/// <summary>
/// Remove points and add intersection points in case of backward order
/// </summary>
/// <param name="X">x values of curve</param>
/// <param name="Y">y values of curve</param>
/// <param name="lower">if true, algorithm for lower tube curve is used;<para>
/// if false, algorithm for upper tube curve is used</para></param>
/// <return>Number of loops, that are removed.</return>
private static int removeLoop(List<double> X, List<double> Y, bool lower)
{
// Visualization of working of removeLoop
bool visualize = false;
List<double> XLoops = new List<double>(X);
List<double> YLoops = new List<double>(Y);
int j = 1;
int countLoops = 0;
#if GUI
// Visualization
if (visualize)
{
ChartControl chartControl = new ChartControl(600, Int32.MaxValue, true);
chartControl.AddLine("Tube curve with loop", X, Y, Color.FromKnownColor(KnownColor.Cyan));
Application.Run(chartControl);
}
#endif
while (j < X.Count - 2)
{
// Find backward segment (j, j + 1)
if (X[j + 1] < X[j])
{
countLoops++;
// ----------------------------------------------------------------------------------------------------
// 1. Find i,k, such that i <= j < j + 1 <= k - 1 and segment (i - 1, i) intersect segment (k - 1, k)
// ----------------------------------------------------------------------------------------------------
int i, k, iPrevious;
double y;
// for calculation and adding of intersection point
bool addPoint = true;
double ix = 0;
double iy = 0;
i = j;
iPrevious = i;
// Find initial value for i = i_s, such that X[i_s - 1] <= X[j + 1] < X[i_s]
// it holds: i element of interval (i_s, j)
while (X[j + 1] < X[i - 1]) // X[j + 1] < X[i - 1] => (i - 1 > 0 && Y[i - 2] <= Y[i - 1]) (in case of lower)
i--;
// initial value for k
k = j + 1;
y = Y[i - 1]; // X[i - 1] <= X[j + 1] == X[k] < X[i] && in case of lower: y == Y[i - 1] <= Y[i] <= Y[j] == Y[j + 1] == Y[k]
// Find k
while (((lower && y < Y[k]) || (!lower && Y[k] < y)) && k + 1 < Y.Count)// y < Y[k] => k < X.Count
{
iPrevious = i;
k++;
while (X[i] < X[k] && i < j)
i++;
// it holds X[i - 1] < X[k] <= X[i], particularly X[i] != X[i - 1]
// linear interpolation of (x, y) = (X[k], y) on segment (i - 1, i)
y = (Y[i] - Y[i - 1]) / (X[i] - X[i - 1]) * (X[k] - X[i - 1]) + Y[i - 1];
}
// k located: intersection point is on segment (k - 1, k)
// i approximately located: intersection point is on polygonal line (iPrevoius - 1, i)
// Regular case
if (iPrevious > 1)
i = iPrevious - 1;
// Special case handling: assure, that i - 1 >= 0
else
i = iPrevious;
if (X[k] != X[k - 1])
// linear interpolation of (x, y) = (X[i], y) on segment (k - 1, k)
y = (Y[k] - Y[k - 1]) / (X[k] - X[k - 1]) * (X[i] - X[k - 1]) + Y[k - 1];
// it holds Y[i] = Y[iPrevious - 1] < Y[k - 1]
// Find i
while ((X[k] != X[k - 1] && ((lower && Y[i] < y) || (!lower && y < Y[i]))) || (X[k] == X[k - 1] && X[i] < X[k]))
{
i++;
if (X[k] != X[k - 1])
// linear interpolation of (x, y) = (X[i], y) on segment (k - 1, k)
y = (Y[k] - Y[k - 1]) / (X[k] - X[k - 1]) * (X[i] - X[k - 1]) + Y[k - 1];
}
// ----------------------------------------------------------------------------------------------------
// 2. Calculate intersection point (ix, iy) of segments (i - 1, i) and (k - 1, k)
// ----------------------------------------------------------------------------------------------------
double a1 = 0;
double a2 = 0;
// both branches vertical
if (X[i] == X[i - 1] && X[k] == X[k - 1])
{
// add no point; check if case occur: slopes have different signs
addPoint = false;
}
// case i-branch vertical
else if (X[i] == X[i - 1])
{
ix = X[i];
iy = Y[k - 1] + ((X[i] - X[k - 1]) * (Y[k] - Y[k - 1])) / (X[k] - X[k - 1]);
}
// case k-branch vertical
else if (X[k] == X[k - 1])
{
ix = X[k];
iy = Y[i - 1] + ((X[k] - X[i - 1]) * (Y[i] - Y[i - 1])) / (X[i] - X[i - 1]);
}
// common case
else
{
a1 = (Y[i] - Y[i - 1]) / (X[i] - X[i - 1]); // slope of segment (i - 1, i)
a2 = (Y[k] - Y[k - 1]) / (X[k] - X[k - 1]); // slope of segment (k - 1, k)
// common case: no equal slopes
if (a1 != a2)
{
ix = (a1 * X[i - 1] - a2 * X[k - 1] - Y[i - 1] + Y[k - 1]) / (a1 - a2);
if (Math.Abs(a1) > Math.Abs(a2))
// calculate y on segment (k - 1, k)
