Vulkan-Docs/doc/specs/vulkan/chapters/primsrast.txt

// Copyright (c) 2015-2016 The Khronos Group Inc.
// Copyright notice at https://www.khronos.org/registry/speccopyright.html

[[primsrast]]
= Rasterization

Rasterization is the process by which a primitive is converted to a
two-dimensional image. Each point of this image contains associated data
such as depth, color, or other attributes.

Rasterizing a primitive begins by determining which squares of an integer
grid in framebuffer coordinates are occupied by the primitive, and assigning
one or more depth values to each such square. This process is described
below for points, lines, and polygons.

A grid square, including its latexmath:[$(x,y)$] framebuffer coordinates,
latexmath:[$z$] (depth), and associated data added by fragment
shaders, is called a fragment. A fragment is located by its upper left
corner, which lies on integer grid coordinates.

Rasterization operations also refer to a fragment's sample locations, which
are offset by subpixel fractional values from its upper left corner. The
rasterization rules for points, lines, and triangles involve testing whether
each sample location is inside the primitive. Fragments need not actually be
square, and rasterization rules are not affected by the aspect ratio of
fragments. Display of non-square grids, however, will cause rasterized
points and line segments to appear fatter in one direction than the other.

We assume that fragments are square, since it simplifies antialiasing and
texturing. After rasterization, fragments are processed by the
<<fragops-early,early per-fragment tests>>, if enabled.

Several factors affect rasterization, including the members of
sname:VkPipelineRasterizationStateCreateInfo and
sname:VkPipelineMultisampleStateCreateInfo.

The sname:VkPipelineRasterizationStateCreateInfo structure is defined as:

include::../structs/VkPipelineRasterizationStateCreateInfo.txt[]

  * pname:sType is the type of this structure.
  * pname:pNext is `NULL` or a pointer to an extension-specific structure.
  * pname:flags is reserved for future use.
  * pname:depthClampEnable controls whether to clamp the fragment's depth
    values instead of clipping primitives to the z planes of the frustum, as
    described in <<vertexpostproc-clipping,Primitive Clipping>>.
  * pname:rasterizerDiscardEnable controls whether primitives are discarded
    immediately before the rasterization stage.
  * pname:polygonMode is the triangle rendering mode. See
    elink:VkPolygonMode.
  * pname:cullMode is the triangle facing direction used for primitive
    culling. See elink:VkCullModeFlagBits.
  * pname:frontFace is the front-facing triangle orientation to be used for
    culling. See elink:VkFrontFace.
  * pname:depthBiasEnable controls whether to bias fragment depth values.
  * pname:depthBiasConstantFactor is a scalar factor controlling the
    constant depth value added to each fragment.
  * pname:depthBiasClamp is the maximum (or minimum) depth bias of a
    fragment.
  * pname:depthBiasSlopeFactor is a scalar factor applied to a fragment's
    slope in depth bias calculations.
  * pname:lineWidth is the width of rasterized line segments.

include::../validity/structs/VkPipelineRasterizationStateCreateInfo.txt[]

The sname:VkPipelineMultisampleStateCreateInfo structure is defined as:

include::../structs/VkPipelineMultisampleStateCreateInfo.txt[]

The members of the sname:VkPipelineMultisampleStateCreateInfo structure are
as follows:

  * pname:sType is the type of this structure.
  * pname:pNext is `NULL` or a pointer to an extension-specific structure.
  * pname:flags is reserved for future use.
  * pname:rasterizationSamples is a elink:VkSampleCountFlagBits specifying
    the number of samples per pixel used in rasterization.
  * pname:sampleShadingEnable specifies that fragment shading executes
    per-sample if ename:VK_TRUE, or per-fragment if ename:VK_FALSE, as
    described in <<primsrast-sampleshading,Sample Shading>>.
  * pname:minSampleShading is the minimum number of unique samples to shade
    for each fragment.
  * pname:pSampleMask is a bitmask of static coverage information that is
    ANDed with the coverage information generated during rasterization, as
    described in <<fragops-samplemask,Sample Mask>>.
  * pname:alphaToCoverageEnable controls whether a temporary coverage value
    is generated based on the value of the alpha component of the fragment's
    first color output as specified in the <<fragops-covg,Multisample
    Coverage>> section.
  * pname:alphaToOneEnable controls whether the value of the alpha component of
    the fragment's first color output is replaced with one as described in
    <<fragops-covg,Multisample Coverage>>.

