How to take dot product of every element in a list?

Question

I have a list of coordinates (coord):

{{59.1915, 42.0843}, {62.0695, 71.2996}, {48.4124, 52.1423}, 
 {51.3325,60.8624}, {54.8544, 44.72}, {55.39, 63.9032},
 {59.1566,67.2319}, {48.1271, 57.0733}, {63.0908, 76.0996}, 
 {50.644, 47.5917}}

Using:

diffcoord = Differences[coord]

I can calculate the difference between consecutive coordinates:

{{2.87796, 29.2153}, {-13.6571, -19.1573}, {2.9201, 8.7201}, 
 {3.52192, -16.1424}, {0.53561, 19.1832}, {3.76663, 3.32865}, 
 {-11.0295, -10.1585}, {14.9637, 19.0262}, {-12.4468, -28.5079}}

Now, I want to calculate:

d = dx1*dx2 + dy1*dy2

for each pair of differenced coordinate in the list. where $dx_1 = x_2-x_1, dy_1=y_2-y_1$ (which is the first element of the diffcoord list)

I was trying to use Dotbut I found out that it's not going to do I want. Is there a command like Differences or Accumulate that does what I need.

More info:

I am trying to calculate the correlation between unit tangent vectors (uni-leipzig.de/~pwm/web/?section=introduction&page=polymers). To calculate unit tangent vector, I need to take the derivative of the curve at that point. Since it's cubic spline, I can't do so. I am trying to approximate it by (iiserpune.ac.in/~cathale/lects/bio322-phybio1/2013bio322/…) . Do you know of any better way to do it?

This question is a follow-up to this question: How to find a length of a curve constructed using Spline?

Answer 1

Here's how I'd do it. Starting with diffcoord

diffcoord = {{2.87796, 29.2153}, {-13.6571, -19.1573}, {2.9201, 
  8.7201}, {3.52192, -16.1424}, {0.53561, 19.1832}, {3.76663, 
  3.32865}, {-11.0295, -10.1585}, {14.9637, 
  19.0262}, {-12.4468, -28.5079}};

Pair each element with the next and Dot the pairs.

Dot@@@(Transpose@{diffcoord[[1;;-2]],diffcoord[[2;;-1]]})

The output is

{-598.991, -206.934, -130.479, -307.777, 65.8716, -75.3581, -358.32,
-728.647}

There is certainly a more clever way to combine the two if you need to do this on huge lists, but this makes sense to me.

Edit: @Guess who it is reminded me that Partition has a 3rd argument to specify an offset, which makes this a lot nicer looking!

Dot@@@Partition[diffcoord,2,1]

Answer 2

lst = {{59.1915, 42.0843}, {62.0695, 71.2996}, {48.4124, 
    52.1423}, {51.3325, 60.8624}, {54.8544, 44.72}, {55.39, 
    63.9032}, {59.1566, 67.2319}, {48.1271, 57.0733}, {63.0908, 
    76.0996}, {50.644, 47.5917}};

BlockMap[Apply[Dot], Differences@lst, 2, 1]

{-598.991, -206.934, -130.479, -307.777, 65.8725, -75.3586, -358.323,
-728.65}

As of v10.2 PartitionMap has been superseded by BlockMap.

Answer 3

You can use PartitionMap from Developer package, e.g.

lst = {{59.1915, 42.0843}, {62.0695, 71.2996}, {48.4124, 
    52.1423}, {51.3325, 60.8624}, {54.8544, 44.72}, {55.39, 
    63.9032}, {59.1566, 67.2319}, {48.1271, 57.0733}, {63.0908, 
    76.0996}, {50.644, 47.5917}};
Developer`PartitionMap[Dot @@ # &, Differences@lst, 2, 1]

yields:

{-598.991, -206.934, -130.479, -307.777, 65.8725, -75.3586, -358.323, \
-728.65}

A recent answer by Mr. Wizard brought this to my attention. I post in case others not aware.

Answer 4

Your wording is very ambiguous. You start with coordinates, then calculate the differences between successive points, and then claim you want to calculate dx1*dx2 + dxy*dy2 for each pair of coordinate [sic] in the list. Do you mean for each successive pair of two-element lists? Or the outer product, i.e., the differences between the each two-element list and every other two element list?

Please be clear.

You might mean you want the below (but I'm not sure):

Norm[#]^2 & /@ 
 Differences[{{2.87796, 29.2153}, {-13.6571, -19.1573}, {2.9201, 
    8.7201}, {3.52192, -16.1424}, {0.53561, 19.1832}, {3.76663, 
    3.32865}, {-11.0295, -10.1585}, {14.9637, 
    19.0262}, {-12.4468, -28.5079}}]