SourceClass CubicBezierShape ⧉

Find out the t value for the point where the curve is intersected with the base line. The base line is the line from P0 to P3. If the curve only has two intersection points with the base line, they should be 0.0 and 1.0. In this case, the "fill" will be simple since the curve is a convex line. If the curve has more than two intersection points with the base line, the "fill" will be a problem. We need to find out where is the 3rd t value (0<t<1) And the original cubic curve will be split into two curves (0.0..t and t..1.0). B(t) = (1-t)^3P0 + 3t(1-t)^2P1 + 3t^2(1-t)P2 + t^3P3 or B(t) = (P3 - 3P2 + 3P1 - P0)t^3 + (3P2 - 6P1 + 3P0)t^2 + (3P1 - 3P0)t + P0 this B(t) should be on the line between P0 and P3. Therefore: (B.x - P0.x)/(P3.x - P0.x) = (B.y - P0.y)/(P3.y - P0.y), or: B.x * (P3.y - P0.y) - B.y * (P3.x - P0.x) + P0.x * (P0.y - P3.y) + P0.y * (P3.x - P0.x) = 0 B.x = (P3.x - 3 * P2.x + 3 * P1.x - P0.x) * t^3 + (3 * P2.x - 6 * P1.x + 3 * P0.x) * t^2 + (3 * P1.x - 3 * P0.x) * t + P0.x B.y = (P3.y - 3 * P2.y + 3 * P1.y - P0.y) * t^3 + (3 * P2.y - 6 * P1.y + 3 * P0.y) * t^2 + (3 * P1.y - 3 * P0.y) * t + P0.y Combine the above three equations and iliminate B.x and B.y, we get:

or a * t^3 + b * t^2 + c * t + d = 0

let x = t - b / (3 * a), then we have:

when p > 0, there will be one real root, two complex roots when p = 0, there will be two real roots, when p=q=0, there will be three real roots but all 0. when p < 0, there will be three unique real roots. this is what we need. (x1, x2, x3) t = x + b / (3 * a), then we have: t1, t2, t3. the one between 0.0 and 1.0 is what we need. <https://baike.baidu.com/item/%E4%B8%80%E5%85%83%E4%B8%89%E6%AC%A1%E6%96%B9%E7%A8%8B/8388473 />

find a set of points that approximate the cubic Bézier curve. the number of points is determined by the tolerance. the points may not be evenly distributed in the range [0.0,1.0] (t value)

find a set of points that approximate the cubic Bézier curve. the number of points is determined by the tolerance. the points may not be evenly distributed in the range [0.0,1.0] (t value) this api will check whether the curve will cross the base line or not when closed = true. The result will be a vec of vec of Pos2. it will store two closed aren in different vec. The epsilon is used to compare a float value.

Iterates through the curve invoking a callback at each point.

Creates a cubic Bézier curve based on 4 points and stroke.

The first point is the starting point and the last one is the ending point of the curve. The middle points are the control points.

Logical bounding rectangle (ignoring stroke width)

Calculate the point (x,y) at t based on the cubic Bézier curve equation. t is in [0.0,1.0] Bézier Curve

split the original cubic curve into a new one within a range.

Convert the cubic Bézier curve to one or two PathShape's. When the curve is closed and it has to intersect with the base line, it will be converted into two shapes. Otherwise, it will be converted into one shape. The tolerance will be used to control the max distance between the curve and the base line. The epsilon is used when comparing two floats.

Transform the curve with the given transform.

The visual bounding rectangle (includes stroke width)