative point P moves along a differentiable curve, the unit tangent vector, , changes its direction as the curve bends since it must remain tangent to the curve at P. At any point P on C, the rate of change in by the parameter s is called the curvature, k(s) (a scalar quantity) of the curve C at its point P, expressed as ) s ( T ) s ( T ds ) s ( T d ) s (

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ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010 On the other hand, if one considers the unit tangent vectors at two points P 1 and P 2 which are infinitesimally close (i.e., ds apart) to each other along the curve C as shown in the Figure, the difference between the unit tangent vector at P 2 and the unit tangent vector at P 1 has to be perpendicular to and in the direction at which rotates (or towards the curve C bends). ) s ( T d ) ds s ( T ) s ( T ) s ( T ) s ( T ds ) s ( T d ds ) s ( T d ) s ( N is called the unit normal vector (or the principal normal ) of the curve C at its point P. The unit vector in the direction of /ds defined as ) s ( N ) s ( T d
ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010 P 1 ds P 2 x y 0 z T(s+ds)=T(s)+dT(s) v+dv Curve C T(s) v N(s) dT(s)/ds Circle of curvature with a radius tangent to C at P 1 Change in Unit Tangent Vector and Curvature

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ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010 Using this definition, one can also write ) s ( N ) s ( ds ) s ( T d ) s ( N ) s ( 1 ds ) s ( T d or where ρ(s) is called as the radius of curvature , which can be considered as the radius of a circle that is tangent to the curve C at P whose curvature is identical to the curvature of the curve C at P. This circle is referred to as the circle of curvature .
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Curvature and Torsion - ME 210 Applied Mathematics for...

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