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QuadCurve2D.java
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1401 lines (1297 loc) · 46.7 KB
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/*
* Copyright (c) 1997, 2013, Oracle and/or its affiliates. All rights reserved.
* ORACLE PROPRIETARY/CONFIDENTIAL. Use is subject to license terms.
*
*
*
*
*
*
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*
*
*/
package java.awt.geom;
import java.awt.Shape;
import java.awt.Rectangle;
import java.io.Serializable;
import sun.awt.geom.Curve;
/**
* The <code>QuadCurve2D</code> class defines a quadratic parametric curve
* segment in {@code (x,y)} coordinate space.
* <p>
* This class is only the abstract superclass for all objects that
* store a 2D quadratic curve segment.
* The actual storage representation of the coordinates is left to
* the subclass.
*
* @author Jim Graham
* @since 1.2
*/
public abstract class QuadCurve2D implements Shape, Cloneable {
/**
* A quadratic parametric curve segment specified with
* {@code float} coordinates.
*
* @since 1.2
*/
public static class Float extends QuadCurve2D implements Serializable {
/**
* The X coordinate of the start point of the quadratic curve
* segment.
* @since 1.2
* @serial
*/
public float x1;
/**
* The Y coordinate of the start point of the quadratic curve
* segment.
* @since 1.2
* @serial
*/
public float y1;
/**
* The X coordinate of the control point of the quadratic curve
* segment.
* @since 1.2
* @serial
*/
public float ctrlx;
/**
* The Y coordinate of the control point of the quadratic curve
* segment.
* @since 1.2
* @serial
*/
public float ctrly;
/**
* The X coordinate of the end point of the quadratic curve
* segment.
* @since 1.2
* @serial
*/
public float x2;
/**
* The Y coordinate of the end point of the quadratic curve
* segment.
* @since 1.2
* @serial
*/
public float y2;
/**
* Constructs and initializes a <code>QuadCurve2D</code> with
* coordinates (0, 0, 0, 0, 0, 0).
* @since 1.2
*/
public Float() {
}
/**
* Constructs and initializes a <code>QuadCurve2D</code> from the
* specified {@code float} coordinates.
*
* @param x1 the X coordinate of the start point
* @param y1 the Y coordinate of the start point
* @param ctrlx the X coordinate of the control point
* @param ctrly the Y coordinate of the control point
* @param x2 the X coordinate of the end point
* @param y2 the Y coordinate of the end point
* @since 1.2
*/
public Float(float x1, float y1,
float ctrlx, float ctrly,
float x2, float y2)
{
setCurve(x1, y1, ctrlx, ctrly, x2, y2);
}
/**
* {@inheritDoc}
* @since 1.2
*/
public double getX1() {
return (double) x1;
}
/**
* {@inheritDoc}
* @since 1.2
*/
public double getY1() {
return (double) y1;
}
/**
* {@inheritDoc}
* @since 1.2
*/
public Point2D getP1() {
return new Point2D.Float(x1, y1);
}
/**
* {@inheritDoc}
* @since 1.2
*/
public double getCtrlX() {
return (double) ctrlx;
}
/**
* {@inheritDoc}
* @since 1.2
*/
public double getCtrlY() {
return (double) ctrly;
}
/**
* {@inheritDoc}
* @since 1.2
*/
public Point2D getCtrlPt() {
return new Point2D.Float(ctrlx, ctrly);
}
/**
* {@inheritDoc}
* @since 1.2
*/
public double getX2() {
return (double) x2;
}
/**
* {@inheritDoc}
* @since 1.2
*/
public double getY2() {
return (double) y2;
}
/**
* {@inheritDoc}
* @since 1.2
*/
public Point2D getP2() {
return new Point2D.Float(x2, y2);
}
/**
* {@inheritDoc}
* @since 1.2
*/
public void setCurve(double x1, double y1,
double ctrlx, double ctrly,
double x2, double y2)
{
this.x1 = (float) x1;
this.y1 = (float) y1;
this.ctrlx = (float) ctrlx;
this.ctrly = (float) ctrly;
this.x2 = (float) x2;
this.y2 = (float) y2;
}
/**
* Sets the location of the end points and control point of this curve
* to the specified {@code float} coordinates.
