/*
* Bezier :
* Evaluate a Bezier curve at a particular parameter value
* Fill in control points for resulting sub-curves if "Left" and
* "Right" are non-null.
*
int degree; Degree of bezier curve
Point *V; Control pts
double t; Parameter value
Point *Left; RETURN left half ctl pts
Point *Right; RETURN right half ctl pts
*/
public static Point bezier(Point[] V, int degree, double t, Point[] Left, Point[] Right)
{
int i, j; /* Index variables */
Point[,] Vtemp = new Point[W_DEGREE + 1, W_DEGREE + 1];
/* Copy control points */
for (j = 0; j <= degree; j++)
{
Vtemp[0, j] = V[j];
}
/* Triangle computation */
for (i = 1; i <= degree; i++)
{
for (j = 0; j <= degree - i; j++)
{
// Vtemp[i, j].X =
// (1.0 - t) * Vtemp[i - 1, j].X + t * Vtemp[i - 1, j + 1].X;
// Vtemp[i, j].Y =
// (1.0 - t) * Vtemp[i - 1, j].Y + t * Vtemp[i - 1, j + 1].Y;
Vtemp[i, j] = new Point(
(1.0 - t) * Vtemp[i - 1, j].X + t * Vtemp[i - 1, j + 1].X,
(1.0 - t) * Vtemp[i - 1, j].Y + t * Vtemp[i - 1, j + 1].Y);
}
}
if (Left != null)
{
for (j = 0; j <= degree; j++)
{
Left[j] = Vtemp[j, 0];
}
}
if (Right != null)
{
for (j = 0; j <= degree; j++)
{
Right[j] = Vtemp[degree - j, j];
}
}
return (Vtemp[degree, 0]);
}
public void Line(Point p1, Point p2)
{
if (Stroking)
_target.DrawLine(Import.Point(p1), Import.Point(p2), _strokeBrush.Brush, StrokeWeight);
}
public static Point nearestPointOnCubicBezier(Point[] V, Point P)
{
double t = nearestTOnCubicBezier(V, P);
return bezier(V, DEGREE, t, null, null);
}
static double V2SquaredLength(Point v)
{
return v.Vector.SquaredLength();
}
static Point V2Sub(Point a, Point b, ref Point c)
{
c = (a.Vector - b.Vector).ToPoint();
return c;
}
public static void Bezier(this IGeometryFigures _, Point start, Point span1, Point span2, Point end)
{
_.Bezier(new CubicBezier(start, span1, span2, end));
}
static Point V2ScaleII(Point v, double s)
{
return new Point(v.X * s, v.Y * s);
}
/*
* CrossingCount :
* Count the number of times a Bezier control polygon
* crosses the 0-axis. This number is >= the number of roots.
Point *V; Control pts of Bezier curve
int degree; Degreee of Bezier curve
*
*/
static int CrossingCount(Point[] V, int degree)
{
int i;
int n_crossings = 0; /* Number of zero-crossings */
int sign, old_sign; /* Sign of coefficients */
sign = old_sign = SGN(V[0].Y);
for (i = 1; i <= degree; i++)
{
sign = SGN(V[i].Y);
if (sign != old_sign) n_crossings++;
old_sign = sign;
}
return n_crossings;
}
/*
* FindRoots :
* Given a 5th-degree equation in Bernstein-Bezier form, find
* all of the roots in the interval [0, 1]. Return the number
* of roots found.
Point *w; The control points
int degree; The degree of the polynomial
double *t; RETURN candidate t-values
int depth; The depth of the recursion
*/
static int FindRoots(Point[] w, int degree, double[] t, int depth)
{
int i;
Point[] Left = new Point[W_DEGREE + 1]; /* New left and right */
Point[] Right = new Point[W_DEGREE + 1]; /* control polygons */
int left_count, /* Solution count from */
right_count; /* children */
double[] left_t = new double[W_DEGREE + 1]; /* Solutions from kids */
double[] right_t = new double[W_DEGREE + 1];
switch (CrossingCount(w, degree))
{
case 0:
{ /* No solutions here */
return 0;
}
case 1:
{ /* Unique solution */
/* Stop recursion when the tree is deep enough */
/* if deep enough, return 1 solution at midpoint */
if (depth >= MAXDEPTH)
{
t[0] = (w[0].X + w[W_DEGREE].X) / 2.0;
return 1;
}
if (ControlPolygonFlatEnough(w, degree))
{
t[0] = ComputeXIntercept(w, degree);
return 1;
}
break;
}
}
/* Otherwise, solve recursively after */
/* subdividing control polygon */
bezier(w, degree, 0.5, Left, Right);
left_count = FindRoots(Left, degree, left_t, depth + 1);
right_count = FindRoots(Right, degree, right_t, depth + 1);
/* Gather solutions together */
for (i = 0; i < left_count; i++)
{
t[i] = left_t[i];
}
for (i = 0; i < right_count; i++)
{
t[i + left_count] = right_t[i];
}
/* Send back total number of solutions */
return (left_count + right_count);
}
/*
* ControlPolygonFlatEnough :
* Check if the control polygon of a Bezier curve is flat enough
* for recursive subdivision to bottom out.
