/// <summary>
/// Checks whether five points are co-cubic within tolerance
/// </summary>
/// <param name="data">In: Data points</param>
/// <param name="i">In: Array of 5 indices</param>
/// <param name="fitError">In: tolerated error - squared</param>
/// <returns>Return true if extended</returns>
private static bool CoCubic(CuspData data, int[] i, double fitError)
{
/* Our error estimate is (t[4]-t[0])^4 times the 4th divided difference
* of the points with resect to the nodes. The divided difference is
* equal to Sum(c(i)*p[i]), where c(i)=Product(t[i]-t[j]: j != i)
* (See Conte & deBoor's Elementary Numerical Analysis, Excercise 2.2-1).
* We multiply each factor in the product by t[4]-t[0].
*/
double d04 = data.Node(i[4]) - data.Node(i[0]);
double d01 = d04 / (data.Node(i[1]) - data.Node(i[0]));
double d02 = d04 / (data.Node(i[2]) - data.Node(i[0]));
double d03 = d04 / (data.Node(i[3]) - data.Node(i[0]));
double d12 = d04 / (data.Node(i[2]) - data.Node(i[1]));
double d13 = d04 / (data.Node(i[3]) - data.Node(i[1]));
double d14 = d04 / (data.Node(i[4]) - data.Node(i[1]));
double d23 = d04 / (data.Node(i[3]) - data.Node(i[2]));
double d24 = d04 / (data.Node(i[4]) - data.Node(i[2]));
double d34 = d04 / (data.Node(i[4]) - data.Node(i[3]));
Vector P = d01 * d02 * d03 * data.XY(i[0]) -
d01 * d12 * d13 * d14 * data.XY(i[1]) +
d02 * d12 * d23 * d24 * data.XY(i[2]) -
d03 * d13 * d23 * d34 * data.XY(i[3]) +
d14 * d24 * d34 * data.XY(i[4]);
return((P * P) < fitError);
}
/// <summary>
/// Extend the current bezier segment if possible
/// </summary>
/// <param name="error">Fitting error sqaure</param>
/// <param name="data">Data points</param>
/// <param name="from">Starting index</param>
/// <param name="next_cusp">NExt cusp index</param>
/// <param name="to">Index of the last index, updated here</param>
/// <param name="cusp">Whether there is a cusp at the end</param>
/// <param name="done">Whether end of the stroke is reached</param>
/// <returns>Whether the the segment was extended</returns>
private bool ExtendingRange(double error, CuspData data, int from, int next_cusp, ref int to, ref bool cusp, ref bool done)
{
to++;
cusp = true; // Presumed guilty
done = to >= data.Count - 1;
if (done)
{
to = data.Count - 1;
cusp = true;
return(false);
}
cusp = to >= next_cusp;
if (cusp)
{
to = next_cusp;
return(false);
}
Debug.Assert(to - from >= 4);
int d = (to - from) / 4;
int[] i = { from, from + d, (to + from) / 2, to - d, to };
// Test for "cubicness"
return(CoCubic(data, i, error));
}
/// <summary>
/// Add Line to the bezier
/// </summary>
/// <param name="data">In: Data points</param>
/// <param name="from">In: The index of the line's first point</param>
/// <param name="to">In: The index of the line's last point</param>
private void AddLine(CuspData data, int from, int to)
{
const double third = 1.0d / 3.0d;
AddBezierPoint((2 * data.XY(from) + data.XY(to)) * third);
AddBezierPoint((data.XY(from) + 2 * data.XY(to)) * third);
AddSegmentPoint(data, to);
}
/// <summary>
/// Construct bezier control points from points
/// </summary>
/// <param name="stylusPoints">Original StylusPointCollection</param>
/// <param name="fitError">Fitting error</param>
/// <returns>Whether the algorithm succeeded</returns>
internal bool ConstructBezierState(StylusPointCollection stylusPoints, double fitError)
{
// If the point count is zero, the curve cannot be constructed
if ((null == stylusPoints) || (stylusPoints.Count == 0))
return false;
// Compile list of distinct points and their nodes
CuspData dat = new CuspData();
dat.Analyze(stylusPoints,
fitError /*typically zero*/);
return ConstructFromData(dat, fitError);
}
/// <summary>
/// Construct bezier control points from points
