15.2. GEOMETRY OF SPACE CURVES∗ 279

and soN =

1√5(−1,0,2) .

This was pretty easy.

Example 15.1.6 Find a formula for the curvature of the curve given by the graph of y =f (x) for x ∈ [a,b]. Assume whatever you like about smoothness of f .

You need to write this as a parametric curve. This is most easily accomplished by lettingt = x. Thus a parametrization is (t, f (t) ,0) : t ∈ [a,b] . Then you can use the formula givenabove. The acceleration is (0, f ′′ (t) ,0) and the velocity is (1, f ′ (t) ,0). Therefore,

a×v =(0, f ′′ (t) ,0

)×(1, f ′ (t) ,0

)=(0,0,− f ′′ (t)

).

Therefore, the curvature is given by

|a×v||v|3

=| f ′′ (t)|(

1+ f ′ (t)2)3/2 .

Sometimes curves do not come to you parametrically. This is unfortunate when itoccurs but you can sometimes find a parametric description of such curves. It should beemphasized that it is only sometimes when you can actually find a parametrization. Generalsystems of nonlinear equations cannot be solved using algebra.

Example 15.1.7 Find a parametrization for the intersection of the surfaces

y+3z = 2x2 +4 and y+2z = x+1.

You need to solve for x and y in terms of x. This yields

z = 2x2− x+3, y =−4x2 +3x−5.

Therefore, letting t = x, the parametrization is

(x,y,z) =(t,−4t2−5+3t,−t +3+2t2) .

Example 15.1.8 Find a parametrization for the straight line joining (3,2,4) and (1,10,5).

(x,y,z) = (3,2,4)+ t (−2,8,1) = (3−2t,2+8t,4+ t) where t ∈ [0,1]. Note where thiscame from. The vector (−2,8,1) is obtained from (1,10,5)− (3,2,4). Now you shouldcheck to see this works.

15.2 Geometry Of Space Curves∗

If you are interested in more on space curves, you should read this section. Otherwise,proceed to the exercises. Denote by R(s) the function which takes s to a point on this curvewhere s is arc length. Thus R(s) equals the point on the curve which occurs when you havetraveled a distance of s along the curve from one end. This is known as the parametrizationof the curve in terms of arc length. Note also that it incorporates an orientation on the curvebecause there are exactly two ends you could begin measuring length from. In this section,assume anything about smoothness and continuity to make the following manipulationsvalid. In particular, assume that R′ exists and is continuous.