Line Segments

Atualizado em 2023/10/21
Tempos estimado de leitura: 6 min
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To dynamically set the line width, call:

void AfxCmdSetLineWidth
    afxDrawScript dscr, /// is the command buffer into which the command will be recorded.
    afxReal lineWidth /// is the width of rasterized line segments.

This command sets the line width for subsequent drawing commands when the graphics pipeline is created with VK_DYNAMIC_STATE_LINE_WIDTH set in VkPipelineDynamicStateCreateInfo::pDynamicStates. Otherwise, this state is specified by the VkPipelineRasterizationStateCreateInfo::lineWidth value used to create the currently active pipeline.

Not all line widths need be supported for line segment rasterization, but width 1.0 antialiased segments must be provided. The range and gradations are obtained from the lineWidthRange and lineWidthGranularity members of VkPhysicalDeviceLimits. If, for instance, the size range is from 0.1 to 2.0 and the gradation size is 0.1, then the sizes 0.1, 0.2, …, 1.9, 2.0 are supported. Additional line widths may also be supported. There is no requirement that these widths be equally spaced. If an unsupported width is requested, the nearest supported width is used instead.

Basic Line Segment Rasterization

Rasterized line segments produce fragments which intersect a rectangle centered on the line segment. Two of the edges are parallel to the specified line segment; each is at a distance of one-half the current width from that segment in directions perpendicular to the direction of the line. The other two edges pass through the line endpoints and are perpendicular to the direction of the specified line segment. Coverage bits that correspond to sample points that intersect the rectangle are 1, other coverage bits are 0.

Next we specify how the data associated with each rasterized fragment are obtained. Let pr = (xd, yd) be the framebuffer coordinates at which associated data are evaluated. This may be the center of a fragment or the location of a sample within the fragment. When rasterizationSamples is VK_SAMPLE_COUNT_1_BIT, the fragment center must be used. Let pa = (xa, ya) and pb = (xb,yb) be initial and final endpoints of the line segment, respectively. Set

t = ((pr – pa) * (pb – pa)) / (||(pb – pa)||^2)

(Note that t = 0 at pa and t = 1 at pb. Also note that this calculation projects the vector from pa to pr onto the line, and thus computes the normalized distance of the fragment along the line.)

If strictLines is TRUE, line segments are rasterized using perspective or linear interpolation.

Perspective interpolation for a line segment interpolates two values in a manner that is correct when taking the perspective of the viewport into consideration, by way of the line segment’s clip coordinates. An interpolated value f can be determined by

f = ((1 – t) * fa / wa + t * fb / wb) / (1 – t) / wa + t / wb

where fa and fb are the data associated with the starting and ending endpoints of the segment, respectively; wa and wb are the clip w coordinates of the starting and ending endpoints of the segment, respectively.

Linear interpolation for a line segment directly interpolates two values, and an interpolated value f can be determined by

f = (1 – t) * fa t * fb

where fa and fb are the data associated with the starting and ending endpoints of the segment, respectively.

The clip coordinate w for a sample is determined using perspective interpolation. The depth value z for a sample is determined using linear interpolation. Interpolation of fragment shader input values are determined by Interpolation decorations.

The above description documents the preferred method of line rasterization, and must be used when the implementation advertises the strictLines limit in VkPhysicalDeviceLimits as VK_TRUE.

When strictLines is FALSE, the edges of the lines are generated as a parallelogram surrounding the original line. The major axis is chosen by noting the axis in which there is the greatest distance between the line start and end points. If the difference is equal in both directions then the X axis is chosen as the major axis. Edges 2 and 3 are aligned to the minor axis and are centered on the endpoints of the line as in Non strict lines, and each is lineWidth long. Edges 0 and 1 are parallel to the line and connect the endpoints of edges 2 and 3. Coverage bits that correspond to sample points that intersect the parallelogram are 1, other coverage bits are 0.

Samples that fall exactly on the edge of the parallelogram follow the polygon rasterization rules.

Interpolation occurs as if the parallelogram was decomposed into two triangles where each pair of vertices at each end of the line has identical attributes.

Only when strictLines is FALSE implementations may deviate from the non-strict line algorithm described above in the following ways:

  • Implementations may instead interpolate each fragment according to the formula in Basic Line Segment Rasterization using the original line segment endpoints.
  • Rasterization of non-antialiased non-strict line segments may be performed using the rules defined in Bresenham Line Segment Rasterization.

