Similar to parametric surfaces, paths are typically defined by using a parametric function mapping a single input, t to any number of dimensions through a vector valued function. However, instead of visualizing t as time, we instead visualize it by tracing out the entire path along the domain of t


One question we might have about a path is how long it is. We can use integrals to work out how long a path is by breaking the problem into smaller sub-problems.

Say for instance we take a very small section of that path like dt. You could imagine approximating that path using a vector, or a series of vectors in their place.

Because we can reduce the problem into a sum of vector lengths over a defined interval, we can also express the problem as an integral! To do this we integrate over the length of each vector for dt. We determine what each vector is by taking the gradient vector of the path at each point t. Then measure the length.


So for instance, on our function above, we would have:


Now considering that a negative squared is a positive, we can rewrite as follows:


And remembering that sin2(θ)+cos2(θ)=1 we can simplify: