What is minimum homotopy area between two curves? consider we have two x-monotone curves that have the same endpoints and we want to compute the min homotopy area between two curves. How can I compute the minimum homotopy area between two curves?
I saw a lot of papers and links but there were confusing.
I want to know what is the minimum homotopy area between two green and blue curves (ignore the purple curve)
 A: For the above two curves, The criterion of homotopy area is equal to the sum of the integral of the faces and for regular geometric shapes is equal to the sum of the area faces.
the minimum homotopy area $HA(µ, τ ) = ∑_{i=1}^k \left|W(δ_i)\right| \,dp$.
Since the blue and red paths are uniform, the number of coils at each point on a face is one or negative.
So in the example above, the homotopy area between two paths is equal to the sum of the area between the faces created by the two green and blue paths.
For You can find the answer in this link:
https://research.tue.nl/nl/publications/homotopy-measures-for-representative-trajectories
Given a point p let $ω(p, δ)$ denote the winding number of p with respect to an oriented closed curve δ. We say that δ is atomic if $ω(p, δ)$ is either all non-negative, or all non-positive, for all points $p ∈ R^2$. Furthermore, let $W(δ) = \int_{p ∈ R^2} ω(p.δ) \,dx$ denote the total winding number of curve δ. Let µ and τ be two curves from s to t, let $δ = loop(µ, τ )$ denote the closed curve obtained by concatenating µ and the reverse of τ, and let $s = p_1,.., p_L= t $ denote the intersection points between µ and τ, ordered along µ. there is a (not necessarily contiguous) subsequence of the intersection points {$p_i$} that decompose δ into a set of atomic closed curves $∆(µ, τ ) = δ_1,.., δ_k$, such that the minimum homotopy area $HA(µ, τ ) = ∑_{i=1}^k \left|W(δ_i)\right| \,dp$.
A vertical line L_x with x-coordinate x intersects (the faces of) G (above image) in a set of intervals $I (x) = I_1,.., I_n$. So, for any curve δ that uses only edges of μ, τ, all points (values) in an interval $I_i$ have the same winding number $ω(I_i, δ)$. Since the trajectories are x-monotone, so is µ. The curves $δ ∈ ∆(µ, τi)$ are built by concatenating a piece of µ and a reversed piece of τi. Hence, any vertical line $L_x$ intersects δ in exactly two points: µ(x) and τi(x). Therefore, any point p on L_xthat lies in between these points has winding number one or minus one with respect to δ. Any point outside the interval defined by µ(x) and τi(x) has winding number zero, Thus
$HA(µ, τ )$ is minimum homotopy area between μ, τ.
$$HA(µ, τ ) = \int_{x∈R} \left|μ(x)-τ(x)\right| \,dx$$
