## DefinitionA space curve is said to be a plane curve, if it is completly contained in a plane. By choosing the coordinate system with i and j on the plane of the curve we can always simplify its parametrization to the form: |

> alias(v=vector):i:=v([1,0,0]):j:=v([0,1,0]):k:=v([0,0,1]):

> len := proc(u) sqrt(innerprod(evalm(u),evalm(u))); end:

The arc length from x=a to x=b is, |

> L := Int( len( diff( x*i + f(x)*j, x) ), x=a..b );

b / | / / d \2\1/2 L := | |1 + |---- f(x)| | dx | \ \ dx / / / a

In order to deduce the formula for the curvature at x, we need first the unit tangent vector at x, |

> Dr := diff(x*i + f(x)*j, x); T := Dr / len( Dr );

/ d \ Dr := i + |---- f(x)| j \ dx / / d \ i + |---- f(x)| j \ dx / T := --------------------- / / d \2\1/2 |1 + |---- f(x)| | \ \ dx / /

The formula for the curvature is then, |

> K := len( diff(T,x) ) / len( Dr );

/ / 2 \2 \1/2 | | d | | | |----- f(x)| | | | 2 | | | \ dx / | |-------------------| |/ / d \2\2| ||1 + |---- f(x)| | | \\ \ dx / / / K := ------------------------ / / d \2\1/2 |1 + |---- f(x)| | \ \ dx / /

To help maple simplify the denominators we do, |

> K := sqrt( simplify(K^2) );

/ / 2 \2 \1/2 | | d | | | |----- f(x)| | | | 2 | | | \ dx / | K := |-------------------| |/ / d \2\3| ||1 + |---- f(x)| | | \\ \ dx / / /

Another way to write the same thing is For example the curvature of the parabola y = x^2 at the points (0,0) and (1,1) can be easily computed with: |

> f := x -> x^2;

2 f := x -> x> k := unapply(K,x);

1/2 / 1 \1/2 k := x -> 4 |-----------| | 2 3| \(1 + 4 x ) /> k0 := simplify(k(0)); k1 := simplify(k(1));

k0 := 2 1/2 k1 := 2/25 5

the curvature at x=y=1 is then about, |

> k1f := evalf(k1,2);

k1f := .18

Notice that, |

> Limit(k(x),x=infinity) = limit(k(x),x=infinity);

1/2 / 1 \1/2 Limit 4 |-----------| = 0 x -> infinity | 2 3| \(1 + 4 x ) /

This agrees with the fact that the parabola gets straighter and straighter (curvature = 0) as x gets larger. |

Link to the commands in this file

Carlos Rodriguez <carlos@math.albany.edu> Last modified: Wed Oct 23 14:05:35 EDT 1996