Bresenham Line Segment Rasterization

Non-strict lines may also follow these rasterization rules for non-antialiased lines.

Line segment rasterization begins by characterizing the segment as either x-major or y-major. xmajor line segments have slope in the closed interval [-1,1]; all other line segments are y-major (slope is determined by the segment’s endpoints). We specify rasterization only for x-major segments except in cases where the modifications for y-major segments are not self-evident.

Ideally, Vulkan uses a diamond-exit rule to determine those fragments that are produced by rasterizing a line segment. For each fragment f with center at framebuffer coordinates xf and yf, define a diamond-shaped region that is the intersection of four half planes:

Rf = {(x, y) | abs(x – xf) + abs(y – yf) < (1/2)}

Essentially, a line segment starting at pa and ending at pb produces those fragments f for which the segment intersects Rf, except if pb is contained in Rf.

To avoid difficulties when an endpoint lies on a boundary of Rf we (in principle) perturb the supplied endpoints by a tiny amount. Let pa and pb have framebuffer coordinates (xa , ya) and (xb, yb), respectively. Obtain the perturbed endpoints pa ‘ given by (xa , ya) – (ε, ε 2 ) and pb’ given by (xb, yb) – (ε, ε 2 ). Rasterizing the line segment starting at pa and ending at pb produces those fragments f for which the segment starting at pa ‘ and ending on pb’ intersects Rf , except if pb’ is contained in Rf . ε is chosen to be so small that rasterizing the line segment produces the same fragments when δ is substituted for ε for any 0 < δ ≤ ε.

When pa and pb lie on fragment centers, this characterization of fragments reduces to Bresenham’s algorithm with one modification: lines produced in this description are “half-open”, meaning that the final fragment (corresponding to pb) is not drawn. This means that when rasterizing a series of connected line segments, shared endpoints will be produced only once rather than twice (as would occur with Bresenham’s algorithm).

Implementations may use other line segment rasterization algorithms, subject to the following rules:

  • The coordinates of a fragment produced by the algorithm must not deviate by more than one unit in either x or y framebuffer coordinates from a corresponding fragment produced by the diamond-exit rule.
  • The total number of fragments produced by the algorithm must not differ from that produced by the diamond-exit rule by no more than one.
  • For an x-major line, two fragments that lie in the same framebuffer-coordinate column must not be produced (for a y-major line, two fragments that lie in the same framebuffer-coordinate row must not be produced).
  • If two line segments share a common endpoint, and both segments are either x-major (both left-to-right or both right-to-left) or y-major (both bottom-to-top or both top-to-bottom), then rasterizing both segments must not produce duplicate fragments. Fragments also must not be omitted so as to interrupt continuity of the connected segments.

The actual width w of Bresenham lines is determined by rounding the line width to the nearest integer, clamping it to the implementation-dependent lineWidthRange (with both values rounded to the nearest integer), then clamping it to be no less than 1.

Bresenham line segments of width other than one are rasterized by offsetting them in the minor direction (for an x-major line, the minor direction is y, and for a y-major line, the minor direction is x) and producing a row or column of fragments in the minor direction. If the line segment has endpoints given by (x0, y0) and (x1, y1) in framebuffer coordinates, the segment with endpoints (x0, y0 – ((w – 1) / 2)) and (x1, y1 – ((w – 1) / 2)) is rasterized, but instead of a single fragment, a column of fragments of height w (a row of fragments of length w for a y-major segment) is produced at each x (y for ymajor) location. The lowest fragment of this column is the fragment that would be produced by rasterizing the segment of width 1 with the modified coordinates.

The preferred method of attribute interpolation for a wide line is to generate the same attribute values for all fragments in the row or column described above, as if the adjusted line was used for interpolation and those values replicated to the other fragments, except for FragCoord which is interpolated as usual. Implementations may instead interpolate each fragment according to the formula in Basic Line Segment Rasterization, using the original line segment endpoints.

When Bresenham lines are being rasterized, sample locations may all be treated as being at the pixel center (this may affect attribute and depth interpolation).


The sample locations described above are not used for determining coverage, they are only used for things like attribute interpolation. The rasterization rules that determine coverage are defined in terms of whether the line intersects pixels, as opposed to the point sampling rules used for other primitive types. So these rules are independent of the sample locations. One consequence of this is that Bresenham lines cover the same pixels regardless of the number of rasterization samples, and cover all samples in those pixels (unless masked out or killed).

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