*
* @param x1 the X coordinate of the start point
* @param y1 the Y coordinate of the start point
* @param ctrlx the X coordinate of the control point
* @param ctrly the Y coordinate of the control point
* @param x2 the X coordinate of the end point
* @param y2 the Y coordinate of the end point
* @since 1.2
*/
public void setCurve(float x1, float y1,
float ctrlx, float ctrly,
float x2, float y2)
{
this.x1 = x1;
this.y1 = y1;
this.ctrlx = ctrlx;
this.ctrly = ctrly;
this.x2 = x2;
this.y2 = y2;
}
/**
* {@inheritDoc}
* @since 1.2
*/
public Rectangle2D getBounds2D() {
float left = Math.min(Math.min(x1, x2), ctrlx);
float top = Math.min(Math.min(y1, y2), ctrly);
float right = Math.max(Math.max(x1, x2), ctrlx);
float bottom = Math.max(Math.max(y1, y2), ctrly);
return new Rectangle2D.Float(left, top,
right - left, bottom - top);
}
/*
* JDK 1.6 serialVersionUID
*/
private static final long serialVersionUID = -8511188402130719609L;
}
/**
* A quadratic parametric curve segment specified with
* {@code double} coordinates.
*
* @since 1.2
*/
public static class Double extends QuadCurve2D implements Serializable {
/**
* The X coordinate of the start point of the quadratic curve
* segment.
* @since 1.2
* @serial
*/
public double x1;
/**
* The Y coordinate of the start point of the quadratic curve
* segment.
* @since 1.2
* @serial
*/
public double y1;
/**
* The X coordinate of the control point of the quadratic curve
* segment.
* @since 1.2
* @serial
*/
public double ctrlx;
/**
* The Y coordinate of the control point of the quadratic curve
* segment.
* @since 1.2
* @serial
*/
public double ctrly;
/**
* The X coordinate of the end point of the quadratic curve
* segment.
* @since 1.2
* @serial
*/
public double x2;
/**
* The Y coordinate of the end point of the quadratic curve
* segment.
* @since 1.2
* @serial
*/
public double y2;
/**
* Constructs and initializes a <code>QuadCurve2D</code> with
* coordinates (0, 0, 0, 0, 0, 0).
* @since 1.2
*/
public Double() {
}
/**
* Constructs and initializes a <code>QuadCurve2D</code> from the
* specified {@code double} coordinates.
