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. |