include::../validity/structs/VkPipelineMultisampleStateCreateInfo.txt[]

Rasterization only produces fragments corresponding to pixels in the
framebuffer. Fragments which would be produced by application of any of the
primitive rasterization rules described below but which lie outside the
framebuffer are not produced, nor are they processed by any later stage of
the pipeline, including any of the early per-fragment tests described in
<<fragops-early,Early Per-Fragment Tests>>.

Surviving fragments are processed by fragment shaders. Fragment shaders
determine associated data for fragments, and can: also modify or replace
their assigned depth values.

If the subpass for which this pipeline is being created uses
color and/or depth/stencil attachments, then the value of
pname:rasterizationSamples must: be the same as the sample count for
those subpass attachments. Otherwise, the value of
pname:rasterizationSamples must: follow the rules for a
<<renderpass-noattachments, zero-attachment subpass>>.


[[primsrast-discard]]
== Discarding Primitives Before Rasterization

Primitives are discarded before rasterization if the
pname:rasterizerDiscardEnable member of
slink:VkPipelineRasterizationStateCreateInfo is enabled. When enabled,
primitives are discarded after they are processed by the last active shader
stage in the pipeline before rasterization.


[[primsrast-multisampling]]
== Multisampling

Multisampling is a mechanism to antialias all {apiname} primitives: points,
lines, and polygons. The technique is to sample all primitives multiple
times at each pixel. Each sample in each framebuffer attachment has storage
for a color, depth, and/or stencil value, such that per-fragment operations
apply to each sample independently. The color sample values can: be later
_resolved_ to a single color (see <<copies-resolve,Resolving Multisample
Images>> and the <<renderpass,Render Pass>> chapter for more details on how
to resolve multisample images to non-multisample images).

{apiname} defines rasterization rules for single-sample modes in a way that
is equivalent to a multisample mode with a single sample in the center of
each pixel.

Each fragment includes a coverage value with pname:rasterizationSamples bits
(see <<fragops-samplemask,Sample Mask>>). Each fragment includes
pname:rasterizationSamples depth values and sets of associated data. An
implementation may: choose to assign the same associated data to more than
one sample. The location for evaluating such associated data may: be
anywhere within the pixel including the pixel center or any of the sample
locations. When pname:rasterizationSamples is ename:VK_SAMPLE_COUNT_1_BIT,
the pixel center must: be used. The different associated data values need
not all be evaluated at the same location. Each pixel fragment thus consists
of integer x and y grid coordinates, pname:rasterizationSamples depth values
and sets of associated data, and a coverage value with
pname:rasterizationSamples bits.

It is understood that each pixel has pname:rasterizationSamples locations
associated with it. These locations are exact positions, rather than regions
or areas, and each is referred to as a sample point. The sample points
associated with a pixel must: be located inside or on the boundary of the
unit square that is considered to bound the pixel. Furthermore, the relative
locations of sample points may: be identical for each pixel in the
framebuffer, or they may: differ. If the current pipeline includes a
fragment shader with one or more variables in its interface decorated with
code:Sample and code:Input, the data associated with those variables will be
assigned independently for each sample. The values for each sample must: be
evaluated at the location of the sample. The data associated with any other
variables not decorated with code:Sample and code:Input need not be
evaluated independently for each sample.