iy = a2 * (ix - X[k - 1]) + Y[k - 1];
else
// calculate y on segment (i - 1, i)
iy = a1 * (ix - X[i - 1]) + Y[i - 1];
}
else
// case equal slopes: add no point
addPoint = false;
}
// ----------------------------------------------------------------------------------------------------
// 3. Delete points i until (including) k - 1
// ----------------------------------------------------------------------------------------------------
int count = k - i;
X.RemoveRange(i, count);
Y.RemoveRange(i, count);
// ----------------------------------------------------------------------------------------------------
// 4. Add intersection point
// ----------------------------------------------------------------------------------------------------
// add intersection point, if it isn`t already there
if (addPoint && (X[i] != ix || Y[i] != iy))
{
X.Insert(i, ix);
Y.Insert(i, iy);
}
// ----------------------------------------------------------------------------------------------------
// 5. set j = i
// ----------------------------------------------------------------------------------------------------
j = i;
// ----------------------------------------------------------------------------------------------------
// 6. Delete points that are doubled
// ----------------------------------------------------------------------------------------------------
if (X[i - 1] == X[i] && Y[i - 1] == Y[i])
{
X.RemoveAt(i);
Y.RemoveAt(i);
j = i - 1;
}
}
j++;
}
#if GUI
// Visualization
if (visualize)
{
ChartControl control = new ChartControl(500, Int32.MaxValue, true);
control.AddLine("Tube curve with loop", XLoops, YLoops, Color.FromKnownColor(KnownColor.Cyan));
control.AddLine("Tube curve without loop", X, Y, Color.FromKnownColor(KnownColor.Red));
Application.Run(control);
}
#endif
return countLoops;
}
/// <summary>
/// Remove points and add intersection points in case of backward order
/// </summary>
/// <param name="X">x values of curve</param>
/// <param name="Y">y values of curve</param>
/// <param name="lower">if true, algorithm for lower tube curve is used;<para>
/// if false, algorithm for upper tube curve is used</para></param>
/// <return>Number of loops, that are removed.</return>
private static int removeLoop(List <double> X, List <double> Y, bool lower)
{
// Visualization of working of removeLoop
bool visualize = false;
List <double> XLoops = new List <double>(X);
List <double> YLoops = new List <double>(Y);
int j = 1;
int countLoops = 0;
#if GUI
// Visualization
if (visualize)
{
ChartControl chartControl = new ChartControl(600, Int32.MaxValue, true);
chartControl.AddLine("Tube curve with loop", X, Y, Color.FromKnownColor(KnownColor.Cyan));
Application.Run(chartControl);
}
#endif
while (j < X.Count - 2)
{
// Find backward segment (j, j + 1)
if (X[j + 1] < X[j])
{
countLoops++;
// ----------------------------------------------------------------------------------------------------
// 1. Find i,k, such that i <= j < j + 1 <= k - 1 and segment (i - 1, i) intersect segment (k - 1, k)
// ----------------------------------------------------------------------------------------------------
int i, k, iPrevious;
double y;
// for calculation and adding of intersection point
bool addPoint = true;
double ix = 0;
double iy = 0;
int kMax;
i = j;
iPrevious = i;
// Find initial value for i = i_s, such that X[i_s - 1] <= X[j + 1] < X[i_s]
// it holds: i element of interval (i_s, j)
while (X[j + 1] < X[i - 1]) // X[j + 1] < X[i - 1] => (i - 1 > 0 && Y[i - 2] <= Y[i - 1]) (in case of lower)
{
i--;
}
// j + 1 < k <= kMax
kMax = j + 1;
while (X[kMax] < X[j] && kMax < Y.Count - 1)
{
kMax++;
}
// initial value for k
k = j + 1;
y = Y[i - 1]; // X[i - 1] <= X[j + 1] == X[k] < X[i] && in case of lower: y == Y[i - 1] <= Y[i] <= Y[j] == Y[j + 1] == Y[k]
// Find k
while (((lower && y < Y[k]) || (!lower && Y[k] < y)) && k < kMax)// y < Y[k] => k < X.Count
{
iPrevious = i;
k++;
//while ((X[i] < X[k] || (X[i] == X[k] && Y[i] < Y[k])) && i < j)
while ((X[i] < X[k] || (lower && X[i] == X[k] && Y[i] < Y[k] && !(k + 1 < X.Count && X[k] == X[k + 1] && Y[k + 1] < Y[k])) || (!lower && X[i] == X[k] && Y[i] > Y[k] && !(k + 1 < X.Count && X[k] == X[k + 1] && Y[k + 1] > Y[k]))) && i < j)