*
* @param x1 the X coordinate of the start point
* @param y1 the Y coordinate of the start point
* @param ctrlx the X coordinate of the control point
* @param ctrly the Y coordinate of the control point
* @param x2 the X coordinate of the end point
* @param y2 the Y coordinate of the end point
* @since 1.2
*/
public Double(double x1, double y1,
double ctrlx, double ctrly,
double x2, double y2)
{
setCurve(x1, y1, ctrlx, ctrly, x2, y2);
}
/**
* {@inheritDoc}
* @since 1.2
*/
public double getX1() {
return x1;
}
/**
* {@inheritDoc}
* @since 1.2
*/
public double getY1() {
return y1;
}
/**
* {@inheritDoc}
* @since 1.2
*/
public Point2D getP1() {
return new Point2D.Double(x1, y1);
}
/**
* {@inheritDoc}
* @since 1.2
*/
public double getCtrlX() {
return ctrlx;
}
/**
* {@inheritDoc}
* @since 1.2
*/
public double getCtrlY() {
return ctrly;
}
/**
* {@inheritDoc}
* @since 1.2
*/
public Point2D getCtrlPt() {
return new Point2D.Double(ctrlx, ctrly);
}
/**
* {@inheritDoc}
* @since 1.2
*/
public double getX2() {
return x2;
}
/**
* {@inheritDoc}
* @since 1.2
*/
public double getY2() {
return y2;
}
/**
* {@inheritDoc}
* @since 1.2
*/
public Point2D getP2() {
return new Point2D.Double(x2, y2);
}
/**
* {@inheritDoc}
* @since 1.2
*/
public void setCurve(double x1, double y1,
double ctrlx, double ctrly,
double x2, double y2)
{
this.x1 = x1;
this.y1 = y1;
this.ctrlx = ctrlx;
this.ctrly = ctrly;
this.x2 = x2;
this.y2 = y2;
}
/**
* {@inheritDoc}
* @since 1.2
*/
public Rectangle2D getBounds2D() {
double left = Math.min(Math.min(x1, x2), ctrlx);
double top = Math.min(Math.min(y1, y2), ctrly);
double right = Math.max(Math.max(x1, x2), ctrlx);
double bottom = Math.max(Math.max(y1, y2), ctrly);
return new Rectangle2D.Double(left, top,
right - left, bottom - top);
}
/*
* JDK 1.6 serialVersionUID
*/
private static final long serialVersionUID = 4217149928428559721L;
}
/**
* This is an abstract class that cannot be instantiated directly.
* Type-specific implementation subclasses are available for
* instantiation and provide a number of formats for storing
* the information necessary to satisfy the various accessor
* methods below.
*
* @see java.awt.geom.QuadCurve2D.Float
* @see java.awt.geom.QuadCurve2D.Double
* @since 1.2
*/
protected QuadCurve2D() {
}
/**
* Returns the X coordinate of the start point in
* <code>double</code> in precision.
* @return the X coordinate of the start point.
* @since 1.2
*/
public abstract double getX1();
/**
* Returns the Y coordinate of the start point in
* <code>double</code> precision.
* @return the Y coordinate of the start point.
* @since 1.2
*/
public abstract double getY1();
/**
* Returns the start point.
* @return a <code>Point2D</code> that is the start point of this
* <code>QuadCurve2D</code>.
* @since 1.2
*/
public abstract Point2D getP1();
/**
* Returns the X coordinate of the control point in
* <code>double</code> precision.
* @return X coordinate the control point
* @since 1.2
*/
public abstract double getCtrlX();
/**
* Returns the Y coordinate of the control point in
* <code>double</code> precision.
* @return the Y coordinate of the control point.
* @since 1.2
*/
public abstract double getCtrlY();
/**
* Returns the control point.
* @return a <code>Point2D</code> that is the control point of this
* <code>Point2D</code>.
* @since 1.2
*/
public abstract Point2D getCtrlPt();
/**
* Returns the X coordinate of the end point in
* <code>double</code> precision.
* @return the x coordinate of the end point.
* @since 1.2
*/
public abstract double getX2();
/**
* Returns the Y coordinate of the end point in
* <code>double</code> precision.
* @return the Y coordinate of the end point.
* @since 1.2
*/
public abstract double getY2();
/**
* Returns the end point.
* @return a <code>Point</code> object that is the end point
* of this <code>Point2D</code>.
* @since 1.2
*/
public abstract Point2D getP2();
/**
* Sets the location of the end points and control point of this curve
* to the specified <code>double</code> coordinates.
*
* @param x1 the X coordinate of the start point
* @param y1 the Y coordinate of the start point
* @param ctrlx the X coordinate of the control point
* @param ctrly the Y coordinate of the control point
* @param x2 the X coordinate of the end point
* @param y2 the Y coordinate of the end point
* @since 1.2
*/
public abstract void setCurve(double x1, double y1,
double ctrlx, double ctrly,
double x2, double y2);
/**
* Sets the location of the end points and control points of this
* <code>QuadCurve2D</code> to the <code>double</code> coordinates at
* the specified offset in the specified array.