/// </summary>
/// <param name="stylusPoints">Original StylusPointCollection</param>
/// <param name="fitError">Fitting error</param>
/// <returns>Whether the algorithm succeeded</returns>
internal bool ConstructBezierState(StylusPointCollection stylusPoints, double fitError)
{
// If the point count is zero, the curve cannot be constructed
if ((null == stylusPoints) || (stylusPoints.Count == 0))
{
return(false);
}
// Compile list of distinct points and their nodes
CuspData dat = new CuspData();
dat.Analyze(stylusPoints,
fitError /*typically zero*/);
return(ConstructFromData(dat, fitError));
}
/// <summary>
/// Add a bezier segment to the bezier buffer
/// </summary>
/// <param name="data">In: Data points</param>
/// <param name="from">In: Index of the first point</param>
/// <param name="tanStart">In: Unit tangent vector at the start</param>
/// <param name="to">In: Index of the last point, updated here</param>
/// <param name="tanEnd">In: Unit tangent vector at the end</param>
/// <returns>True if the segment was added</returns>
private bool AddBezierSegment(CuspData data, int from, ref Vector tanStart, int to, ref Vector tanEnd)
{
switch (to - from)
{
case 1:
AddLine(data, from, to);
return(true);
case 2:
AddParabola(data, from);
return(true);
}
// We have at least 4 points, compute a least squares cubic
return(AddLeastSquares(data, from, ref tanStart, to, ref tanEnd));
}
/// <summary>
/// Add parabola to the bezier
/// </summary>
/// <param name="data">In: Data points</param>
/// <param name="from">In: The index of the parabola's first point</param>
private void AddParabola(CuspData data, int from)
{
/* Denote s = 1-t. We construct the parabola with Bezier points A,B,C that
* goes thru the point P at parameter value t, that is
* P = s^2A + 2stB + t^2C
*
* We know A and C, and we solve for B:
* B = (P - s^2A - t^2C) / 2st.
*
* Elevating the degree to cubic replaces B with 2 points, the first at
* 2B/3 + A/3, and the second at 2B/3 + C/3.
*
* That is, one point at
* (P/(st) - Ct/s + A(-s/t + 1)) / 3
* and the other point at
* (P/(st) + C(-t/s + 1) - As/t) / 3
*/
// By the way the nodes were constructed:
//ASSERT(data.Node(from+2) - data.Node(from) >
// data.Node(from+1) - data.Node(from));
double t = (data.Node(from + 1) - data.Node(from)) / (data.Node(from + 2) - data.Node(from));
double s = 1 - t;
if (t < .001 || s < .001)
{
// A straight line will be a better approximation
AddLine(data, from, from + 2);
return;
}
double tt = 1 / t;
double ss = 1 / s;
const double third = 1.0d / 3.0d;
Vector P = (tt * ss) * data.XY(from + 1);
Vector B = third * (P + (1 - s * tt) * data.XY(from) - (t * ss) * data.XY(from + 1));
AddBezierPoint(B);
B = third * (P - (s * tt) * data.XY(from) + (1 - t * ss) * data.XY(from + 2));
AddBezierPoint(B);
AddSegmentPoint(data, from + 2);
}
/// <summary>
/// Add least square fit curve to the bezier
/// </summary>
/// <param name="data">In: Data points</param>
/// <param name="from">In: Index of the first point</param>
/// <param name="V">In: Unit tangent vector at the start</param>
/// <param name="to">In: Index of the last point, updated here</param>
/// <param name="W">In: Unit tangent vector at the end</param>
/// <returns>Return true segment added</returns>
private bool AddLeastSquares(CuspData data, int from, ref Vector V, int to, ref Vector W)
{
/* To do: When there is a cusp at either one of the ends, we'll get a
* better approximation if we use a construction without a prescribed
* tangent there */
/*
* The Bezier points of this segment are A, A+sV, B+uW, and B, where A,B are the
* endpoints, and V,W are the end tangents. For the node tj, denote f0j=(1-tj)^3,
* f1j=3(1-tj)^2tj, f2j=3(1-tj)tj^2, f3j=tj^3. Let Pj be the jth point.