If the pname:standardSampleLocations member of
slink:VkPhysicalDeviceFeatures is ename:VK_TRUE, then the sample counts
ename:VK_SAMPLE_COUNT_1_BIT, ename:VK_SAMPLE_COUNT_2_BIT,
ename:VK_SAMPLE_COUNT_4_BIT, ename:VK_SAMPLE_COUNT_8_BIT, and
ename:VK_SAMPLE_COUNT_16_BIT have sample locations as listed in the
following table, with the latexmath:[$i$]th entry in the table corresponding
to bit latexmath:[$i$] in the sample masks. ename:VK_SAMPLE_COUNT_32_BIT and
ename:VK_SAMPLE_COUNT_64_BIT do not have standard sample locations.
Locations are defined relative to an origin in the upper left corner of the
pixel.

<<<

.Standard sample locations
[align="center"]
|========================================
|ename:VK_SAMPLE_COUNT_1_BIT|ename:VK_SAMPLE_COUNT_2_BIT|ename:VK_SAMPLE_COUNT_4_BIT|ename:VK_SAMPLE_COUNT_8_BIT|ename:VK_SAMPLE_COUNT_16_BIT
|
latexmath:[$(0.5,0.5)$]
|
latexmath:[$(0.25,0.25)$] +
latexmath:[$(0.75,0.75)$]
|
latexmath:[$( 0.375, 0.125)$] +
latexmath:[$( 0.875, 0.375)$] +
latexmath:[$( 0.125, 0.625)$] +
latexmath:[$( 0.625, 0.875)$]
|
latexmath:[$( 0.5625, 0.3125)$] +
latexmath:[$( 0.4375, 0.6875)$] +
latexmath:[$( 0.8125, 0.5625)$] +
latexmath:[$( 0.3125, 0.1875)$] +
latexmath:[$( 0.1875, 0.8125)$] +
latexmath:[$( 0.0625, 0.4375)$] +
latexmath:[$( 0.6875, 0.9375)$] +
latexmath:[$( 0.9375, 0.0625)$]
|
latexmath:[$( 0.5625, 0.5625)$] +
latexmath:[$( 0.4375, 0.3125)$] +
latexmath:[$( 0.3125, 0.625)$] +
latexmath:[$( 0.75,   0.4375)$] +
latexmath:[$( 0.1875, 0.375)$] +
latexmath:[$( 0.625,  0.8125)$] +
latexmath:[$( 0.8125, 0.6875)$] +
latexmath:[$( 0.6875, 0.1875)$] +
latexmath:[$( 0.375,  0.875)$] +
latexmath:[$( 0.5,    0.0625)$] +
latexmath:[$( 0.25,   0.125)$] +
latexmath:[$( 0.125,  0.75)$] +
latexmath:[$( 0.0,    0.5)$] +
latexmath:[$( 0.9375, 0.25)$] +
latexmath:[$( 0.875,  0.9375)$] +
latexmath:[$( 0.0625, 0.0)$]
|========================================


[[primsrast-sampleshading]]
== Sample Shading

Sample shading can: be used to specify a minimum number of unique samples to
process for each fragment. Sample shading is controlled by the
pname:sampleShadingEnable member of
slink:VkPipelineMultisampleStateCreateInfo. If pname:sampleShadingEnable is
ename:VK_FALSE, sample shading is considered disabled and has no effect.
Otherwise, an implementation must: provide a minimum of
latexmath:[$\max(\lceil{minSampleShading \times rasterizationSamples}\rceil,
1)$] unique associated data for each fragment, where pname:minSampleShading
is the minimum fraction of sample shading and pname:rasterizationSamples is
the number of samples requested in
slink:VkPipelineMultisampleStateCreateInfo. These are associated with the
samples in an implementation-dependent manner. pname:minSampleShading must:
be in the range latexmath:[$[0,1\]$]. When the sample shading fraction is
1.0, a separate set of associated data are evaluated for each sample, and
each set of values is evaluated at the sample location.


[[primsrast-points]]
== Points

A point is drawn by generating a set of fragments in the shape of a square
centered around the vertex of the point. Each vertex has an associated point
size that controls the width/height of that square. The point size is taken
from the (potentially clipped) shader built-in code:PointSize written by:

  * the geometry shader, if active;
  * the tessellation evaluation shader, if active and no geometry shader is
    active;
  * the tessellation control shader, if active and no geometry or
    tessellation evaluation shader is active; or
  * the vertex shader, otherwise

and clamped to the implementation-dependent point size range
latexmath:[$[pointSizeRange[0\],pointSizeRange[1\]\]$]. If the value written
to code:PointSize is less than or equal to zero, or if no value was written
to code:PointSize, results are undefined.

