A detailed example was worked out [Transparancies 2 and 3], showing
how to find the speed, tangential acceleration, normal acceleration, and
curvature as a function of "t", for a given space curve. The curvature was
found to be infinite at t=0, and to decrease as 1/5t when "t" increases.
The curve was plotted in Maple using the following commands:
x:= sin(t) - t*cos(t); y:= cos(t) + t*sin(t); z:= t^2; with(plots): spacecurve([x,y,z],t=0..8*Pi);The picture at right confirms the decreasing curvature as the spiral travels up the z-axis.
The binormal vector was defined [Transparancy 4] as the cross product of T x N . It was shown that the cross product of velocity vector with the acceleration vector yields an expression involving the binormal vector. This can be rearranged to obtain an alternative expression for curvature.
The three Frenet Formulas were given, and were justified by geometrical arguments. The unit tangent, principal normal, and binormal vectors form a right handed triplet which rotates as one moves along a spacecurve. For planar motion the binormal vector remains perpendicular to the plane, and doesn't tilt unless the spacecurve leaves the plane. When this happens, the spacecurve is said to have "torsion".
A more rigorous proof of the Frenet Formulas is also given [Transparancies 5, 6, and 7].
The lecture finished off with an introduction to planar motion in Polar Coordinates. An orthonormal coordinate system was defined by introducing a radial unit vector, and an angular unit vector [Transparancy 7]. This is a "moving coordinate system", which rotates as one moves along the spacecurve. In taking derivatives, one must also consider the derivatives of the coordinate system, ie. the derivatives of the radial and angular unit vectors [Transparancy 8]. Using the chain rule leads to the velocity vector being written in terms of radial velocity and angular velocity.
