[chronojump] Added the calculation of sprint using only 2 positions

[Xavier Padullés <xpadulles src gnome org>]

commit 6b18d3e2aace593ea4703a8fe024840a88985ffd
Author: Xavier Padullés <x padulles gmail com>
Date:   Fri Mar 31 11:52:32 2017 +0200

    Added the calculation of sprint using only 2 positions

 r-scripts/sprintUtil.R |   25 ++++++++++++++++++++++++-
 1 files changed, 24 insertions(+), 1 deletions(-)
---
diff --git a/r-scripts/sprintUtil.R b/r-scripts/sprintUtil.R
index 531b2ce..59bd73d 100644
--- a/r-scripts/sprintUtil.R
+++ b/r-scripts/sprintUtil.R
@@ -80,8 +80,8 @@ splitTime <- function(Vmax, K, position, tolerance = 0.001, initTime = 1)
 {
         #Trying to find the solution of Position(time) = f(time)
         #We have to find the time where y = 0.
-        y = Vmax*(initTime + (1/K)*exp(-K*initTime)) -Vmax/K - position
         t = initTime
+        y = Vmax*(t + (1/K)*exp(-K*t)) -Vmax/K - position
         while (abs(y) > tolerance){
                 v = Vmax*(1 - exp(-K*t))
                 t = t - y / v
@@ -90,6 +90,29 @@ splitTime <- function(Vmax, K, position, tolerance = 0.001, initTime = 1)
         return(t)
 }
 
+#Given x(t) = Vmax*(t + (1/K)*exp(-K*t)) -Vmax - 1/K
+# x1 = x(t1)    eq. (1)
+# x2 = x(t2)    eq. (2)
+#Isolating Vmax from the first expressiona and sustituting in the second one we have:
+# x2*(t1 + exp(-K*t1)/K - 1/K) = x1*(t2 + exp(-K*t2)/K -1/K)    eq. (3)
+#Passing all the terms of (3) at the left of the equation to have the form y(K) = 0
+#Using the iterative Newton's method of the tangent aproximation to find K
+#Derivative: y'(K) =  (x2/x1)*(t1 - exp(-K*t1)*(K*t1 + 1)/K^2 + 1/K^2) + exp(-K*t2)*(K*t2 +1) - 1/K^2
+getSprintFrom2SplitTimes <- function(x1, x2, t1, t2, tolerance = 0.0001, initK = 1)
+{
+        #We have to find the K where y = 0.
+        K = initK
+        y = (x2/x1)*( t1 + exp(-K*t1)/K - 1/K) - t2 - exp(-K*t2)/K + 1/K
+        while (abs(y) > tolerance){
+                derivY = (x2/x1)*(t1 - exp(-K*t1)*(K*t1 + 1)/K^2 + 1/K^2) + exp(-K*t2)*(K*t2 +1) - 1/K^2
+                K = K - y / derivY
+                y = (x2/x1)*( t1 + exp(-K*t1)/K - 1/K) - t2 - exp(-K*t2)/K + 1/K
+        }
+        #Calculing Vmax substituting the K found in the eq. (1)
+        Vmax = x1/(t1 + exp(-K*t1)/K -1/K)
+        return(list(K = K, Vmax = Vmax))
+}
+
 
 prepareGraph <- function(os, pngFile, width, height)
 {