The semantics of Vg

The following notations and definitions give precise meaning to images and their combinators.

Colors

The semantics of colors is the one ascribed to Gg.color: colors are in a linearized sRGBA space.

Color stops

A value of type Gg.Color.stops specifies a color at each point of the 1D unit space. It is defined by a list of pairs (t_i, c_i) where t_i is a value from 0 to 1 and c_i the corresponding color at that value. Colors at points between t_i and t_i+1 are linearly interpolated between c_i and c_i+1. If t_i lies outside 0 to 1 or if t_i-1 >= t_i the semantics is undefined.

Given a stops value stops = [(t₀, c₀); (t₁,c₁); ... (t_n, c_n)] and any point t of 1D space, the semantic function:

[] : Gg.Color.stops -> float -> Gg.color

maps them to a color value written [stops]_t as follows.

• [stops]_t = (0, 0, 0, 0) for any t if stops = []
• [stops]_t = c₀ if t < t₀.
• [stops]_t = c_n if t >= t_n.
• [stops]_t = (1-u)c_i + uc_i+1 with u = (t - t_i)/(t_i+1-t_i) if t_i <= t < t_i+1

Images

Values of type Vg.image represent maps from the infinite 2D euclidian space to colors. Given an image i and a point pt of the plane the semantic function

[]: image -> Gg.p2 -> Gg.color

maps them to a color value written [i]_pt representing the image's color at this point.

Paths and areas

A value of type Vg.path is a list of subpaths. A subpath is a list of directed and connected curved segments. Subpaths are disconnected from each other and may (self-)intersect.

A path and an area specification of type Vg.P.area define a finite area of the 2D euclidian space. Given an area specification a, a path p and a point pt, the semantic function:

[]: P.area -> path -> Gg.p2 -> bool

maps them to a boolean value written [a, p]_pt that indicates whether pt belongs to the area or not.

The semantics of area rules is as follows:

• [`Anz, p]_pt is true iff the winding number of p around pt is non zero. To determine the winding number cast a ray from pt to infinity in any direction (just make sure the ray doesn't intersect p tangently or at a singularity). Starting with zero add one for each intersection with a counter-clockwise oriented segment of p and substract one for each clockwise ones. The resulting sum is the winding number. This is usually refered to as the non-zero winding rule and is the default for Vg.I.cut.
• [`Aeo, p]_pt is true iff the number of intersections of p with a ray cast from pt to infinity in any direction is odd (just make sure the ray doesn't intersect p tangently or at a singularity). This is usually refered to as the even-odd rule.
• [`O o, p]_pt is true iff pt is in the outline area of p as defined by the value o of type Vg.P.outline. See Outline areas, Segment jointures, Subpath caps, Outline dashes.

Outline areas

The outline area of a path specified by a value o of type Vg.P.outline is the union of its subpaths outline areas.

A subpath outline area is inside the parallel curves at a distance o.width / 2 of its path segments that are joined accoring to the join style o.join (see below) and closed at the subpath end points with a cap style o.cap (see below). The outline area of a subpath can also be chopped at regular intervals according to the o.dashes parameter (see below).

Segment jointures

The shape of subpath segment jointures is specified in o.join by a value of type Vg.P.join. From left to right:

• `Miter, the outer parallel curves are extended until they meet unless the joining angle is smaller than o.miter_angle in which case the join is converted to a bevel.
• `Round, joins the outer parallel curves by a semicircle centered at the end point with a diameter equal to o.width.
• `Bevel, joins the outer parallel curves by a segment.

Subpath caps

The shape of subpath (or dashes) end points is specified in o.cap by a value of type Vg.P.cap. From left to right:

• `Butt, end points are square and extend only to the exact end point of the path.
• `Round, end points are rounded by a semicircle at the end point with a diameter equal to o.width.
• `Square, end points are square and extend by a distance equal to half o.width.

Outline dashes

The path outline area can be chopped at regular intervals by specifying a value (off, pat) of type Vg.P.dashes in o.dashes.

The dash pattern pat is a list of lengths that specify the length of alternating dashes and gaps (starting with dashes). The dash offset off is a positive offset that indicates where to start in the dash pattern at the beginning of a subpath.