# Produce one smooth curve on one triangle mesh

I hope to get one smooth curve on one triangle mesh. I get one path on the mesh at first. The path consists of vertices of the mesh. I can see the path from the image below.

Each one green dot indicates one vertex on the mesh. The method that I am using is to fit one smooth curve using the vertices and project the smooth curve on the mesh. To project the smooth curve, I find the closest point on the mesh for each vertex of the smooth curve.

However the curve is not on the mesh because it intersects with the mesh. I realize that the vertices need to move along the edge of the triangle shared with the previous vertex.

In my view, there are two kinds of methods to produce the smooth curve. The first kind is to move the points of the curve along the mesh to make the curve smooth. The second kind is to fit the points to obtain one smooth curve and project the smooth curve on the mesh,but the projection should be complex.

Can you give me some advices or links to websites or papers? Thanks in advance.

• Can you clarify what you mean by "smooth". Mathematically, a smooth curve would has at least some chosen number of continuous derivatives. The simplest way to enforce this would be to just use a spline of appropriate order through your original points. Also, by "on the mesh" do you mean that the points lie on the vertices or edges of the mesh? Of course if you restrict the entire curve to lie on the (piecewise linear) triangulation it must be piecewise linear and therefore not smooth. Commented Oct 15, 2014 at 13:08
• It seems like what you actually want is to project a curve (which you define as your smooth curve) onto the mesh in some nice way. What "nice" means may be up for interpretation. Commented Oct 15, 2014 at 13:10
• Considering that your geometry is made with linear interpolators (triangles), I would say that you should accept piecewise-linear as smooth enough. Commented Oct 15, 2014 at 17:22
• I realize that the points of the smooth curve need to be located on the edges of the mesh after I got the result of the second figure. Then I search over the internet and find the following paper: wwwisg.cs.uni-magdeburg.de/visual/files/publications/2014/…. Currently I am reading the paper. Commented Oct 16, 2014 at 1:17
• @DougLipinski By reading the paper, curve fitting should be done on manifold not Euclidean space. Commented Oct 16, 2014 at 2:41