B-スプラインをベジェ曲線に変換して描画する（その1 OpenとClosed）

\boldsymbol{S}(t)=\sum_{i=0}^{m-1}b_{i,n}(t)\boldsymbol{P}_i

\begin{align}
&b_{i,0}(t) = 
\left
\{
\begin{array}{ll}
1 & if \quad t_i \leq t < t_{i+1} \\
0 & otherwise
\end{array}
\right. \\
&b_{i,k}(t) =
\frac{t-t_i}{t_{i+k}-t_i} b_{i,k-1}(t) +
\frac{t_{i+k+1}-t}{t_{i+k+1}-t_{i+1}}  b_{i+1,k-1}(t)
\end{align}

\boldsymbol{B}(t)=\sum_{i=0}^{n}b_{i,n}(t)\boldsymbol{P}_i \quad , \qquad 0\leq t<1

b_{i,n}(t)=
\begin{pmatrix}
n \\
i
\end{pmatrix}
t^i(1-t)^{n-i} \quad , \qquad t=0, \cdots ,n

var ('t')
t0=-3
t1=-2
t2=-1
t3=0
t4=1
t5=2
t6=3
t7=4
b20=0
b30=1
b40=0
b11=0
b21=(t-t2)/(t3-t2)*b20+(t4-t)/(t4-t3)*b30
b31=(t-t3)/(t4-t3)*b30+(t5-t)/(t5-t4)*b40
b41=0
b02=0
b12=(t-t1)/(t3-t1)*b11+(t4-t)/(t4-t2)*b21
b22=(t-t2)/(t4-t2)*b21+(t5-t)/(t5-t3)*b31
b32=(t-t3)/(t5-t3)*b31+(t6-t)/(t6-t4)*b41
b42=0
b03=(t-t0)/(t3-t0)*b02+(t4-t)/(t4-t1)*b12
b13=(t-t1)/(t4-t1)*b12+(t5-t)/(t5-t2)*b22
b23=(t-t2)/(t5-t2)*b22+(t6-t)/(t6-t3)*b32
b33=(t-t3)/(t6-t3)*b32+(t7-t)/(t7-t4)*b42
b03.coefficients(t)
b13.coefficients(t)
b23.coefficients(t)
b33.coefficients(t)

[[1/6, 0], [-1/2, 1], [1/2, 2], [-1/6, 3]]
[[2/3, 0], [-1, 2], [1/2, 3]]
[[1/6, 0], [1/2, 1], [1/2, 2], [-1/2, 3]]
[[1/6, 3]]

var ('t')
b03=binomial(3,0)*t^0*(1-t)^(3-0)
b13=binomial(3,1)*t^1*(1-t)^(3-1)
b23=binomial(3,2)*t^2*(1-t)^(3-2)
b33=binomial(3,3)*t^3*(1-t)^(3-3)
b03.coefficients(t)
b13.coefficients(t)
b23.coefficients(t)
b33.coefficients(t)

[[1, 0], [-3, 1], [3, 2], [-1, 3]]
[[3, 1], [-6, 2], [3, 3]]
[[3, 2], [-3, 3]]
[[1, 3]]

\boldsymbol{S}(t)=
\begin{bmatrix}
t^3 & t^2 & t & 1
\end{bmatrix}
\boldsymbol{R_{bsp}}
\begin{bmatrix}
{\boldsymbol{P_{bsp}}}_0 \\
{\boldsymbol{P_{bsp}}}_1 \\
{\boldsymbol{P_{bsp}}}_2 \\
{\boldsymbol{P_{bsp}}}_3
\end{bmatrix}

\boldsymbol{R_{bsp}}=
\frac{1}{6}
\begin{bmatrix}
-1 & 3 & -3 & 1 \\
3 & -6 & 3 & 0 \\
-3 & 0 & 3 & 0 \\
1 & 4 & 1 & 0
\end{bmatrix}

