CHAPTER 3 DIFFERENTIATION VECTORS OF cos d dTˆ dt aw2 sin And  wtˆ ds dt a w  b  2  ds aw2 dTˆ *  So that a w b  ds K 2 (*) b=0, z=0, Two limiting cases of are worth noting. First if then and the helix a xy-plane, (*) reduces to a circle of radius in the while equation reduces to 1a .  K a=0, x=y=0 z=bt. Secondly, if then and This tells us that the point moves along the z-axis. In this case the curvature k=0 i.e. a straight line (see following figure) Z y X Definition: We defined a vector perpendicular to both and by the formula Bˆ Tˆ Nˆ   Bˆ Tˆ Nˆ This vector is called the binormal vector of the curve. The three vectors and (in this order) define a moving right-handed coordinate Tˆ Nˆ Bˆ system at any specified point of C. Definition: A plane to a curve at a point P containing the tangent and principal P normal at is called an osculating plane. P Definition: The normal plane is the plane through perpendicular to the tangent. Definition: The rectifying plane is the plane through P which is perpendicular to the principal normal. ds the torsion Definition: The arc of rotation of the binormal vector (i.e. ) d Bˆ vector of the curve and its magnitude is denoted by .  is called  1
CHAPTER 3 DIFFERENTIATION OF VECTORS And d ds  dTˆ dt ds dt aw2 a w  b 2 cos wtˆ  sin  So that K  dTˆ ds aw2 a w  b 2 * Two limiting cases of (*) are worth noting. First if b=0, then z=0, and the helix reduces to a circle of radius a in the xy-plane, while equation (*) reduces to K  1a . Secondly, if a=0, then x=y=0 and z=bt. This tells us that the point moves along the z-axis. In this case the curvature k=0 i.e. a straight line (see following figure) Z y X Definition: We defined a vector Bˆ perpendicular to both Tˆ and Nˆ by the formula Bˆ  Tˆ  Nˆ This vector is called the binormal vector of the curve. The three vectors Tˆ Nˆ and Bˆ (in this order) define a moving right-handed coordinate system at any specified point of C. Definition: A plane to a curve at a point P containing the tangent and principal normal at P is called an osculating plane. Definition: The normal plane is the plane through P perpendicular to the tangent. Definition: The rectifying plane is the plane through P which is perpendicular to the principal normal. Definition: The arc of rotation of the binormal vector (i.e. d Bˆ ds ) the torsion vector of the curve and its magnitude is denoted by .   1 is called