Arc length in three dimensions is a very natural extension of Arc length in two dimensions .

As in two dimensions, the speed of a parametrized curve at a point t is defined to be the length of the velocity vector at the point, which is the square root of the dot product of the velocity vector with itself.

[!--@char sqrt--][X^'(t)·X ^'(t)]

As before, we designate the speed by s^'(t) or dsdt , indicating the rate of change of distance along the curve with respect to time. We can then integrate the speed from t=a to t=b to get the total length of the curve.

s(b)-s(a) =Integral[a,b][!--@char sqrt--][X ^'(t) ·X^'( t)dt]

More generally, the arclength of a curve between two values of the parameter t₀ and t₁ is given by

s(t)-s(t ₀)=Integral[t,t₀][!--@char sqrt--][X^'(u)·X^' (u)du]

Note that the value of the integral can be approximated by using a summation, and this is closely related to the geometric notion of approximating the length of a curve by using inscribed polygons.

Demonstration 4: Speed and arclength in Three-space

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In this demo we can enter a curve in the control window and see three graphs: the graph of the curve, the graph of s^'(t) - the velocity curve, and the graph of s(t) - the distance along the curve versus the variable t . If the tapedeck is moving forward, a point will move along the velocity curve, leaving vertical lines as it goes. This illustrates the notion that the distance along the curve is equal to the area under between the function graph of the speed and the t-axis.

What can we say about the speed and arclength of a circular helix? (You can calculate s(t) for yourself and compare your results with the graph in the demo.)At which points would a car on the curve be moving fastest?

What can we say about the speed and arclength of an exponential spiral on a cone? At which points would a car on the curve be moving fastest?

Same question for an ellipse, or for a space cardiod. What about the twisted cubic?

What happens on a curve if the distance function has a horizontal tangent? Find an example of such a curve.