# Estimating cline width at multiple points along a trait gradient

In the map below I have 66 (green) points along the midpoint (0.5) for an allele frequency cline across the region (see map). I created the points by placing a point every 25 km along the 0.5 allele frequency contour. The thick red lines represent the 0.22 and 0.78 allele frequency contours. I would like to estimate the width of the cline at each of the 66 points. The "estimated" width should be based on the shortest distance between the two contours when the line runs through the point. This would allow me to compare the cline width at 25 km increments.

The result would look something like the map below. Note that I hand-drew only of 3 of the 66 desired lines and my lines are probably not the shortest distance between the two contours for each of the 3 points. Each of the line must be a straight and unbroken (i.e. no bend at the point). I would also get the total distance of the line, plus the distance for each segment (0.22 to 0.5 and 0.5 to 0.78).

This is a map created using the "Near" function with my points as the "input feature" and contours as the "near feature". I created the map by running the near function for each contour separately. As you can see the map is not quite right. Any other suggestions?

• It is hard to understand what is needed here. First, a "point" cannot be "bisected." Second, what is a "trait contour"? (It's probably not the coastal outline or the country boundaries, but is it one of the solid thick gray lines? Or perhaps the sequence of dots?) At the end, to what do you refer where you write "each point"? These ambiguities allow so many different interpretations it's anybody's guess what you want to do. Could you please clarify this? (A simple diagram of what the solution would look like might help immensely.) May 4, 2012 at 14:33
• Well done--that is a very helpful clarification. May I follow up, though, by asking why you seek the shortest distance along line (segments) through the points? Conceptually, these three curves represent a continuous surface. In many applications it would be natural to measure widths in terms of distances along curves orthogonal to the clines: that is, in directions that are always parallel to the gradient of the surface. Are you using line segments to approximate such curves? Or is it truly important that widths be measured along straight lines only? May 4, 2012 at 15:16
• Another point of clarification: when a "line runs through a point" must it be a straight unbroken line segment, or do you mean "line" in the more general sense of a polyline that could potentially bend at the point? May 4, 2012 at 15:19
• The allele cline is a continuous surface and you are absolutely correct that measuring the widths between curves would be ideal. I had searched the literature and could not find a way to do just what you described, so I thought my alternative method would be the best way to demonstrate the heterogeneity of the cline surface. In hybrid zone theory it is traditional to calculate the wdith of the cline as the distance between to 0.22 and 0.78 frequencies. Yes, I mean a straight unbroken line (i.e. no bend at the point). May 4, 2012 at 15:28
• Do you perhaps have the data in a fully gridded form (from which these two clines were extracted)? May 4, 2012 at 15:30