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

// Copyright (c) 2015-2017 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 [eq]#(x,y)# framebuffer coordinates, [eq]#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.

// refBegin VkPipelineRasterizationStateCreateInfo Structure specifying parameters of a newly created pipeline rasterization state

The sname:VkPipelineRasterizationStateCreateInfo structure is defined as:

include::../api/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.

ifdef::VK_AMD_rasterization_order[]
The application can: also chain a
sname:VkPipelineRasterizationStateRasterizationOrderAMD structure to the
sname:VkPipelineRasterizationStateCreateInfo structure through its
pname:pNext member.
This structure enables selecting the rasterization order to use when
rendering with the corresponding graphics pipeline as described in
<<primrast-order, Rasterization Order>>.
endif::VK_AMD_rasterization_order[]

.Valid Usage
****
  * If the <<features-features-depthClamp,depth clamping>> feature is not
    enabled, pname:depthClampEnable must: be ename:VK_FALSE
  * If the <<features-features-fillModeNonSolid,non-solid fill modes>>
    feature is not enabled, pname:polygonMode must: be
    ename:VK_POLYGON_MODE_FILL
****

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

// refBegin VkPipelineMultisampleStateCreateInfo Structure specifying parameters of a newly created pipeline multisample state

The sname:VkPipelineMultisampleStateCreateInfo structure is defined as:

include::../api/structs/VkPipelineMultisampleStateCreateInfo.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: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 fraction of sample shading, as
    described in <<primsrast-sampleshading,Sample Shading>>.
  * 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 alpha component of the fragment's first color
    output as specified in the <<fragops-covg,Multisample Coverage>>
    section.
  * pname:alphaToOneEnable controls whether the alpha component of the
    fragment's first color output is replaced with one as described in
    <<fragops-covg,Multisample Coverage>>.

.Valid Usage
****
  * If the <<features-features-sampleRateShading,sample rate shading>>
    feature is not enabled, pname:sampleShadingEnable must: be
    ename:VK_FALSE
  * If the <<features-features-alphaToOne,alpha to one>> feature is not
    enabled, pname:alphaToOneEnable must: be ename:VK_FALSE
  * pname:minSampleShading must: be in the range [eq]#[0,1]#
****

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 pname:rasterizationSamples must: be the same
as the sample count for those subpass attachments.
Otherwise, 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.


[[primrast-order]]
== Rasterization Order

Within a subpass of a <<renderpass,render pass instance>>, for a given
(x,y,layer,sample) sample location, the following operations are guaranteed
to execute in _rasterization order_, for each separate primitive that
includes that sample location:

  . <<fragops-scissor,Scissor test>>
  . <<fragops-samplemask,Sample mask generation>>)
  . <<fragops-dbt,Depth bounds test>>
  . <<fragops-stencil, Stencil test, stencil op and stencil write>>
  . <<fragops-depth, Depth test and depth write>>
  . <<fragops-samplecount,Sample counting>> for
     <<queries-occlusion,occlusion queries>>
  . <<framebuffer-blending, Blending>>, <<framebuffer-logicop, logic
    operations>>, and color writes

Each of these operations is atomically executed for each primitive and
sample location.

Execution of these operations for each primitive in a subpass occurs in
ifndef::VK_AMD_rasterization_order[]
<<drawing-primitive-order, primitive order>>.
endif::VK_AMD_rasterization_order[]
ifdef::VK_AMD_rasterization_order[]
an order determined by the application.

The application can: select a graphics pipeline to use one of the following
primitive rasterization ordering rules:

include::../api/enums/VkRasterizationOrderAMD.txt[]

  * ename:VK_RASTERIZATION_ORDER_STRICT_AMD indicates that the order of
    these operations for each primitive in a subpass must: occur in
    <<drawing-primitive-order, primitive order>>.
  * ename:VK_RASTERIZATION_ORDER_RELAXED_AMD indicates that the order of
    these operations for each primitive in a subpass may: not occur in
    <<drawing-primitive-order, primitive order>>.

The rasterization order to use for a graphics pipeline is specified by
chaining a sname:VkPipelineRasterizationStateRasterizationOrderAMD structure
through the pname:pNext member to
slink:VkPipelineRasterizationStateCreateInfo.

The sname:VkPipelineRasterizationStateRasterizationOrderAMD structure is
defined as:

include::../api/structs/VkPipelineRasterizationStateRasterizationOrderAMD.txt[]

  * pname:sType is the type of this structure.
  * pname:pNext is `NULL` or a pointer to an extension-specific structure.
  * pname:rasterizationOrder is the primitive rasterization order to use.

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

If the +VK_AMD_rasterization_order+ device extension is not enabled or the
application does not request a particular rasterization order through
specifying a sname:VkPipelineRasterizationStateRasterizationOrderAMD
structure then the rasterization order used by the graphics pipeline
defaults to ename:VK_RASTERIZATION_ORDER_STRICT_AMD.
endif::VK_AMD_rasterization_order[]


[[primsrast-multisampling]]
== Multisampling

Multisampling is a mechanism to antialias all Vulkan 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).

