Biarc

1. In the below examples biarcs $A(J)B$ are subtended by the chord $AB,$ and $J$ is the join point. Tangent vector at the start point $A$ is $\backslash mathbf\{n\}(\backslash alpha)$, and $\backslash mathbf\{n\}(\backslash beta)$ is the tangent at the end point $B:$

   {{NumBlk||$$\backslash mathbf\{n\}(\backslash alpha)\; =\; \backslash left\backslash |\backslash cos\backslash alpha,\backslash ,\backslash sin\backslash alpha\backslash right\backslash |^T,\backslash quad$$

   \mathbf{n}(\beta) = \left\|\cos\beta,\,\sin\beta\right\|^T.|{{EquationRef|1}}}}

2. Fig. 2 shows six examples of biarcs $AJB.$

   • Biarc 1 is drawn with $\backslash alpha=100^\backslash circ,\backslash ;\backslash beta=30^\backslash circ.$ Biarcs 2-6 have $\backslash alpha=100^\backslash circ,\backslash ;\backslash beta=-30^\backslash circ.$
   • In examples 1, 2, 6 curvature changes sign, and the join point $J$ is also the inflection point. Biarc 3 includes the straight line segment $JB$.
   • Biarcs 1–4 are short in the sense that they do not turn near endpoints. Alternatively, biarcs 5,6 are long: turning near one of endpoints means that they intersect the left or the right complement of the chord to the infinite straight line.
   • Biarcs 2–6 share end tangents. They can be found in the lower fragment of Fig. 3, among the family of biarcs with common tangents.

3. Fig. 3 shows two examples of biarc families, sharing end points and end tangents.

4. Fig. 4 shows two examples of biarc families, sharing end points and end tangents, end tangents being parallel: $\backslash alpha=\backslash beta.$

5. Fig. 5 shows specific families with either {{nowrap|$|\backslash alpha|\; =\; \backslash pi$ or $|\backslash beta|\; =\; \backslash pi.$}}

align="left"

valign="bottom" |File:Biarcs_6_Examples.png |File:BiarcsWiki1.png |File:BiarcsWiki2.png

{{clear}}

File:BiarcsWikiPi.png

Different colours in figures 3, 4, 5 are explained below as subfamilies

$\backslash color\{sienna\}\backslash mathcal\{B\}^\{\backslash ,+\}$,

$\backslash color\{blue\}\backslash mathcal\{B\}^\{\backslash ,-\}\_1$,

$\backslash color\{green\}\backslash mathcal\{B\}^\{\backslash ,-\}\_2$.

In particular, for biarcs, shown in brown on shaded background (lens-like or lune-like), the following holds:

• the total rotation (turning angle) of the curve is exactly $\backslash beta-\backslash alpha$ (not $\backslash beta-\backslash alpha\backslash pm\; 2\backslash pi$, which is the rotation for other biarcs);
• $\backslash sgn(\backslash alpha+\backslash beta)=\backslash sgn(k\_2-k\_1)$: the sum $\backslash alpha+\backslash beta$ is the angular width of the lens/lune, covering the biarc, whose sign corresponds to either increasing (+1) or decreasing curvature (−1) of the biarc, according to generalized Vogt's theorem ({{ill|Теорема Фогта#Обобщение теоремы|ru|vertical-align=sup}}).

A family of biarcs with common end points $A\; =\; (-c,0)$, $B\; =\; (c,0)$, and common end tangents (1) is denoted as $\backslash mathcal\{B\}(p;\backslash ,\backslash alpha,\backslash beta,c),$ or, briefly, as $\backslash mathcal\{B\}(p),$ $p$ being the family parameter. Biarc properties are described below in terms of article.{{cite journal | last=Kurnosenko|first=A. I.| year=2013| title=Biarcs and bilens| journal=Computer Aided Geometric Design| volume=30| issue=3| pages=310–330| doi=10.1016/j.cagd.2012.12.002| url=https://www.researchgate.net/publication/257014166| format=PDF}}

1. Constructing of a biarc is possible if

   {{NumBlk|||{{EquationRef|2}}}}

2. Denote

   • $k\_1$, $\backslash theta\_1$ and $L\_1$  the curvature, the turning angle and the length of the arc $AJ$:    $\backslash theta\_1\; =\; k\_1\; L\_1$;
   • $k\_2$, $\backslash theta\_2$ and $L\_2$  the same for the arc $JB$:    $\backslash theta\_2=k\_2L\_2$.

   Then

   $$k\_1(p)\; =\; -\backslash frac\; 1\; c\; \backslash left(\backslash sin\backslash alpha+p^\{-1\}\backslash sin\backslash omega\backslash right),\backslash quad\; k\_2(p)\; =\; \backslash frac\; 1\; c\; \backslash left(\backslash sin\backslash beta+p\backslash sin\backslash omega\backslash right),\backslash quad\backslash text\{where\}\backslash quad\; \backslash omega=\backslash frac\{\backslash alpha+\backslash beta\}2$$

   (due to {{EquationNote|2|(2)}}, $\backslash sin\backslash omega\backslash not=0$).