* @param coords the array containing coordinate values
* @param offset the index into the array from which to start
* getting the coordinate values and assigning them to this
* <code>QuadCurve2D</code>
* @since 1.2
*/
public void setCurve(double[] coords, int offset) {
setCurve(coords[offset + 0], coords[offset + 1],
coords[offset + 2], coords[offset + 3],
coords[offset + 4], coords[offset + 5]);
}
/**
* Sets the location of the end points and control point of this
* <code>QuadCurve2D</code> to the specified <code>Point2D</code>
* coordinates.
* @param p1 the start point
* @param cp the control point
* @param p2 the end point
* @since 1.2
*/
public void setCurve(Point2D p1, Point2D cp, Point2D p2) {
setCurve(p1.getX(), p1.getY(),
cp.getX(), cp.getY(),
p2.getX(), p2.getY());
}
/**
* Sets the location of the end points and control points of this
* <code>QuadCurve2D</code> to the coordinates of the
* <code>Point2D</code> objects at the specified offset in
* the specified array.
* @param pts an array containing <code>Point2D</code> that define
* coordinate values
* @param offset the index into <code>pts</code> from which to start
* getting the coordinate values and assigning them to this
* <code>QuadCurve2D</code>
* @since 1.2
*/
public void setCurve(Point2D[] pts, int offset) {
setCurve(pts[offset + 0].getX(), pts[offset + 0].getY(),
pts[offset + 1].getX(), pts[offset + 1].getY(),
pts[offset + 2].getX(), pts[offset + 2].getY());
}
/**
* Sets the location of the end points and control point of this
* <code>QuadCurve2D</code> to the same as those in the specified
* <code>QuadCurve2D</code>.
* @param c the specified <code>QuadCurve2D</code>
* @since 1.2
*/
public void setCurve(QuadCurve2D c) {
setCurve(c.getX1(), c.getY1(),
c.getCtrlX(), c.getCtrlY(),
c.getX2(), c.getY2());
}
/**
* Returns the square of the flatness, or maximum distance of a
* control point from the line connecting the end points, of the
* quadratic curve specified by the indicated control points.
*
* @param x1 the X coordinate of the start point
* @param y1 the Y coordinate of the start point
* @param ctrlx the X coordinate of the control point
* @param ctrly the Y coordinate of the control point
* @param x2 the X coordinate of the end point
* @param y2 the Y coordinate of the end point
* @return the square of the flatness of the quadratic curve
* defined by the specified coordinates.
* @since 1.2
*/
public static double getFlatnessSq(double x1, double y1,
double ctrlx, double ctrly,
double x2, double y2) {
return Line2D.ptSegDistSq(x1, y1, x2, y2, ctrlx, ctrly);
}
/**
* Returns the flatness, or maximum distance of a
* control point from the line connecting the end points, of the
* quadratic curve specified by the indicated control points.
*
* @param x1 the X coordinate of the start point
* @param y1 the Y coordinate of the start point
* @param ctrlx the X coordinate of the control point
* @param ctrly the Y coordinate of the control point
* @param x2 the X coordinate of the end point
* @param y2 the Y coordinate of the end point
* @return the flatness of the quadratic curve defined by the
* specified coordinates.
* @since 1.2
*/
public static double getFlatness(double x1, double y1,
double ctrlx, double ctrly,
double x2, double y2) {
return Line2D.ptSegDist(x1, y1, x2, y2, ctrlx, ctrly);
}
/**
* Returns the square of the flatness, or maximum distance of a
* control point from the line connecting the end points, of the
* quadratic curve specified by the control points stored in the
* indicated array at the indicated index.
* @param coords an array containing coordinate values
* @param offset the index into <code>coords</code> from which to
* to start getting the values from the array
* @return the flatness of the quadratic curve that is defined by the
* values in the specified array at the specified index.