* We are lookig for s,u that minimize
* Sum(A*f0j + (A+sV)*f1j + (B+uW)*f2j + B*f3j - Pj)^2.
*
* Equate the partial derivatives of this w.r.t. s and u to 0:
* Sum(A*f0j + (A+sV)*f1j + (B+uW)*f2j + B*f3j - Pj)*(V*f1j)=0
* Sum(A*f0j + (A+sV)*f1j + (B+uW)*f2j + B*f3j - Pj)*(W*f2j)=0
* hence
*
* s*Sum(V*V*f1j*f1j) + u*Sum(W*V*f1j*f2j)= -Sum(A*(f0j+f1j) + B*(f2j+f3j) - Pj)*V*f1j
* s*Sum(V*W*f1j*f2j) + u*Sum(W*W*f2j*f2j)= -Sum(A*(f0j+f1j) + B*(f2j+f3j) - Pj)*W*f2j
*
* so the equations are
* s*a11 + u*a12 = b1
* s*a12 * u*a22 = b2
*
* with
* a11 = W*W*Sum(f1j^2), a22 = V*V*Sum(f2j^2), a12 = W*V*Sum(f1j*f2j)
* b1 = -V*A*Sum(f0j + f1j)*f1j - V*B*Sum(f2j + f3j)*f1j + Sum(f1j*Pj*V)
* b2 = -W*A*Sum(f0j + f1j)*f2j - W*B*Sum(f2j + f3j)*f2j + Sum(f2j*Pj*W)
*
* V and W ae unit vectors, so V*V = W*W = 1.
* For computational efficiency, we will break b1 and b2 into 3 sums each, and add
* them up at the end
*
* The solution is
* s = (b1*a22 - b2*a12) / det
* u = (b2*a11 - b1*a12) / det
* where det = a11*a22 - a22^2
*/
// Compute the coefficients
double a11 = 0, a12 = 0, a22 = 0, b1 = 0, b2 = 0;
double b11 = 0, b12 = 0, b21 = 0, b22 = 0;
for (int j = checked (from + 1); j < to; j++)
{
// By the way the nodes were constructed -
Debug.Assert(data.Node(to) - data.Node(from) > data.Node(j) - data.Node(from));
double tj = (data.Node(j) - data.Node(from)) / (data.Node(to) - data.Node(from));
double tj2 = tj * tj;
double rj = 1 - tj;
double rj2 = rj * rj;
double f0j = rj2 * rj;
double f1j = 3 * rj2 * tj;
double f2j = 3 * rj * tj2;
double f3j = tj2 * tj;
a11 += f1j * f1j;
a22 += f2j * f2j;
a12 += f1j * f2j;
b11 -= (f0j + f1j) * f1j;
b12 -= (f2j + f3j) * f1j;
b1 += f1j * (data.XY(j) * V);
b21 -= (f0j + f1j) * f2j;
b22 -= (f2j + f3j) * f2j;
b2 += f2j * (data.XY(j) * W);
}
a12 *= (V * W);
b1 += ((V * data.XY(from)) * b11 + (V * data.XY(to)) * b12);
b2 += ((W * data.XY(from)) * b21 + (W * data.XY(to)) * b22);
// Solve the equations
double s = b1 * a22 - b2 * a12;
double u = b2 * a11 - b1 * a12;
double det = a11 * a22 - a12 * a12;
bool accept = (Math.Abs(det) > Math.Abs(s) * DoubleUtil.DBL_EPSILON &&
Math.Abs(det) > Math.Abs(u) * DoubleUtil.DBL_EPSILON);
if (accept)
{
s /= det;
u /= det;
// We'll only accept large enough positive solutions
accept = s > 1.0e-6 && u > 1.0e-6;
}
if (!accept)
{
s = u = (data.Node(to) - data.Node(from)) / 3;
}
AddBezierPoint(data.XY(from) + s * V);
AddBezierPoint(data.XY(to) + u * W);
AddSegmentPoint(data, to);
return(true);
}
/// <summary>
/// Construct bezier curve from data points
/// </summary>
/// <param name="data">In: Data points</param>
/// <param name="fitError">In: tolerated error</param>
/// <returns>Whether bezier construction is possible</returns>
private bool ConstructFromData(CuspData data, double fitError)
{
// Check for empty stroke
if (data.Count < 2)
{
return(false);
}
// Add the first point
AddBezierPoint(data.XY(0));
// Special cases - 2 or 3 points
if (data.Count == 3)
{
AddParabola(data, 0);
return(true);
}
else if (data.Count == 2)
{
AddLine(data, 0, 1);
return(true);
}
// For default case error passed in will be 0.
