I have 6 curve types

![enter image description here][1]

that follow these rules:
<ul><li> 
Type A: If point 1 is positive and point 2 is negative, the curve will not pass through the origin.</li><li>
Type B: If point 1 is negative and point 2 is positive, the curve will pass through the origin twice.</li><li>
Types C-F: If both points are positive or both points are negative, the curve will pass through the origin once.
</li></ul> 

The curves are basically partial (and skewed to some degree) [lima&ccedil;ons][2]:

![enter image description here][3]

What **is** known:

<ul><li> 
The coordinates of points 1 and 2.</li><li>
The slopes at points 1 and 2.</li><li>
The approximate curve type (lima&ccedil;on).
</li></ul>

What **is not** known:

<ul><li> 
The arc length.</li><li>
The degree of skew.
</li></ul>

Data for A-F:

 

    (* A *) {{0.000564367, 0.690525}, {-0.000689501, -0.984192}, -0.111402, 2.45932}
    (* B *) {{-0.000689501, -0.984192}, {0.000664785, 1.07289}, 2.45932, -3.86161}
    (* C *) {{0.000179304, 1.61576}, {0.0000936314, 0.852042}, 2.23535, 0.406903}
    (* D *) {{0.000116063, 0.431337}, {0.000443491, 1.70111}, -0.257071, 6.50847}
    (* E *) {{0.0000347276, 0.190688}, {0.000190634, 1.06651}, -0.730172, 1.75825}
    (* F *) {{-0.000432719, -1.90935}, {-0.000142565, -0.645011}, -4.83652, 2.68761}

in format: `{{point 1}, {point 2}, slope @ point 1, slope at point 2}`

Is it possible to estimate a curve fit (and hence arc length & skew) with only the data given?


  [1]: http://i.stack.imgur.com/4LV2K.png
  [2]: http://en.wikipedia.org/wiki/Lima%C3%A7on
  [3]: http://i.stack.imgur.com/cObuQ.png