* @since 1.2
*/
public static double getFlatnessSq(double coords[], int offset) {
return Line2D.ptSegDistSq(coords[offset + 0], coords[offset + 1],
coords[offset + 4], coords[offset + 5],
coords[offset + 2], coords[offset + 3]);
}
/**
* Returns the flatness, or maximum distance of a
* control point from the line connecting the end points, of the
* quadratic curve specified by the control points stored in the
* indicated array at the indicated index.
* @param coords an array containing coordinate values
* @param offset the index into <code>coords</code> from which to
* start getting the coordinate values
* @return the flatness of a quadratic curve defined by the
* specified array at the specified offset.
* @since 1.2
*/
public static double getFlatness(double coords[], int offset) {
return Line2D.ptSegDist(coords[offset + 0], coords[offset + 1],
coords[offset + 4], coords[offset + 5],
coords[offset + 2], coords[offset + 3]);
}
/**
* Returns the square of the flatness, or maximum distance of a
* control point from the line connecting the end points, of this
* <code>QuadCurve2D</code>.
* @return the square of the flatness of this
* <code>QuadCurve2D</code>.
* @since 1.2
*/
public double getFlatnessSq() {
return Line2D.ptSegDistSq(getX1(), getY1(),
getX2(), getY2(),
getCtrlX(), getCtrlY());
}
/**
* Returns the flatness, or maximum distance of a
* control point from the line connecting the end points, of this
* <code>QuadCurve2D</code>.
* @return the flatness of this <code>QuadCurve2D</code>.
* @since 1.2
*/
public double getFlatness() {
return Line2D.ptSegDist(getX1(), getY1(),
getX2(), getY2(),
getCtrlX(), getCtrlY());
}
/**
* Subdivides this <code>QuadCurve2D</code> and stores the resulting
* two subdivided curves into the <code>left</code> and
* <code>right</code> curve parameters.
* Either or both of the <code>left</code> and <code>right</code>
* objects can be the same as this <code>QuadCurve2D</code> or
* <code>null</code>.
* @param left the <code>QuadCurve2D</code> object for storing the
* left or first half of the subdivided curve
* @param right the <code>QuadCurve2D</code> object for storing the
* right or second half of the subdivided curve
* @since 1.2
*/
public void subdivide(QuadCurve2D left, QuadCurve2D right) {
subdivide(this, left, right);
}
/**
* Subdivides the quadratic curve specified by the <code>src</code>
* parameter and stores the resulting two subdivided curves into the
* <code>left</code> and <code>right</code> curve parameters.
* Either or both of the <code>left</code> and <code>right</code>
* objects can be the same as the <code>src</code> object or
* <code>null</code>.
* @param src the quadratic curve to be subdivided
* @param left the <code>QuadCurve2D</code> object for storing the
* left or first half of the subdivided curve
* @param right the <code>QuadCurve2D</code> object for storing the
* right or second half of the subdivided curve
* @since 1.2
*/
public static void subdivide(QuadCurve2D src,
QuadCurve2D left,
QuadCurve2D right) {
double x1 = src.getX1();
double y1 = src.getY1();
double ctrlx = src.getCtrlX();
double ctrly = src.getCtrlY();
double x2 = src.getX2();
double y2 = src.getY2();
double ctrlx1 = (x1 + ctrlx) / 2.0;
double ctrly1 = (y1 + ctrly) / 2.0;
double ctrlx2 = (x2 + ctrlx) / 2.0;
double ctrly2 = (y2 + ctrly) / 2.0;
ctrlx = (ctrlx1 + ctrlx2) / 2.0;
ctrly = (ctrly1 + ctrly2) / 2.0;
if (left != null) {
left.setCurve(x1, y1, ctrlx1, ctrly1, ctrlx, ctrly);
}
if (right != null) {
right.setCurve(ctrlx, ctrly, ctrlx2, ctrly2, x2, y2);
}
}
/**
* Subdivides the quadratic curve specified by the coordinates
* stored in the <code>src</code> array at indices
* <code>srcoff</code> through <code>srcoff</code> + 5
* and stores the resulting two subdivided curves into the two
* result arrays at the corresponding indices.