Not all point sizes need be supported, but the size 1.0 must: be supported.
The range of supported sizes and the size of evenly-spaced gradations within
that range are implementation-dependent. The range and gradations are
obtained from the pname:pointSizeRange and pname:pointSizeGranularity
members of slink:VkPhysicalDeviceLimits. If, for instance, the size range is
from 0.1 to 2.0 and the gradation size is 0.1, then the size 0.1, 0.2, ...,
1.9, 2.0 are supported. Additional point sizes may: also be supported. There
is no requirement that these sizes be equally spaced. If an unsupported
size is requested, the nearest supported size is used instead.


[[primsrast-points-basic]]
=== Basic Point Rasterization

Point rasterization produces a fragment for each framebuffer pixel with one
or more sample points that intersect a region centered at the point's
latexmath:[$(x_f,y_f)$]. This region is a square with side equal to the
current point size. Coverage bits that correspond to sample points that
intersect the region are 1, other coverage bits are 0.

All fragments produced in rasterizing a point are assigned the same
associated data, which are those of the vertex corresponding to the point.
However, the fragment shader built-in code:PointCoord contains point sprite
texture coordinates. The latexmath:[$s$] and latexmath:[$t$] point sprite
texture coordinates vary from zero to one across the point horizontally
left-to-right and top-to-bottom, respectively. The following formulas are
used to evaluate latexmath:[$s$] and latexmath:[$t$]:

[latexmath]
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
\[
  s = {1 \over 2} + { \left( x_p - x_f \right) \over size }
\]
\[
  t = {1 \over 2} + { \left( y_p - y_f \right) \over size }.
\]
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

where size is the point's size, latexmath:[$(x_p,y_p)$] is the location at
which the point sprite coordinates are evaluated - this may: be the
framebuffer coordinates of the pixel center (i.e. at the half-integer) or
the location of a sample, and latexmath:[$(x_f,y_f)$] is the exact,
unrounded framebuffer coordinate of the vertex for the point. When
pname:rasterizationSamples is ename:VK_SAMPLE_COUNT_1_BIT, the pixel center
must: be used.


[[primsrast-lines]]
== Line Segments

A line is drawn by generating a set of fragments overlapping a rectangle
centered on the line segment. Each line segment has an associated width that
controls the width of that rectangle.

The line width is set by the pname:lineWidth property of
slink:VkPipelineRasterizationStateCreateInfo in the currently active
pipeline if the pipeline was not created with
ename:VK_DYNAMIC_STATE_LINE_WIDTH enabled. Otherwise, the line width is set
by calling fname:vkCmdSetLineWidth:

include::../protos/vkCmdSetLineWidth.txt[]

  * pname:commandBuffer is the command buffer into which the command will be
    recorded.
  * pname:lineWidth is the width of rasterized line segments.

include::../validity/protos/vkCmdSetLineWidth.txt[]

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 pname:lineWidthRange and pname:lineWidthGranularity
members of slink:VkPhysicalDeviceLimits. If, for instance, the size range is
from 0.1 to 2.0 and the gradation size is 0.1, then the size 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.


