B.2.1. Elliptical arc endpoint syntax

An elliptical arc, as represented in the SVG path command,is described by the following parameters in order:

(x₁, y₁) are the absolute coordinates of thecurrent point on the path, obtained from the last twoparameters of the previous path command.

r_xandr_yare the radii of the ellipse (also known as its semi-major andsemi-minor axes).

φ is the angle from the x-axis of the current coordinatesystem to the x-axis of the ellipse.

f_A isthe large arc flag, and is 0if an arc spanning less than or equal to 180 degrees is chosen, or 1if an arc spanning greater than 180 degrees is chosen.

f_S isthe sweep flag, and is 0 ifthe line joining center to arc sweeps through decreasingangles, or 1 if it sweepsthrough increasing angles.

(x₂, y₂) are the absolute coordinates of thefinal point of the arc.

B.2.2. Parameterization alternatives

An arbitrary point (x, y) on the ellipticalarc can be described by the 2-dimensional matrix equation:

    x = rx*cos(θ)*cos(φ) - ry*sin(θ)*sin(φ) + cx    y = rx*cos(θ)*sin(φ) + ry*sin(θ)*cos(φ) + cy

(c_x, c_y) arethe coordinates of the center of the ellipse.

r_x andr_yare the radii of the ellipse (also known as its semi-major andsemi-minor axes).

If one thinks of an ellipse as a circle that has beenstretched and then rotated, thenθ₁,θ₂ and Δθare the start angle, end angle and sweep angle, respectivelyof the arc prior to the stretch and rotate operations.This leads to an alternate parameterization which is commonamong graphics APIs, which will be referred to ascenterparameterization. In the next sections, formulas aregiven for mapping in both directions between centerparameterization and endpoint parameterization.

B.2.3. Conversion from center to endpoint parameterization

Given the following variables:

c_xc_yr_xr_yφθ₁ Δθ

the task is to find:

x₁y₁x₂y₂f_Af_S

This can be achieved using the following formulas:

B.2.4. Conversion from endpoint to center parameterization

Given the following variables:

x₁y₁x₂y₂f_Af_Sr_xr_yφ

the task is to find:

c_xc_yθ₁ Δθ

The equations simplify after a translation whichplaces the origin at the midpoint of the line joining(x₁, y₁) to(x₂, y₂), followed bya rotation to line up the coordinate axes with the axes of the ellipse.All transformed coordinates will be written with primes. They arecomputed as intermediate values on the way toward finding the requiredcenter parameterization variables. This procedure consists of thefollowing steps:

• Step 1: Compute (x₁′, y₁′)

  $$\left(\begin{array}{c}{x_{\mathit{1}}}^{\mathit{\prime }}\\ {y_{\mathit{1}}}^{\mathit{\prime }}\end{array}\right)=\left(\begin{array}{cc}\cos \mathit{\phi }& \sin \mathit{\phi }\\ -\sin \mathit{\phi }& \cos \mathit{\phi }\end{array}\right)\cdot \left(\begin{array}{c}\frac{x_{\mathit{1}}\mathit{-}{x}_{\mathit{2}}}{\mathit{2}}\\ \frac{y_{\mathit{1}}\mathit{-}{y}_{\mathit{2}}}{\mathit{2}}\end{array}\right)$$

  (eq. 5.1)

• Step 2: Compute (c_x′, c_y′)

  $$\left(\begin{array}{c}{c_x}^{\prime }\\ {c_y}^{\prime }\end{array}\right)=\pm \sqrt{\frac{{r_x}^2{r_y}^2-{r_x}^2{\left({y_1}^{\prime }\right)}^2-{r_y}^2{\left({x_1}^{\prime }\right)}^2}{{r_x}^2{\left({y_1}^{\prime }\right)}^2+{r_y}^2{\left({x_1}^{\prime }\right)}^2}}\left(\begin{array}{c}\frac{r_x{y_1}^{\prime }}{r_y}\\ -\frac{r_y{x_1}^{\prime }}{r_x}\end{array}\right)$$

  (eq. 5.2)

  where the + sign is chosen iff_A ≠ f_S, and the − sign is chosen iff_A = f_S.