* Either or both of the <code>left</code> and <code>right</code>
* arrays can be <code>null</code> or a reference to the same array
* and offset as the <code>src</code> array.
* Note that the last point in the first subdivided curve is the
* same as the first point in the second subdivided curve. Thus,
* it is possible to pass the same array for <code>left</code> and
* <code>right</code> and to use offsets such that
* <code>rightoff</code> equals <code>leftoff</code> + 4 in order
* to avoid allocating extra storage for this common point.
* @param src the array holding the coordinates for the source curve
* @param srcoff the offset into the array of the beginning of the
* the 6 source coordinates
* @param left the array for storing the coordinates for the first
* half of the subdivided curve
* @param leftoff the offset into the array of the beginning of the
* the 6 left coordinates
* @param right the array for storing the coordinates for the second
* half of the subdivided curve
* @param rightoff the offset into the array of the beginning of the
* the 6 right coordinates
* @since 1.2
*/
public static void subdivide(double src[], int srcoff,
double left[], int leftoff,
double right[], int rightoff) {
double x1 = src[srcoff + 0];
double y1 = src[srcoff + 1];
double ctrlx = src[srcoff + 2];
double ctrly = src[srcoff + 3];
double x2 = src[srcoff + 4];
double y2 = src[srcoff + 5];
if (left != null) {
left[leftoff + 0] = x1;
left[leftoff + 1] = y1;
}
if (right != null) {
right[rightoff + 4] = x2;
right[rightoff + 5] = y2;
}
x1 = (x1 + ctrlx) / 2.0;
y1 = (y1 + ctrly) / 2.0;
x2 = (x2 + ctrlx) / 2.0;
y2 = (y2 + ctrly) / 2.0;
ctrlx = (x1 + x2) / 2.0;
ctrly = (y1 + y2) / 2.0;
if (left != null) {
left[leftoff + 2] = x1;
left[leftoff + 3] = y1;
left[leftoff + 4] = ctrlx;
left[leftoff + 5] = ctrly;
}
if (right != null) {
right[rightoff + 0] = ctrlx;
right[rightoff + 1] = ctrly;
right[rightoff + 2] = x2;
right[rightoff + 3] = y2;
}
}
/**
* Solves the quadratic whose coefficients are in the <code>eqn</code>
* array and places the non-complex roots back into the same array,
* returning the number of roots. The quadratic solved is represented
* by the equation:
* <pre>
* eqn = {C, B, A};
* ax^2 + bx + c = 0
* </pre>
* A return value of <code>-1</code> is used to distinguish a constant
* equation, which might be always 0 or never 0, from an equation that
* has no zeroes.
* @param eqn the array that contains the quadratic coefficients
* @return the number of roots, or <code>-1</code> if the equation is
* a constant
* @since 1.2
*/
public static int solveQuadratic(double eqn[]) {
return solveQuadratic(eqn, eqn);
}
/**
* Solves the quadratic whose coefficients are in the <code>eqn</code>
* array and places the non-complex roots into the <code>res</code>
* array, returning the number of roots.
* The quadratic solved is represented by the equation:
* <pre>
* eqn = {C, B, A};
* ax^2 + bx + c = 0
* </pre>
* A return value of <code>-1</code> is used to distinguish a constant
* equation, which might be always 0 or never 0, from an equation that
* has no zeroes.
* @param eqn the specified array of coefficients to use to solve
* the quadratic equation
* @param res the array that contains the non-complex roots
* resulting from the solution of the quadratic equation
* @return the number of roots, or <code>-1</code> if the equation is
* a constant.