   Turning angles:

   $$$$

   \theta_1(p)=2\arg\left(e^{-i\alpha} +p^{-1}{e^{-i\omega}}\right),\quad

   \theta_2(p)=2\arg\left(e^{ i\beta } +p\, e^{i\omega}\right).

3. The locus of join points $J$ is the circle

   $$$$

   X_J(p)=\frac{c(p^2-1)}{{p^2+2p\cos\gamma+1}},\quad

   Y_J(p)=\frac{2cp\sin\gamma}{p^2+2p\cos\gamma+1},\quad\text{where}\quad

   \gamma=\frac{\alpha-\beta}2

   (shown dashed in Fig.3, Fig.5).

   This circle (straight line if $\backslash gamma=0$, Fig.4) passes through points $A,B,$ the tangent at $A$ being $\backslash mathbf\{n\}(\backslash gamma).$

   Biarcs intersect this circle under the constant angle  $-\backslash omega.$

4. Tangent vector to the biarc $\backslash mathcal\{B\}(p)$ at the join point is $\backslash mathbf\{n\}\backslash left(\backslash tau\_\{\{\}\_J\}\backslash right)$, where

   $$$$

   \tau_{\scriptscriptstyle J}(p)={-2} \arctan \dfrac{p\sin\frac{\alpha}2+\sin\frac{\beta}2}{p\cos\frac{\alpha}2+\cos\frac{\beta}2}.

5. Biarcs with $p\; =\; \backslash pm1$ have the join point on the Y-axis $(X\_J=0),$ and yield the minimal curvature jump, $\backslash min\backslash left|k\_2(p)-k\_1(p)\backslash right|,$ at $J.$

6. Degenerate biarcs are:

   • Biarc $\backslash mathcal\{B\}(0)$: as $p\; \backslash to\; 0$, $J(p)\; \backslash to\; A$, arc $AJ$ vanishes.
   • Biarc $\backslash mathcal\{B\}(\backslash infty)$: as $p\; \backslash to\; \backslash pm\backslash infty$, $J(p)\; \backslash to\; B$, arc $JB$ vanishes.
   • Discontinuous biarc $\backslash mathcal\{B\}(p^\backslash ast)$ includes straight line $AP\_\backslash infty\; J$ or $JP\_\backslash infty\; B,$ and passes through the infinite point $P\_\backslash infty$: $$$$

   p^\ast=

   \begin{cases}

   -\dfrac{\sin\omega}{\sin\alpha}, & \text{if}\; |\alpha|\geqslant|\beta|\quad(|\alpha|=\pi\;\Longrightarrow\; p^\ast=-\infty),\\[1ex]

   -\dfrac{\sin\beta}{\sin\omega}, & \text{if}\; |\alpha|\leqslant|\beta|\quad(|\beta|=\pi\;\Longrightarrow\; p^\ast=0).

   \end{cases}

   Darkened lens-like region in Figs.3,4 is bounded by biarcs $\backslash mathcal\{B\}(0),\backslash ,\backslash mathcal\{B\}(\backslash infty).$ It covers biarcs with $p>0.$

   Discontinuous biarc is shown by red dash-dotted line.

7. The whole family $\backslash mathcal\{B\}(p;\backslash ,\backslash alpha,\backslash beta,c)$ can be subdivided into three subfamilies of non-degenerate biarcs:

   $$$$

   \begin{array}{l}

   \mathcal{B}^{\,+}(p){:}\quad p\in(0;\infty);\\

   \mathcal{B}^{\,-}_1(p){:}\quad p\in(p^\ast;0);\\

   \mathcal{B}^{\,-}_2(p){:}\quad p\in(-\infty;p^\ast);\\

   \left[\mathcal{B}^{\,-}(p) = \mathcal{B}^{\,-}_1(p) \cup \mathcal{B}^{\,-}_2(p)\right].

   \end{array}

   Subfamily $\backslash mathcal\{B\}^\{\backslash ,-\}\_1$ vanishes if $p^\backslash ast=0$   $(|\backslash beta|=\backslash pi).$

   Subfamily $\backslash mathcal\{B\}^\{\backslash ,-\}\_2$ vanishes if $p^\backslash ast=-\backslash infty$ $(|\backslash alpha|=\backslash pi).$

   In figures 3, 4, 5

   biarcs $\backslash color\{sienna\}\backslash mathcal\{B\}^\{\backslash ,+\}$ are shown in brown,

   biarcs $\backslash color\{blue\}\backslash mathcal\{B\}^\{\backslash ,-\}\_1$ in blue,

   and biarcs $\backslash color\{green\}\backslash mathcal\{B\}^\{\backslash ,-\}\_2$ in green.

{{Reflist}}

• {{cite book|last1=Nutbourne|first1=A. W.|last2=Martin|first2=R. R.|title=Differential geometry applied to curve and surface design. Vol.1: Foundations|publisher=Ellis Horwood|year=1988|isbn=978-0132118224}}

• [https://www.ryanjuckett.com/biarc-interpolation/ Biarc Interpolation]
• [https://www.redblobgames.com/articles/curved-paths/#biarcs Interactive visualization and link list]

Category:Piecewise-circular curves

Category:Splines (mathematics)

Category:Circles