// 3% is the default value
if (DoubleUtil.DBL_EPSILON > fitError)
{
fitError = 0.03f * (data.Distance() * StrokeCollectionSerializer.HimetricToAvalonMultiplier);
}
data.SetTanLinks(0.5f * fitError);
// otherwise use the value specified in the drawing attribute
// get (error)^2
fitError *= (fitError);
bool done = false;
int to = 0;
int next_cusp = 0;
int prev_cusp = 0;
bool is_a_cusp = true;
Vector tanEnd = new Vector(0, 0);
Vector tanStart = new Vector(0, 0);
for (int from = 0; !done; from = to)
{
if (is_a_cusp)
{
prev_cusp = next_cusp;
next_cusp = data.GetNextCusp(from);
if (!data.Tangent(ref tanStart, from, prev_cusp, next_cusp, false, true))
{
return(false);
}
}
else
{
tanStart.X = -tanEnd.X;
tanStart.Y = -tanEnd.Y;
}
to = from + 3;
// No meat in this loop, just extending the index range
while (ExtendingRange(fitError, data, from, next_cusp, ref to, ref is_a_cusp, ref done))
{
;
}
// Find the tangent
if (!data.Tangent(ref tanEnd, to, prev_cusp, next_cusp, true, is_a_cusp))
{
return(false);
}
// Add bezier segment
if (!AddBezierSegment(data, from, ref tanStart, to, ref tanEnd))
{
return(false);
}
}
return(true);
}
/// <summary>
/// Construct bezier curve from data points
/// </summary>
/// <param name="data">In: Data points</param>
/// <param name="fitError">In: tolerated error</param>
/// <returns>Whether bezier construction is possible</returns>
private bool ConstructFromData(CuspData data, double fitError)
{
// Check for empty stroke
if (data.Count < 2)
{
return false;
}
// Add the first point
AddBezierPoint(data.XY(0));
// Special cases - 2 or 3 points
if (data.Count == 3)
{
AddParabola(data, 0);
return true;
}
else if (data.Count == 2)
{
AddLine(data, 0, 1);
return true;
}
// For default case error passed in will be 0.