[[primsrast-lines-basic]]
=== 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 latexmath:[$\mathbf{p}_r = (x_d, y_d)$] be the
framebuffer coordinates at which associated data are evaluated. This may: be
the pixel center of a fragment or the location of a sample within the
fragment. When pname:rasterizationSamples is ename:VK_SAMPLE_COUNT_1_BIT,
the pixel center must: be used. Let latexmath:[$\mathbf{p}_a = (x_a, y_a)$]
and latexmath:[$\mathbf{p}_b = (x_b,y_b)$] be initial and final endpoints of
the line segment, respectively. Set

// Equation {linet:eq}
[latexmath]
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
\[
  t = {{( \mathbf{p}_r - \mathbf{p}_a ) \cdot ( \mathbf{p}_b - \mathbf{p}_a )}
      \over {\| \mathbf{p}_b - \mathbf{p}_a \|^2 }}
\]
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

(Note that latexmath:[$t=0$] at latexmath:[$\mathbf{p}_a$] and
latexmath:[$t=1$] at latexmath:[$\mathbf{p}_b$]. Also note that this
calculation projects the vector from latexmath:[$\mathbf{p}_a$] to
latexmath:[$\mathbf{p}_r$] onto the line, and thus computes the normalized
distance of the fragment along the line.)

The value of an associated datum latexmath:[$f$] for the fragment, whether
it be a shader output or the clip latexmath:[$w$] coordinate, is found as

[[line_perspective_interpolation,Equation line_perspective_interpolation]]
.line_perspective_interpolation
[latexmath]
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
\[
  f = {{ (1-t) {f_a / w_a} + t { f_b / w_b} } \over
      {(1-t) / w_a + t / w_b }}
\]
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

where latexmath:[$f_a$] and latexmath:[$f_b$] are the data associated with
the starting and ending endpoints of the segment, respectively;
latexmath:[$w_a$] and latexmath:[$w_b$] are the clip latexmath:[$w$]
coordinates of the starting and ending endpoints of the segments,
respectively. However, depth values for lines must: be interpolated by

[[line_noperspective_interpolation,Equation line_noperspective_interpolation]]
.line_noperspective_interpolation
[latexmath]
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
\[ z = (1-t) z_a + t z_b \]
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

where latexmath:[$z_a$] and latexmath:[$z_b$] are the depth values of the
starting and ending endpoints of the segment, respectively.

The code:NoPerspective and code:Flat
<<shaders-interpolation-decorations,interpolation decorations>> can: be used
with fragment shader inputs to declare how they are interpolated. When
neither decoration is applied, interpolation is performed as described in
<<line_perspective_interpolation>>. When the code:NoPerspective decoration
is used, interpolation is performed in the same fashion as for depth values,
as described in <<line_noperspective_interpolation>>. When the code:Flat
decoration is used, no interpolation is performed, and outputs are taken
from the corresponding input value of the
<<vertexpostproc-flatshading,provoking vertex>> corresponding to that
primitive.

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

When pname:strictLines is ename:VK_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
<<fig-non-strict-lines>>, and each is pname: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.

[[fig-non-strict-lines]]
.Non strict lines
image:images/non_strict_lines.{svgpdf}["Non strict lines",{fullimagewidth}]


[[primsrast-polygons]]
== Polygons

A polygon results from the decomposition of a triangle strip, triangle fan
or a series of independent triangles. Like points and line segments,
polygon rasterization is controlled by several variables in the
slink:VkPipelineRasterizationStateCreateInfo structure.


[[primsrast-polygons-basic]]
=== Basic Polygon Rasterization

The first step of polygon rasterization is to determine whether the triangle
is _back-facing_ or _front-facing_. This determination is made based on the
sign of the (clipped or unclipped) polygon's area computed in framebuffer
coordinates. One way to compute this area is:

[latexmath]
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
\[
  a = -{1 \over 2}\sum_{i=0}^{n-1}
        x_f^i y_f^{i \oplus 1} -
        x_f^{i \oplus 1} y_f^i
\]
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

where latexmath:[$x_f^i$] and latexmath:[$y_f^i$] are the latexmath:[$x$]
and latexmath:[$y$] framebuffer coordinates of the latexmath:[$i$]th vertex
of the latexmath:[$n$]-vertex polygon (vertices are numbered starting at
zero for the purposes of this computation) and latexmath:[$i \oplus 1$] is
latexmath:[$(i + 1)~ \textrm{mod}~ n$]. The interpretation of the sign of
this value is determined by the pname:frontFace property of the
slink:VkPipelineRasterizationStateCreateInfo in the currently active
pipeline, which takes the following values:

include::../enums/VkFrontFace.txt[]

When this is set to ename:VK_FRONT_FACE_COUNTER_CLOCKWISE, a triangle with
positive area is considered front-facing. When it is set to
ename:VK_FRONT_FACE_CLOCKWISE, a triangle with negative area is considered
front-facing. Any triangle which is not front-facing is back-facing,
including zero-area triangles.

