Changes between Version 19 and Version 20 of u/erica/scratch

Timestamp:

02/08/16 12:25:47 (9 years ago)

Author:

Erica Kaminski

Comment:

—

u/erica/scratch

@@ -9 +9 @@
 So given these 2 constraints, we have the following equation,
 
-[[latex($E_{acc} = \sum_{i=0}^{4} k(~e^{-dx*i} - e^{-4dx} )~E_{acc}  $)]]
+[[latex($E_{acc} = \sum_{i=0}^{N} k(~e^{-dx*i} - e^{-Ndx} )~E_{acc}  $)]]
 
 which gives the normalization constant,
 
-[[latex($ \boxed{k = \frac{1}{~\Sigma ~e^{-dx*i}-e^{-4dx}}} $)]]
+[[latex($ \boxed{k = \frac{1}{~\Sigma ~e^{-dx*i}-e^{-Ndx}}} $)]]
 
-Note that the sum runs over cells i=0 to 1, where the '0th' cell is the cell the sink is in, and the '4th' cell is the furthest cell from the sink in the kernel, that the exponential goes to zero at this boundary, and dx*i gives a position (which here assumes the sink is at the cell center -- the actual function is described below).
+where N is the max number of cells in the kernel. Note that the sum runs over cells i=0 to 1, where the '0th' cell is the cell the sink is in, and the '4th' cell is the furthest cell from the sink in the kernel, that the exponential goes to zero at this boundary, and dx*i gives a position (which here assumes the sink is at the cell center -- the actual function is described below).
 
+
+Now, 2 things effect the shape of this smoothing function, 1. the number of cells in the kernel, and 2. dx.
+
+Here is a plot showing, for dx=.1, a kernel of 4 cells (top), and a kernel of 40 cells (bottom):
+
+[[Image()]]
+
+[[Image()]]