// 3% is the default value
if (DoubleUtil.DBL_EPSILON > fitError)
fitError = 0.03f * (data.Distance() * StrokeCollectionSerializer.HimetricToAvalonMultiplier);
data.SetTanLinks(0.5f * fitError);
// otherwise use the value specified in the drawing attribute
// get (error)^2
fitError *= (fitError);
bool done = false;
int to = 0;
int next_cusp = 0;
int prev_cusp = 0;
bool is_a_cusp = true;
Vector tanEnd = new Vector(0, 0);
Vector tanStart = new Vector(0, 0);
for (int from = 0; !done; from = to)
{
if (is_a_cusp)
{
prev_cusp = next_cusp;
next_cusp = data.GetNextCusp(from);
if (!data.Tangent(ref tanStart, from, prev_cusp, next_cusp, false, true))
{
return false;
}
}
else
{
tanStart.X = -tanEnd.X;
tanStart.Y = -tanEnd.Y;
}
to = from + 3;
// No meat in this loop, just extending the index range
while (ExtendingRange(fitError, data, from, next_cusp, ref to, ref is_a_cusp, ref done));
// Find the tangent
if (!data.Tangent(ref tanEnd, to, prev_cusp, next_cusp, true, is_a_cusp))
{
return false;
}
// Add bezier segment
if (!AddBezierSegment(data, from, ref tanStart, to, ref tanEnd))
{
return false;
}
}
return true;
}
/// <summary>
/// Extend the current bezier segment if possible
/// </summary>
/// <param name="error">Fitting error sqaure</param>
/// <param name="data">Data points</param>
/// <param name="from">Starting index</param>
/// <param name="next_cusp">NExt cusp index</param>
/// <param name="to">Index of the last index, updated here</param>
/// <param name="cusp">Whether there is a cusp at the end</param>
/// <param name="done">Whether end of the stroke is reached</param>
/// <returns>Whether the the segment was extended</returns>
private bool ExtendingRange(double error, CuspData data, int from, int next_cusp, ref int to, ref bool cusp, ref bool done)
{
to++;
cusp = true; // Presumed guilty
done = to >= data.Count - 1;
if (done)
{
to = data.Count - 1;
cusp = true;
return false;
}
cusp = to >= next_cusp;
if (cusp)
{
to = next_cusp;
return false;
}
Debug.Assert(to - from >= 4);
int d = (to - from) / 4;
int[] i = { from, from + d, (to + from) / 2, to - d, to };
// Test for "cubicness"
return CoCubic(data, i, error);
}
/// <summary>
/// Add segment point
/// </summary>
/// <param name="data">In: Interpolation data</param>
/// <param name="index">In: The index of the point to add</param>
private void AddSegmentPoint(CuspData data, int index)
{
_bezierControlPoints.Add((Point)data.XY(index));
}
/// <summary>
/// Checks whether five points are co-cubic within tolerance
/// </summary>
/// <param name="data">In: Data points</param>
/// <param name="i">In: Array of 5 indices</param>
/// <param name="fitError">In: tolerated error - squared</param>
/// <returns>Return true if extended</returns>
private static bool CoCubic(CuspData data, int[] i, double fitError)
{
/* Our error estimate is (t[4]-t[0])^4 times the 4th divided difference
* of the points with resect to the nodes. The divided difference is
* equal to Sum(c(i)*p[i]), where c(i)=Product(t[i]-t[j]: j != i)
* (See Conte & deBoor's Elementary Numerical Analysis, Excercise 2.2-1).
* We multiply each factor in the product by t[4]-t[0].
*/
double d04 = data.Node(i[4]) - data.Node(i[0]);
double d01 = d04 / (data.Node(i[1]) - data.Node(i[0]));
double d02 = d04 / (data.Node(i[2]) - data.Node(i[0]));
double d03 = d04 / (data.Node(i[3]) - data.Node(i[0]));
double d12 = d04 / (data.Node(i[2]) - data.Node(i[1]));
double d13 = d04 / (data.Node(i[3]) - data.Node(i[1]));
double d14 = d04 / (data.Node(i[4]) - data.Node(i[1]));
double d23 = d04 / (data.Node(i[3]) - data.Node(i[2]));
double d24 = d04 / (data.Node(i[4]) - data.Node(i[2]));
double d34 = d04 / (data.Node(i[4]) - data.Node(i[3]));
Vector P = d01 * d02 * d03 * data.XY(i[0]) -
d01 * d12 * d13 * d14 * data.XY(i[1]) +
d02 * d12 * d23 * d24 * data.XY(i[2]) -
d03 * d13 * d23 * d34 * data.XY(i[3]) +
d14 * d24 * d34 * data.XY(i[4]);
return ((P * P) < fitError);
}
/// <summary>
/// Add least square fit curve to the bezier
/// </summary>
/// <param name="data">In: Data points</param>
/// <param name="from">In: Index of the first point</param>
/// <param name="V">In: Unit tangent vector at the start</param>
/// <param name="to">In: Index of the last point, updated here</param>
/// <param name="W">In: Unit tangent vector at the end</param>
/// <returns>Return true segment added</returns>
private bool AddLeastSquares(CuspData data, int from, ref Vector V, int to, ref Vector W)
{
/* To do: When there is a cusp at either one of the ends, we'll get a
better approximation if we use a construction without a prescribed
tangent there */
/*
The Bezier points of this segment are A, A+sV, B+uW, and B, where A,B are the
endpoints, and V,W are the end tangents. For the node tj, denote f0j=(1-tj)^3,
f1j=3(1-tj)^2tj, f2j=3(1-tj)tj^2, f3j=tj^3. Let Pj be the jth point.