Once the orientation of triangles is determined, they are culled according
to the setting of pname:cullMode property in the
slink:VkPipelineRasterizationStateCreateInfo of the currently active
pipeline, which takes the following values:

include::../enums/VkCullModeFlagBits.txt[]

If the pname:cullMode is set to ename:VK_CULL_MODE_NONE no triangles are
discarded, if it is set to ename:VK_CULL_MODE_FRONT_BIT front-facing
triangles are discarded, if it is set to ename:VK_CULL_MODE_BACK_BIT then
back-facing triangles are discarded and if it is set to
ename:VK_CULL_MODE_FRONT_AND_BACK then all triangles are discarded.
Following culling, fragments are produced for any triangles which have not
been discarded.

The rule for determining which fragments are produced by polygon
rasterization is called _point sampling_. The two-dimensional projection
obtained by taking the x and y framebuffer coordinates of the polygon's
vertices is formed. Fragments are produced for any pixels for which any
sample points lie inside of this polygon. Coverage bits that correspond to
sample points that satisfy the point sampling criteria are 1, other coverage
bits are 0. Special treatment is given to a sample whose sample location
lies on a polygon edge. In such a case, if two polygons lie on either side
of a common edge (with identical endpoints) on which a sample point lies,
then exactly one of the polygons must: result in a covered sample for that
fragment during rasterization. As for the data associated with each fragment
produced by rasterizing a polygon, we begin by specifying how these values
are produced for fragments in a triangle. Define _barycentric coordinates_
for a triangle. Barycentric coordinates are a set of three numbers,
latexmath:[$a$], latexmath:[$b$], and latexmath:[$c$], each in the range
latexmath:[$\lbrack 0, 1\rbrack$], with latexmath:[$a + b + c = 1$]. These
coordinates uniquely specify any point latexmath:[$p$] within the triangle
or on the triangle's boundary as

[latexmath]
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
\[ p = ap_a + bp_b + cp_c \]
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

where latexmath:[$p_a$], latexmath:[$p_b$], and latexmath:[$p_c$] are the
vertices of the triangle. latexmath:[$a$], latexmath:[$b$], and
latexmath:[$c$] are determined by:

[latexmath]
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
\[
    a = {{\rm A}(p p_b p_c) \over {\rm A}(p_a p_b p_c)}, \quad
    b = {{\rm A}(p p_a p_c) \over {\rm A}(p_a p_b p_c)}, \quad
    c = {{\rm A}(p p_a p_b) \over {\rm A}(p_a p_b p_c)},
\]
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

where latexmath:[$A(lmn)$] denotes the area in framebuffer coordinates of
the triangle with vertices latexmath:[$l$], latexmath:[$n$], and
latexmath:[$n$].

Denote an associated datum at latexmath:[$p_a$], latexmath:[$p_b$], or
latexmath:[$p_c$] as latexmath:[$f_a$], latexmath:[$f_b$], or
latexmath:[$f_c$], respectively. Then the value latexmath:[$f$] of a datum
at a fragment produced by rasterizing a triangle is given by:

[[triangle_perspective_interpolation,Equation triangle_perspective_interpolation]]
.triangle_perspective_interpolation
[latexmath]
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
\[
  f = {{a{f_a / w_a} + b{f_b / w_b} + c{f_c / w_c}} \over
       {a / w_a} + {b / w_b} + {c / w_c}}
\]
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

where latexmath:[$w_a$], latexmath:[$w_b$], and latexmath:[$w_c$] are the
clip latexmath:[$w$] coordinates of latexmath:[$p_a$], latexmath:[$p_b$],
and latexmath:[$p_c$], respectively. latexmath:[$a$], latexmath:[$b$], and
latexmath:[$c$] are the barycentric coordinates of the location at which
the data are produced - this must: be a pixel center or the location of
a sample. When pname:rasterizationSamples is
ename:VK_SAMPLE_COUNT_1_BIT, the pixel center must: be used. Depth values
for triangles must: be interpolated by