\boldsymbol{B}(t)=
\begin{bmatrix}
t^3 & t^2 & t & 1
\end{bmatrix}
\boldsymbol{R_{bz}}
\begin{bmatrix}
{\boldsymbol{P_{bz}}}_0 \\
{\boldsymbol{P_{bz}}}_1 \\
{\boldsymbol{P_{bz}}}_2 \\
{\boldsymbol{P_{bz}}}_3
\end{bmatrix}

\boldsymbol{R_{bz}}=
\begin{bmatrix}
-1 & 3 & -3 & 1 \\
3 & -6 & 3 & 0 \\
-3 & 3 & 0 & 0 \\
1 & 0 & 0 & 0
\end{bmatrix}

\boldsymbol{B}(t)=
\boldsymbol{S}(t)

\boldsymbol{R_{bz}}
\begin{bmatrix}
{\boldsymbol{P_{bz}}}_0 \\
{\boldsymbol{P_{bz}}}_1 \\
{\boldsymbol{P_{bz}}}_2 \\
{\boldsymbol{P_{bz}}}_3
\end{bmatrix}
=
\boldsymbol{R_{bsp}}
\begin{bmatrix}
{\boldsymbol{P_{bsp}}}_0 \\
{\boldsymbol{P_{bsp}}}_1 \\
{\boldsymbol{P_{bsp}}}_2 \\
{\boldsymbol{P_{bsp}}}_3
\end{bmatrix}

\begin{bmatrix}
{\boldsymbol{P_{bz}}}_0 \\
{\boldsymbol{P_{bz}}}_1 \\
{\boldsymbol{P_{bz}}}_2 \\
{\boldsymbol{P_{bz}}}_3
\end{bmatrix}
=
\boldsymbol{R_{bz}}^{-1}
\boldsymbol{R_{bsp}}
\begin{bmatrix}
{\boldsymbol{P_{bsp}}}_0 \\
{\boldsymbol{P_{bsp}}}_1 \\
{\boldsymbol{P_{bsp}}}_2 \\
{\boldsymbol{P_{bsp}}}_3
\end{bmatrix}

\boldsymbol{R_{bz}}^{-1}
\boldsymbol{R_{bsp}}=
\frac{1}{6}
\begin{bmatrix}
1 & 4 & 1 & 0 \\
0 & 4 & 2 & 0 \\
0 & 2 & 4 & 0 \\
0 & 1 & 4 & 1
\end{bmatrix}

private:
	QGraphicsScene scene;
	QPainterPath pp;
	QList<QPointF> cps;	// コントロールポイント
	int m;				// コントロールポイント数
	bool bClosed;
	void drawBsp();
	void drawBspSegment(QPointF p0, QPointF p1, QPointF p2, QPointF p3);

// B-スプラインを描画する
void MainWindow::drawBsp()
{
	pp.clear();
	for(int i = 0; i < m - 3; i++)
		drawBspSegment(cps[i], cps[i + 1], cps[i + 2], cps[i + 3]);
	if(bClosed){
		drawBspSegment(cps[m - 3], cps[m - 2], cps[m - 1], cps[0]);
		drawBspSegment(cps[m - 2], cps[m - 1], cps[0], cps[1]);
		drawBspSegment(cps[m - 1], cps[0], cps[1], cps[2]);
	}
	scene.addPath(pp);
}

// B-スプラインの1セグメントを描画する
void MainWindow::drawBspSegment(QPointF p0, QPointF p1, QPointF p2, QPointF p3)
{
	if(pp.isEmpty()){
		// 最初のセグメントのみ
		QPointF bzp0 = (p0 + 4 * p1 + p2) / 6;
		pp.moveTo(bzp0);
	}else{
		// bzp0は前回のbzp3と一致するので、moveTo不要
	}
	QPointF bzp1 = (4 * p1 + 2 * p2) / 6;
	QPointF bzp2 = (2 * p1 + 4 * p2) / 6;
	QPointF bzp3 = (p1 + 4 * p2 + p3) / 6;
	pp.cubicTo(bzp1, bzp2, bzp3);
}