We are lookig for s,u that minimize
Sum(A*f0j + (A+sV)*f1j + (B+uW)*f2j + B*f3j - Pj)^2.
Equate the partial derivatives of this w.r.t. s and u to 0:
Sum(A*f0j + (A+sV)*f1j + (B+uW)*f2j + B*f3j - Pj)*(V*f1j)=0
Sum(A*f0j + (A+sV)*f1j + (B+uW)*f2j + B*f3j - Pj)*(W*f2j)=0
hence
s*Sum(V*V*f1j*f1j) + u*Sum(W*V*f1j*f2j)= -Sum(A*(f0j+f1j) + B*(f2j+f3j) - Pj)*V*f1j
s*Sum(V*W*f1j*f2j) + u*Sum(W*W*f2j*f2j)= -Sum(A*(f0j+f1j) + B*(f2j+f3j) - Pj)*W*f2j
so the equations are
s*a11 + u*a12 = b1
s*a12 * u*a22 = b2
with
a11 = W*W*Sum(f1j^2), a22 = V*V*Sum(f2j^2), a12 = W*V*Sum(f1j*f2j)
b1 = -V*A*Sum(f0j + f1j)*f1j - V*B*Sum(f2j + f3j)*f1j + Sum(f1j*Pj*V)
b2 = -W*A*Sum(f0j + f1j)*f2j - W*B*Sum(f2j + f3j)*f2j + Sum(f2j*Pj*W)
V and W ae unit vectors, so V*V = W*W = 1.
For computational efficiency, we will break b1 and b2 into 3 sums each, and add
them up at the end
The solution is
s = (b1*a22 - b2*a12) / det
u = (b2*a11 - b1*a12) / det
where det = a11*a22 - a22^2
*/
// Compute the coefficients
double a11 = 0, a12 = 0, a22 = 0, b1 = 0, b2 = 0;
double b11 = 0, b12 = 0, b21 = 0, b22 = 0;
for (int j = checked(from + 1); j < to; j++)
{
// By the way the nodes were constructed -
Debug.Assert(data.Node(to) - data.Node(from) > data.Node(j) - data.Node(from));
double tj = (data.Node(j) - data.Node(from)) / (data.Node(to) - data.Node(from));
double tj2 = tj * tj;
double rj = 1 - tj;
double rj2 = rj * rj;
double f0j = rj2 * rj;
double f1j = 3 * rj2 * tj;
double f2j = 3 * rj * tj2;
double f3j = tj2 * tj;
a11 += f1j * f1j;
a22 += f2j * f2j;
a12 += f1j * f2j;
b11 -= (f0j + f1j) * f1j;
b12 -= (f2j + f3j) * f1j;
b1 += f1j * (data.XY(j) * V);
b21 -= (f0j + f1j) * f2j;
b22 -= (f2j + f3j) * f2j;
b2 += f2j * (data.XY(j) * W);
}
a12 *= (V * W);
b1 += ((V * data.XY(from)) * b11 + (V * data.XY(to)) * b12);
b2 += ((W * data.XY(from)) * b21 + (W * data.XY(to)) * b22);
// Solve the equations
double s = b1 * a22 - b2 * a12;
double u = b2 * a11 - b1 * a12;
double det = a11 * a22 - a12 * a12;
bool accept = (Math.Abs(det) > Math.Abs(s) * DoubleUtil.DBL_EPSILON &&
Math.Abs(det) > Math.Abs(u) * DoubleUtil.DBL_EPSILON);
if (accept)
{
s /= det;
u /= det;
// We'll only accept large enough positive solutions
accept = s > 1.0e-6 && u > 1.0e-6;
}
if (!accept)
s = u = (data.Node(to) - data.Node(from)) / 3;
AddBezierPoint(data.XY(from) + s * V);
AddBezierPoint(data.XY(to) + u * W);
AddSegmentPoint(data, to);
return true;
}
/// <summary>
/// Add Line to the bezier
/// </summary>
/// <param name="data">In: Data points</param>
/// <param name="from">In: The index of the line's first point</param>
/// <param name="to">In: The index of the line's last point</param>