[[triangle_noperspective_interpolation,Equation triangle_noperspective_interpolation]]
.triangle_noperspective_interpolation
[latexmath]
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
\[ z = a z_a + b z_b + c z_c \]
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

where latexmath:[$z_a$], latexmath:[$z_b$], and latexmath:[$z_c$] are the
depth values of latexmath:[$p_a$], latexmath:[$p_b$], and latexmath:[$p_c$],
respectively.

The code:NoPerspective and code:Flat
<<shaders-interpolation-decorations,interpolation decorations>> can: be used
with fragment shader inputs to declare how they are interpolated. When
neither decoration is applied, interpolation is performed as described in
<<triangle_perspective_interpolation>>. When the code:NoPerspective
decoration is used, interpolation is performed in the same fashion as for
depth values, as described in <<triangle_noperspective_interpolation>>. When
the code:Flat decoration is used, no interpolation is performed, and outputs
are taken from the corresponding input value of the
<<vertexpostproc-flatshading,provoking vertex>> corresponding to that
primitive.

For a polygon with more than three edges, such as are produced by clipping a
triangle, a convex combination of the values of the datum at the polygon's
vertices must: be used to obtain the value assigned to each fragment
produced by the rasterization algorithm. That is, it must: be the case that
at every fragment

[latexmath]
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
\[ f = \sum_{i=1}^{n} a_i f_i \]
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

where latexmath:[$n$] is the number of vertices in the polygon and
latexmath:[$f_i$] is the value of the latexmath:[$f$] at vertex
latexmath:[$i$]. For each latexmath:[$i$], latexmath:[$0 \leq a_i \leq 1$]
and latexmath:[$\sum_{i=1}^{n}a_i = 1$]. The values of latexmath:[$a_i$]
may: differ from fragment to fragment, but at vertex latexmath:[$i$],
latexmath:[$a_i = 1$] and latexmath:[$a_j = 0$] for latexmath:[$j \neq i$].

[NOTE]
.Note
====
One algorithm that achieves the required behavior is to triangulate a
polygon (without adding any vertices) and then treat each triangle
individually as already discussed. A scan-line rasterizer that linearly
interpolates data along each edge and then linearly interpolates data across
each horizontal span from edge to edge also satisfies the restrictions (in
this case, the numerator and denominator of equation
<<triangle_perspective_interpolation>> are iterated independently and
a division performed for each fragment).
====


[[primsrast-polygonmode]]
=== Polygon Mode

The interpretation of polygons for rasterization is controlled using the
pname:polygonMode member of slink:VkPipelineRasterizationStateCreateInfo,
which takes the following values:

include::../enums/VkPolygonMode.txt[]

The pname:polygonMode selects which method of rasterization is used for
polygons. If pname:polygonMode is ename:VK_POLYGON_MODE_POINT, then the
vertices of polygons are treated, for rasterization purposes, as if they had
been drawn as points. ename:VK_POLYGON_MODE_LINE causes polygon edges to be
drawn as line segments. ename:VK_POLYGON_MODE_FILL causes polygons to render
using the polygon rasterization rules in this section.

Note that these modes affect only the final rasterization of polygons: in
particular, a polygon's vertices are shaded and the polygon is clipped and
possibly culled before these modes are applied.