Point *V; Control points
int degree; Degree of polynomiaL
*
*/
static bool ControlPolygonFlatEnough(Point[] V, int degree)
{
int i; /* Index variable */
double[] distance; /* Distances from pts to line */
double max_distance_above; /* maximum of these */
double max_distance_below;
double error; /* Precision of root */
double intercept_1,
intercept_2,
left_intercept,
right_intercept;
double a, b, c; /* Coefficients of implicit */
/* eqn for line from V[0]-V[deg]*/
/* Find the perpendicular distance */
/* from each interior control point to */
/* line connecting V[0] and V[degree] */
distance = new double[degree + 1]; // (double*)malloc((unsigned)(degree + 1) * sizeof(double));
{
double abSquared;
/* Derive the implicit equation for line connecting first *'
/* and last control points */
a = V[0].Y - V[degree].Y;
b = V[degree].X - V[0].X;
c = V[0].X * V[degree].Y - V[degree].X * V[0].Y;
abSquared = (a * a) + (b * b);
for (i = 1; i < degree; i++)
{
/* Compute distance from each of the points to that line */
distance[i] = a * V[i].X + b * V[i].Y + c;
if (distance[i] > 0.0)
{
distance[i] = (distance[i] * distance[i]) / abSquared;
}
if (distance[i] < 0.0)
{
distance[i] = -((distance[i] * distance[i]) / abSquared);
}
}
}
/* Find the largest distance */
max_distance_above = 0.0;
max_distance_below = 0.0;
for (i = 1; i < degree; i++)
{
if (distance[i] < 0.0)
{
max_distance_below = Math.Min(max_distance_below, distance[i]);
};
if (distance[i] > 0.0)
{
max_distance_above = Math.Max(max_distance_above, distance[i]);
}
}
// free((char*)distance);
{
double det, dInv;
double a1, b1, c1, a2, b2, c2;
/* Implicit equation for zero line */
a1 = 0.0;
b1 = 1.0;
c1 = 0.0;
/* Implicit equation for "above" line */
a2 = a;
b2 = b;
c2 = c + max_distance_above;
det = a1 * b2 - a2 * b1;
dInv = 1.0 / det;
intercept_1 = (b1 * c2 - b2 * c1) * dInv;
/* Implicit equation for "below" line */
a2 = a;
b2 = b;
c2 = c + max_distance_below;
det = a1 * b2 - a2 * b1;
dInv = 1.0 / det;
intercept_2 = (b1 * c2 - b2 * c1) * dInv;
}
/* Compute intercepts of bounding box */
left_intercept = Math.Min(intercept_1, intercept_2);
right_intercept = Math.Max(intercept_1, intercept_2);
error = 0.5 * (right_intercept - left_intercept);
return error < EPSILON;
}
/*
* ConvertToBezierForm :
* Given a point and a Bezier curve, generate a 5th-degree
* Bezier-format equation whose solution finds the point on the
* curve nearest the user-defined point.
P; The point to find t for
V; The control points
*/
static Point[] ConvertToBezierForm(Point P, Point[] V)
{
int i, j, k, m, n, ub, lb;
int row, column; /* Table indices */
Point[] c = new Point[DEGREE + 1]; /* V(i)'s - P */
Point[] d = new Point[DEGREE]; /* V(i+1) - V(i) */
Point[] w; /* Ctl pts of 5th-degree curve */
double[,] cdTable = new double[3, 4]; /* Dot product of c, d */
/*Determine the c's -- these are vectors created by subtracting*/
/* point P from each of the control points */
for (i = 0; i <= DEGREE; i++)
{
V2Sub(V[i], P, ref c[i]);
}
/* Determine the d's -- these are vectors created by subtracting*/
/* each control point from the next */
for (i = 0; i <= DEGREE - 1; i++)
{
d[i] = V2ScaleII(V2Sub(V[i + 1], V[i], ref d[i]), 3.0);
}
/* Create the c,d table -- this is a table of dot products of the */
/* c's and d's */
for (row = 0; row <= DEGREE - 1; row++)
{
for (column = 0; column <= DEGREE; column++)
{
cdTable[row, column] = V2Dot(d[row], c[column]);
}
}
/* Now, apply the z's to the dot products, on the skew diagonal*/
/* Also, set up the x-values, making these "points" */
w = new Point[W_DEGREE + 1]; // Point *)malloc((unsigned)(W_DEGREE+1) * sizeof(Point));
for (i = 0; i <= W_DEGREE; i++)
{
// w[i].Y = 0.0;
// w[i].X = (double)(i) / W_DEGREE;
w[i] = new Point((double)(i) / W_DEGREE, 0.0);
}
n = DEGREE;
m = DEGREE - 1;
for (k = 0; k <= n + m; k++)
{
lb = Math.Max(0, k - m);
ub = Math.Min(k, n);
for (i = lb; i <= ub; i++)
{
j = k - i;
// w[i + j].Y += cdTable[j, i] * z[j, i];
var old = w[i + j];
w[i + j] = new Point(old.X, old.Y + cdTable[j, i] * z[j, i]);
}
}
return (w);
}
/*
* ComputeXIntercept :
* Compute intersection of chord from first control point to last
* with 0-axis.