* @since 1.3
*/
public static int solveQuadratic(double eqn[], double res[]) {
double a = eqn[2];
double b = eqn[1];
double c = eqn[0];
int roots = 0;
if (a == 0.0) {
// The quadratic parabola has degenerated to a line.
if (b == 0.0) {
// The line has degenerated to a constant.
return -1;
}
res[roots++] = -c / b;
} else {
// From Numerical Recipes, 5.6, Quadratic and Cubic Equations
double d = b * b - 4.0 * a * c;
if (d < 0.0) {
// If d < 0.0, then there are no roots
return 0;
}
d = Math.sqrt(d);
// For accuracy, calculate one root using:
// (-b +/- d) / 2a
// and the other using:
// 2c / (-b +/- d)
// Choose the sign of the +/- so that b+d gets larger in magnitude
if (b < 0.0) {
d = -d;
}
double q = (b + d) / -2.0;
// We already tested a for being 0 above
res[roots++] = q / a;
if (q != 0.0) {
res[roots++] = c / q;
}
}
return roots;
}
/**
* {@inheritDoc}
* @since 1.2
*/
public boolean contains(double x, double y) {
double x1 = getX1();
double y1 = getY1();
double xc = getCtrlX();
double yc = getCtrlY();
double x2 = getX2();
double y2 = getY2();
/*
* We have a convex shape bounded by quad curve Pc(t)
* and ine Pl(t).
*
* P1 = (x1, y1) - start point of curve
* P2 = (x2, y2) - end point of curve
* Pc = (xc, yc) - control point
*
* Pq(t) = P1*(1 - t)^2 + 2*Pc*t*(1 - t) + P2*t^2 =
* = (P1 - 2*Pc + P2)*t^2 + 2*(Pc - P1)*t + P1
* Pl(t) = P1*(1 - t) + P2*t
* t = [0:1]
*
* P = (x, y) - point of interest
*
* Let's look at second derivative of quad curve equation:
*
* Pq''(t) = 2 * (P1 - 2 * Pc + P2) = Pq''
* It's constant vector.
*
* Let's draw a line through P to be parallel to this
* vector and find the intersection of the quad curve
* and the line.
*
* Pq(t) is point of intersection if system of equations
* below has the solution.
*
* L(s) = P + Pq''*s == Pq(t)
* Pq''*s + (P - Pq(t)) == 0
*
* | xq''*s + (x - xq(t)) == 0
* | yq''*s + (y - yq(t)) == 0
*
* This system has the solution if rank of its matrix equals to 1.
* That is, determinant of the matrix should be zero.
*
* (y - yq(t))*xq'' == (x - xq(t))*yq''
*
* Let's solve this equation with 't' variable.
* Also let kx = x1 - 2*xc + x2
* ky = y1 - 2*yc + y2
*
* t0q = (1/2)*((x - x1)*ky - (y - y1)*kx) /
* ((xc - x1)*ky - (yc - y1)*kx)
*
* Let's do the same for our line Pl(t):
*
* t0l = ((x - x1)*ky - (y - y1)*kx) /
* ((x2 - x1)*ky - (y2 - y1)*kx)
*
* It's easy to check that t0q == t0l. This fact means
* we can compute t0 only one time.
*
* In case t0 < 0 or t0 > 1, we have an intersections outside
* of shape bounds. So, P is definitely out of shape.
*
* In case t0 is inside [0:1], we should calculate Pq(t0)
* and Pl(t0). We have three points for now, and all of them
* lie on one line. So, we just need to detect, is our point
* of interest between points of intersections or not.
*
* If the denominator in the t0q and t0l equations is
* zero, then the points must be collinear and so the
* curve is degenerate and encloses no area. Thus the
* result is false.
*/
double kx = x1 - 2 * xc + x2;
double ky = y1 - 2 * yc + y2;
double dx = x - x1;
double dy = y - y1;
double dxl = x2 - x1;
double dyl = y2 - y1;