{
i++;
}
// it holds X[i - 1] < X[k] <= X[i], particularly X[i] != X[i - 1]
// for i < j and X[i - 1] < X[k] it holds X[i - 1] < X[k] <= X[i], particularly X[i] != X[i - 1]
// linear interpolation of (x, y) = (X[k], y) on segment (i - 1, i)
if (X[i] - X[i - 1] != 0)
{
y = (Y[i] - Y[i - 1]) / (X[i] - X[i - 1]) * (X[k] - X[i - 1]) + Y[i - 1];
}
else
{
y = Y[i];
}
}
// k located: intersection point is on segment (k - 1, k)
// i approximately located: intersection point is on polygonal line (iPrevoius - 1, i)
// Regular case
if (iPrevious > 1)
{
i = iPrevious - 1;
}
// Special case handling: assure, that i - 1 >= 0
else
{
i = iPrevious;
}
if (X[k] != X[k - 1])
{
// linear interpolation of (x, y) = (X[i], y) on segment (k - 1, k)
y = (Y[k] - Y[k - 1]) / (X[k] - X[k - 1]) * (X[i] - X[k - 1]) + Y[k - 1];
}
// it holds Y[i] = Y[iPrevious - 1] < Y[k - 1]
// Find i
while ((X[k] != X[k - 1] && ((lower && Y[i] < y) || (!lower && y < Y[i]))) || (X[k] == X[k - 1] && X[i] < X[k]))
{
i++;
if (X[k] != X[k - 1])
{
// linear interpolation of (x, y) = (X[i], y) on segment (k - 1, k)
y = (Y[k] - Y[k - 1]) / (X[k] - X[k - 1]) * (X[i] - X[k - 1]) + Y[k - 1];
}
}
// ----------------------------------------------------------------------------------------------------
// 2. Calculate intersection point (ix, iy) of segments (i - 1, i) and (k - 1, k)
// ----------------------------------------------------------------------------------------------------
double a1 = 0;
double a2 = 0;
// both branches vertical
if (X[i] == X[i - 1] && X[k] == X[k - 1])
{
// add no point; check if case occur: slopes have different signs
addPoint = false;
}
// case i-branch vertical
else if (X[i] == X[i - 1])
{
ix = X[i];
iy = Y[k - 1] + ((X[i] - X[k - 1]) * (Y[k] - Y[k - 1])) / (X[k] - X[k - 1]);
}
// case k-branch vertical
else if (X[k] == X[k - 1])
{
ix = X[k];
iy = Y[i - 1] + ((X[k] - X[i - 1]) * (Y[i] - Y[i - 1])) / (X[i] - X[i - 1]);
}
// common case
else
{
a1 = (Y[i] - Y[i - 1]) / (X[i] - X[i - 1]); // slope of segment (i - 1, i)
a2 = (Y[k] - Y[k - 1]) / (X[k] - X[k - 1]); // slope of segment (k - 1, k)
// common case: no equal slopes
if (a1 != a2)
{
ix = (a1 * X[i - 1] - a2 * X[k - 1] - Y[i - 1] + Y[k - 1]) / (a1 - a2);
if (Math.Abs(a1) > Math.Abs(a2))
{
// calculate y on segment (k - 1, k)
iy = a2 * (ix - X[k - 1]) + Y[k - 1];
}
else
{
// calculate y on segment (i - 1, i)
iy = a1 * (ix - X[i - 1]) + Y[i - 1];
}
}
else
{
// case equal slopes: add no point
addPoint = false;
}
}
// ----------------------------------------------------------------------------------------------------
// 3. Delete points i until (including) k - 1
// ----------------------------------------------------------------------------------------------------
int count = k - i;
X.RemoveRange(i, count);
Y.RemoveRange(i, count);
// ----------------------------------------------------------------------------------------------------
// 4. Add intersection point
// ----------------------------------------------------------------------------------------------------
// add intersection point, if it isn`t already there
if (addPoint && (X[i] != ix || Y[i] != iy))
{
X.Insert(i, ix);
Y.Insert(i, iy);
}
// ----------------------------------------------------------------------------------------------------
// 5. set j = i
// ----------------------------------------------------------------------------------------------------
j = i;
// ----------------------------------------------------------------------------------------------------
// 6. Delete points that are doubled
// ----------------------------------------------------------------------------------------------------
if (X[i - 1] == X[i] && Y[i - 1] == Y[i])
{
X.RemoveAt(i);
Y.RemoveAt(i);
j = i - 1;
}
}
j++;
}
#if GUI
// Visualization
if (visualize)
{
ChartControl control = new ChartControl(500, Int32.MaxValue, true);
control.AddLine("Tube curve with loop", XLoops, YLoops, Color.FromKnownColor(KnownColor.Cyan));
control.AddLine("Tube curve without loop", X, Y, Color.FromKnownColor(KnownColor.Red));
Application.Run(control);
}
#endif
return(countLoops);
}