F.6.1 Elliptical arc syntax

An elliptical arc is a particular path command. As such, it is described by the following parameters in order:

(x₁, y₁) are the absolute coordinates of the current point on the path, obtained from the last two parameters of the previous path command.

r_x and r_y are the radii of the ellipse (also known as its semi-major and semi-minor axes).

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

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

f_S is the sweep flag, and is 0 if the line joining center to arc sweeps through decreasing angles, or 1 if it sweeps through increasing angles.

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

This parameterization of elliptical arcs will be referred to as endpoint parameterization. One of the advantages of endpoint parameterization is that it permits a consistent path syntax in which all path commands end in the coordinates of the new "current point". The following notes give rules and formulas to help implementers deal with endpoint parameterization.

F.6.2 Out-of-range parameters

Arbitrary numerical values are permitted for all elliptical arc parameters, but where these values are invalid or out-of-range, an implementation must make sense of them as follows:

If the endpoints (x₁, y₁) and (x₂, y₂) are identical, then this is equivalent to omitting the elliptical arc segment entirely.

If r_x = 0 or r_y = 0 then this arc is treated as a straight line segment (a "lineto") joining the endpoints.

If r_x or r_y have negative signs, these are dropped; the absolute value is used instead.

If r_x, r_y and φ are such that there is no solution (basically, the ellipse is not big enough to reach from (x₁, y₁) to (x₂, y₂)) then the ellipse is scaled up uniformly until there is exactly one solution (until the ellipse is just big enough).

φ is taken mod 360 degrees.

Any nonzero value for either of the flags f_A or f_S is taken to mean the value 1.

This forgiving yet consistent treatment of out-of-range values ensures that:

• The inevitable approximations arising from computer arithmetic cannot cause a valid set of values written by one SVG implementation to be treated as invalid when read by another SVG implementation. This would otherwise be a problem for common boundary cases such as a semicircular arc.
• Continuous animations that cause parameters to pass through invalid values are not a problem. The motion remains continuous.

F.6.3 Parameterization alternatives

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

(F.6.3.1)

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

r_x and r_y are the radii of the ellipse (also known as its semi-major and semi-minor axes).

θ is the angle from the x-axis of the current coordinate system to the x-axis of the ellipse.

θ ranges from:

• θ₁ which is the start angle of the elliptical arc prior to the stretch and rotate operations.
• θ₂ which is the end angle of the elliptical arc prior to the stretch and rotate operations.
• Δθ which is the difference between these two angles.

If one thinks of an ellipse as a circle that has been stretched and then rotated, then θ₁, θ₂ and Δθ are the start angle, end angle and sweep angle, respectively of the arc prior to the stretch and rotate operations. This leads to an alternate parameterization which is common among graphics APIs, which will be referred to as center parameterization. In the next sections, formulas are given for mapping in both directions between center parameterization and endpoint parameterization.

F.6.4 Conversion from center to endpoint parameterization

Given the following variables:

c_x c_y r_x r_y φ θ₁ Δθ

the task is to find:

x₁ y₁ x₂ y₂ f_A f_S

This can be achieved using the following formulas:

	(F.6.4.1)
	(F.6.4.2)
	(F.6.4.3)
	(F.6.4.4)

F.6.5 Conversion from endpoint to center parameterization

Given the following variables:

x₁ y₁ x₂ y₂ f_A f_S r_x r_y φ

the task is to find:

c_x c_y θ₁ Δθ

The equations simplify after a translation which places the origin at the midpoint of the line joining (x₁, y₁) to (x₂, y₂), followed by a rotation to line up the coordinate axes with the axes of the ellipse. All transformed coordinates will be written with primes. They are computed as intermediate values on the way toward finding the required center parameterization variables. This procedure consists of the following steps:

• Step 1: Compute (x₁′, y₁′)

  (F.6.5.1)

• Step 2: Compute (c_x′, c_y′)

  (F.6.5.2)

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

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

  (F.6.5.3)

• Step 4: Compute θ₁ and Δθ

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

  (F.6.5.4)

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

  This angle function can be used to express θ₁ and Δθ as follows:

  	(F.6.5.5)
  	(F.6.5.6)

  where θ₁ is fixed in the range −360° < Δθ < 360° such that:

  if f_S = 0, then Δθ < 0,

  else if f_S = 1, then Δθ > 0.

  In other words, if f_S = 0 and the right side of (F.6.5.6) is greater than 0, then subtract 360°, whereas if f_S = 1 and the right side of (F.6.5.6) is less than 0, then add 360°. In all other cases leave it as is.

F.6.6 Correction of out-of-range radii

This section formalizes the adjustments to out-of-range r_x and r_y mentioned in F.6.2. Algorithmically these adjustments consist of the following steps:

• Step 1: Ensure radii are non-zero

  If r_x = 0 or r_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 of r_x and r_y:

  (F.6.6.1)

• Step 3: Ensure radii are large enough

  Using the primed coordinate values of equation (F.6.5.1), compute

  (F.6.6.2)

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

  (F.6.6.3)

• Step 4: Proceed with computations

  Proceed with the remaining elliptical arc computations, such as those in section F.6.5.  Note: As a consequence of the radii corrections in this section, equation (F.6.5.2) 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 (F.6.6.3), the radicand of (F.6.5.2) is zero and there is exactly one solution for the center of the ellipse.