private void AddLine(CuspData data, int from, int to)
{
const double third = 1.0d / 3.0d;
AddBezierPoint((2 * data.XY(from) + data.XY(to)) * third);
AddBezierPoint((data.XY(from) + 2 * data.XY(to)) * third);
AddSegmentPoint(data, to);
}
/// <summary>
/// Add parabola to the bezier
/// </summary>
/// <param name="data">In: Data points</param>
/// <param name="from">In: The index of the parabola's first point</param>
private void AddParabola(CuspData data, int from)
{
/* Denote s = 1-t. We construct the parabola with Bezier points A,B,C that
goes thru the point P at parameter value t, that is
P = s^2A + 2stB + t^2C
We know A and C, and we solve for B:
B = (P - s^2A - t^2C) / 2st.
Elevating the degree to cubic replaces B with 2 points, the first at
2B/3 + A/3, and the second at 2B/3 + C/3.
That is, one point at
(P/(st) - Ct/s + A(-s/t + 1)) / 3
and the other point at
(P/(st) + C(-t/s + 1) - As/t) / 3
*/
// By the way the nodes were constructed:
//ASSERT(data.Node(from+2) - data.Node(from) >
// data.Node(from+1) - data.Node(from));
double t = (data.Node(from + 1) - data.Node(from)) / (data.Node(from + 2) - data.Node(from));
double s = 1 - t;
if (t < .001 || s < .001)
{
// A straight line will be a better approximation
AddLine(data, from, from + 2);
return;
}
double tt = 1 / t;
double ss = 1 / s;
const double third = 1.0d / 3.0d;
Vector P = (tt * ss) * data.XY(from + 1);
Vector B = third * (P + (1 - s * tt) * data.XY(from) - (t * ss) * data.XY(from + 1));
AddBezierPoint(B);
B = third * (P - (s * tt) * data.XY(from) + (1 - t * ss) * data.XY(from + 2));
AddBezierPoint(B);
AddSegmentPoint(data, from + 2);
}
/// <summary>
/// Add segment point
/// </summary>
/// <param name="data">In: Interpolation data</param>
/// <param name="index">In: The index of the point to add</param>
private void AddSegmentPoint(CuspData data, int index)
{
_bezierControlPoints.Add((Point)data.XY(index));
}
/// <summary>
/// Add a bezier segment to the bezier buffer
/// </summary>
/// <param name="data">In: Data points</param>
/// <param name="from">In: Index of the first point</param>
/// <param name="tanStart">In: Unit tangent vector at the start</param>
/// <param name="to">In: Index of the last point, updated here</param>
/// <param name="tanEnd">In: Unit tangent vector at the end</param>
/// <returns>True if the segment was added</returns>
private bool AddBezierSegment(CuspData data, int from, ref Vector tanStart, int to, ref Vector tanEnd)
{
switch (to - from)
{
case 1 :
AddLine(data, from, to);
return true;
case 2 :
AddParabola(data, from);
return true;
}
// We have at least 4 points, compute a least squares cubic
return AddLeastSquares(data, from, ref tanStart, to, ref tanEnd);
}