• Step 3: Compute (c_x, c_y)from (c_x′, c_y′)

  $$\left(\begin{array}{c}{c}_x\\ c_y\end{array}\right)=\left(\begin{array}{cc}\cos \mathit{\phi }& -\sin \mathit{\phi }\\ \sin \mathit{\phi }& \cos \mathit{\phi }\end{array}\right)\cdot \left(\begin{array}{c}{c_x}^{\text{'}}\\ {c_y}^{\text{'}}\end{array}\right)+\left(\begin{array}{c}\frac{x_1+x_2}{2}\\ \frac{y_1+y_2}{2}\end{array}\right)$$

  (eq. 5.3)

• Step 4: Computeθ₁ and Δθ

  In general, the angle between two vectors (u_x, u_y) and (v_x, v_y) can be computed as

  $$\angle \left(\begin{array}{ccc}\stackrel{\rightharpoonup }{u}& \mathrm{,}& \stackrel{\rightharpoonup }{v}\end{array}\right)=\pm \arccos \left(\frac{\stackrel{\rightharpoonup }{u}\cdot \stackrel{\rightharpoonup }{v}}{‖\stackrel{\rightharpoonup }{u}‖ ‖\stackrel{\rightharpoonup }{v}‖}\right)$$

  (eq. 5.4)

  where the ± sign appearing here is the sign ofu_x v_y − u_y v_x.

  This angle function can be used to expressθ₁ and Δθ as follows:

  $${\mathit{\theta }}_1=\angle \left(\begin{array}{ccc}\left(\begin{array}{c}1\\ 0\end{array}\right)& \mathit{,}& \left(\begin{array}{c}\frac{{x_{\mathit{1}}}^{\prime }-{c_x}^{\prime }}{r_x}\\ \frac{{y_{\mathit{1}}}^{\prime }-{c_y}^{\prime }}{r_y}\end{array}\right)\end{array}\right)$$

  (eq. 5.5)

  $$\Delta \mathit{\theta }\equiv \angle \left(\begin{array}{ccc}\left(\begin{array}{c}\frac{{x_{\mathit{1}}}^{\prime }-{c_x}^{\prime }}{r_x}\\ \frac{{y_{\mathit{1}}}^{\prime }-{c_y}^{\prime }}{r_y}\end{array}\right)& \mathit{,}& \left(\begin{array}{c}\frac{{-x_{\mathit{1}}}^{\prime }-{c_x}^{\prime }}{r_x}\\ \frac{{-y_{\mathit{1}}}^{\prime }-{c_y}^{\prime }}{r_y}\end{array}\right)\end{array}\right)\mathit{ }\mathrm{mod\; 360}°$$

  (eq. 5.6)

  where Δθ is fixed in the range −360° < Δθ < 360° such that:

  iff_S = 0, then Δθ < 0,

  else iff_S = 1, then Δθ > 0.

  In other words, iff_S = 0 and the right side of (eq. 5.6) is greater than 0, then subtract 360°, whereas iff_S = 1 and the right side of (eq. 5.6) is less than 0, then add 360°. In all other cases leave it as is.

B.2.5. Correction of out-of-range radii

This section describes the mathematical adjustments required for out-of-ranger_x andr_y,as described in thePath implementation notes.Algorithmically these adjustments consist of the followingsteps:

• Step 1: Ensure radii are non-zero

  Ifr_x = 0 orr_y = 0, then treat this as a straight line from (x₁, y₁) to (x₂, y₂) and stop. Otherwise,

• Step 2: Ensure radii are positive

  Take the absolute value ofr_x andr_y:

  $$\begin{array}{l}{r}_y\leftarrow \left|r_y\right|\\ \\ r_y\leftarrow \left|r_y\right|\end{array}$$

  (eq. 6.1)

• Step 3: Ensure radii are large enough

  Using the primed coordinate values of equation (eq. 5.1), compute

  $$\mathit{\Lambda }=\frac{{\left({x_{\mathit{1}}}^{\prime }\right)}^{\mathit{2}}}{{r_x}^{\mathit{2}}}+\frac{{\left({y_{\mathit{1}}}^{\prime }\right)}^{\mathit{2}}}{{r_y}^{\mathit{2}}}$$

  (eq. 6.2)

  If the result of the above equation is less than or equal to 1, then no further change need be made tor_x andr_y. If the result of the above equation is greater than 1, then make the replacements

  $$\begin{array}{l}{r}_x\leftarrow \sqrt{\mathit{\Lambda }}\mathit{ }{r}_x\\ \\ r_y\leftarrow \sqrt{\mathit{\Lambda }}\mathit{ }{r}_y\end{array}$$

  (eq. 6.3)

• Step 4: Proceed with computations

  Proceed with the remaining elliptical arc computations, such as those in theConversion from endpoint to center parameterization algorithm. Note: As a consequence of the radii corrections in this section, the equation (5.2) for the center of the ellipse always has at least one solution (i.e. the radicand is never negative). In the case that the radii are scaled up using equation (eq. 6.3), the radicand of (eq. 5.2) is zero and there is exactly one solution for the center of the ellipse.