*
*/
/* NOTE: "T" and "Y" do not have to be computed, and there are many useless
* operations in the following (e.g. "0.0 - 0.0").
Point *V; Control points
int degree; Degree of curve
*/
static double ComputeXIntercept(Point[] V, int degree)
{
double XLK, YLK, XNM, YNM, XMK, YMK;
double det, detInv;
double S;
double X;
XLK = 1.0 - 0.0;
YLK = 0.0 - 0.0;
XNM = V[degree].X - V[0].X;
YNM = V[degree].Y - V[0].Y;
XMK = V[0].X - 0.0;
YMK = V[0].Y - 0.0;
det = XNM * YLK - YNM * XLK;
detInv = 1.0 / det;
S = (XNM * YMK - YNM * XMK) * detInv;
/* T = (XLK*YMK - YLK*XMK) * detInv; */
X = 0.0 + XLK * S;
/* Y = 0.0 + YLK * S; */
return X;
}
public static double nearestTOnCubicBezier(Point[] V, Point P)
{
Point[] w; /* Ctl pts for 5th-degree eqn */
double[] t_candidate = new double[W_DEGREE]; /* Possible roots */
int n_solutions; /* Number of roots found */
double t; /* Parameter value of closest pt*/
/* Convert problem to 5th-degree Bezier form */
w = ConvertToBezierForm(P, V);
/* Find all possible roots of 5th-degree equation */
n_solutions = FindRoots(w, W_DEGREE, t_candidate, 0);
/* Compare distances of P to all candidates, and to t=0, and t=1 */
{
double dist, new_dist;
Point p;
Point v = new Point();
int i;
/* Check distance to beginning of curve, where t = 0 */
dist = V2SquaredLength(V2Sub(P, V[0], ref v));
t = 0.0;
/* Find distances for candidate points */
for (i = 0; i < n_solutions; i++)
{
p = bezier(V, DEGREE, t_candidate[i],
null, null);
new_dist = V2SquaredLength(V2Sub(P, p, ref v));
if (new_dist < dist)
{
dist = new_dist;
t = t_candidate[i];
}
}
/* Finally, look at distance to end point, where t = 1.0 */
new_dist = V2SquaredLength(V2Sub(P, V[DEGREE], ref v));
if (new_dist < dist)
{
dist = new_dist;
t = 1.0;
}
}
return t;
/* Return the point on the curve at parameter value t */
// printf("t : %4.12f\n", t);
}
public static double nearestTOfPoint(this CubicBezier b, Point p)
{
var points = new[] {b.Start, b.Span1, b.Span2, b.End};
return nearestTOnCubicBezier(points, p);
}
public void Polygon(Point[] points)
{
if (points.Length < 2)
return;
if (points.Length == 2)
{
Line(points[0], points[1]);
return;
}
var startPoint = Import.Point(points[0]);
if (Filling)
{
fillPath(startPoint,
sink =>
{
for (int i = 1; i != points.Length; ++i)
sink.AddLine(Import.Point(points[i]));
});
}
if (Stroking)
{
drawClosedPath(startPoint,
sink =>
{
for (int i = 1; i != points.Length; ++i)
sink.AddLine(Import.Point(points[i]));
});
}
}
/* return the dot product of vectors a and b */
static double V2Dot(Point a, Point b)
{
return ((a.X * b.X) + (a.Y * b.Y));
}
public Bounds(Point leftTop, Point rightBottom)
{
LeftTop = leftTop;
RightBottom = rightBottom;
}
public static Vector2 Point(Point p)
{
return Point(p.X, p.Y);
}