[[primsrast-depthbias]]
=== Depth Bias

The depth values of all fragments generated by the rasterization of a
polygon can: be offset by a single value that is computed for that polygon.
This behavior is controlled by the pname:depthBiasEnable,
pname:depthBiasConstantFactor, pname:depthBiasClamp, and
pname:depthBiasSlopeFactor members of
slink:VkPipelineRasterizationStateCreateInfo, or by the corresponding
parameters to the fname:vkCmdSetDepthBias command if depth bias state is
dynamic.

include::../protos/vkCmdSetDepthBias.txt[]

  * pname:commandBuffer is the command buffer into which the command will be
    recorded.
  * pname:depthBiasConstantFactor is a scalar factor controlling the
    constant depth value added to each fragment.
  * pname:depthBiasClamp is the maximum (or minimum) depth bias of a
    fragment.
  * pname:depthBiasSlopeFactor is a scalar factor applied to a fragment's
    slope in depth bias calculations.

include::../validity/protos/vkCmdSetDepthBias.txt[]

If pname:depthBiasEnable is ename:VK_FALSE, no depth bias is applied and the
fragment's depth values are unchanged.

pname:depthBiasSlopeFactor scales the maximum depth slope of the polygon,
and pname:depthBiasConstantFactor scales an implementation-dependent
constant that relates to the usable resolution of the depth buffer. The
resulting values are summed to produce the depth bias value which is then
clamped to a minimum or maximum value specified by pname:depthBiasClamp.
pname:depthBiasSlopeFactor, pname:depthBiasConstantFactor, and
pname:depthBiasClamp can: each be positive, negative, or zero.

The maximum depth slope latexmath:[$m$] of a triangle is

[latexmath]
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
\begin{equation}
m = \sqrt{ \left({\partial z_f \over \partial x_f}\right)^2
 +  \left({\partial z_f \over  \partial y_f}\right)^2}
\end{equation}
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

where latexmath:[$(x_f, y_f, z_f)$] is a point on the triangle.
latexmath:[$m$] may: be approximated as

[latexmath]
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
\begin{equation}
m =  \max \left \{ \left |{\partial z_f \over \partial x_f} \right |,
\left |{\partial z_f \over \partial y_f} \right | \right \}.
\end{equation}
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

The minimum resolvable difference latexmath:[$r$] is an
implementation-dependent parameter that depends on the depth buffer
representation. It is the smallest difference in framebuffer coordinate
latexmath:[$z$] values that is guaranteed to remain distinct throughout
polygon rasterization and in the depth buffer. All pairs of fragments
generated by the rasterization of two polygons with otherwise identical
vertices, but latexmath:[$z_f$] values that differ by $r$, will have
distinct depth values.

For fixed-point depth buffer representations, latexmath:[$r$] is constant
throughout the range of the entire depth buffer. For floating-point depth
buffers, there is no single minimum resolvable difference. In this case, the
minimum resolvable difference for a given polygon is dependent on the
maximum exponent, latexmath:[$e$], in the range of latexmath:[$z$] values
spanned by the primitive. If latexmath:[$n$] is the number of bits in the
floating-point mantissa, the minimum resolvable difference, latexmath:[$r$],
for the given primitive is defined as

[latexmath]
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
\begin{equation}
r = 2^{e - n}
\end{equation}
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

If no depth buffer is present, latexmath:[$r$] is undefined.

The bias value latexmath:[$o$] for a polygon is

[latexmath]
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
\begin{equation}
o =
\begin{cases}
    m \times depthBiasSlopeFactor +
         r \times depthBiasConstantFactor  & depthBiasClamp = 0\ or\ NaN \\
    \min(m \times depthBiasSlopeFactor +
         r \times depthBiasConstantFactor,
         depthBiasClamp)                   & depthBiasClamp > 0  \\
    \max(m \times depthBiasSlopeFactor +
         r \times depthBiasConstantFactor,
         depthBiasClamp)                   & depthBiasClamp < 0  \\
\end{cases}
\end{equation}
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

latexmath:[$m$] is computed as described above. If the depth buffer uses a
fixed-point representation, latexmath:[$m$] is a function of depth values in
the range latexmath:[$[0,1\]$], and latexmath:[$o$] is applied to depth
values in the same range.

For fixed-point depth buffers, fragment depth values are always limited to
the range latexmath:[$[0,1\]$] by clamping after depth bias addition is
performed. Fragment depth values are clamped even when the depth buffer uses
a floating-point representation.