Vulkan 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:VkPhysicalDeviceLimits
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 [eq]##i##th entry in
the table corresponding to bit [eq]#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
|
[eq]#(0.5,0.5)#
|
[eq]#(0.25,0.25)# +
[eq]#(0.75,0.75)#
|
[eq]#(0.375, 0.125)# +
[eq]#(0.875, 0.375)# +
[eq]#(0.125, 0.625)# +
[eq]#(0.625, 0.875)#
|
[eq]#(0.5625, 0.3125)# +
[eq]#(0.4375, 0.6875)# +
[eq]#(0.8125, 0.5625)# +
[eq]#(0.3125, 0.1875)# +
[eq]#(0.1875, 0.8125)# +
[eq]#(0.0625, 0.4375)# +
[eq]#(0.6875, 0.9375)# +
[eq]#(0.9375, 0.0625)#
|
[eq]#(0.5625, 0.5625)# +
[eq]#(0.4375, 0.3125)# +
[eq]#(0.3125, 0.625)# +
[eq]#(0.75,   0.4375)# +
[eq]#(0.1875, 0.375)# +
[eq]#(0.625,  0.8125)# +
[eq]#(0.8125, 0.6875)# +
[eq]#(0.6875, 0.1875)# +
[eq]#(0.375,  0.875)# +
[eq]#(0.5,    0.0625)# +
[eq]#(0.25,   0.125)# +
[eq]#(0.125,  0.75)# +
[eq]#(0.0,    0.5)# +
[eq]#(0.9375, 0.25)# +
[eq]#(0.875,  0.9375)# +
[eq]#(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 [eq]#max({lceil}
pname:minSampleShading {times} pname: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.
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
[eq]#[pname:pointSizeRange[0],pname: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
[eq]#(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 [eq]#s# and [eq]#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 [eq]#s# and [eq]#t#:

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

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

where size is the point's size, [eq]#(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 [eq]#(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.

// refBegin vkCmdSetLineWidth Set the dynamic line width state

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::../api/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.

.Valid Usage
****
  * The currently bound graphics pipeline must: have been created with the
    ename:VK_DYNAMIC_STATE_LINE_WIDTH dynamic state enabled
  * If the <<features-features-wideLines,wide lines>> feature is not
    enabled, pname:lineWidth must: be `1.0`
****

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 [eq]#**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 [eq]#**p**~a~ = (x~a~, y~a~)# and [eq]#**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 [eq]#t = 0# at [eq]#**p**_a# and [eq]#t = 1# at [eq]#**p**~b~#.
Also note that this calculation projects the vector from [eq]#**p**~a~# to
[eq]#**p**~r~# onto the line, and thus computes the normalized distance of
the fragment along the line.)

[[line_perspective_interpolation]]
The value of an associated datum [eq]#f# for the fragment, whether it be a
shader output or the clip [eq]#w# coordinate, must: be determined using
_perspective interpolation_:

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

where [eq]#f~a~# and [eq]#f~b~# are the data associated with the starting
and ending endpoints of the segment, respectively; [eq]#w~a~# and [eq]#w~b~#
are the clip [eq]#w# coordinates of the starting and ending endpoints of the
segments, respectively.

[[line_linear_interpolation]]
Depth values for lines must: be determined using _linear interpolation_:

  :: [eq]#z = (1 - t) z~a~ + t z~b~#

where [eq]#z~a~# and [eq]#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, <<line_perspective_interpolation,
perspective interpolation>> is performed as described above.
When the code:NoPerspective decoration is used, <<line_linear_interpolation,
linear interpolation>> is performed in the same fashion as for depth values,
as described above.
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]]
image::images/non_strict_lines.svg[align="center",title="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

// refBegin VkFrontFace Interpret polygon front-facing orientation

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 [eq]#x# and [eq]#y#
framebuffer coordinates of the [eq]##i##th vertex of the [eq]#n#-vertex
polygon (vertices are numbered starting at zero for the purposes of this
computation) and [eq]#i {oplus} 1# is [eq]#(i + 1) mod n#.

The interpretation of the sign of [eq]#a# is determined by the
slink:VkPipelineRasterizationStateCreateInfo::pname:frontFace property of
the currently active pipeline, which takes the following values:

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

If pname:frontFace is set to ename:VK_FRONT_FACE_COUNTER_CLOCKWISE, a
triangle with positive area is considered front-facing.
If 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.

// refEnd VkFrontFace

// refBegin VkCullModeFlagBits Bitmask controlling triangle culling

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

include::../api/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.

// refEnd VkCullModeFlagBits

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, [eq]#a#, [eq]#b#, and
[eq]#c#, each in the range [eq]#[0,1]#, with [eq]#a + b + c = 1#.
These coordinates uniquely specify any point [eq]#p# within the triangle or
on the triangle's boundary as

  :: [eq]#p = a p~a~ + b p~b~ + c p~c~#

where [eq]#p~a~#, [eq]#p~b~#, and [eq]#p~c~# are the vertices of the
triangle.
[eq]#a#, [eq]#b#, and [eq]#c# are determined by:

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

where [eq]#A(lmn)# denotes the area in framebuffer coordinates of the
triangle with vertices [eq]#l#, [eq]#m#, and [eq]#n#.

Denote an associated datum at [eq]#p~a~#, [eq]#p~b~#, or [eq]#p~c~# as
[eq]#f~a~#, [eq]#f~b~#, or [eq]#f~c~#, respectively.

[[triangle_perspective_interpolation]]
The value of an associated datum [eq]#f# for a fragment produced by
rasterizing a triangle, whether it be a shader output or the clip [eq]#w#
coordinate, must: be determined using 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 [eq]#w~a~#, [eq]#w~b~#, and [eq]#w~c~# are the clip [eq]#w#
coordinates of [eq]#p~a~#, [eq]#p~b~#, and [eq]#p~c~#, respectively.
[eq]#a#, [eq]#b#, and [eq]#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.

[[triangle_linear_interpolation]]
Depth values for triangles must: be determined using linear interpolation:

  :: [eq]#z = a z~a~ + b z~b~ + c z~c~#

where [eq]#z~a~#, [eq]#z~b~#, and [eq]#z~c~# are the depth values of
[eq]#p~a~#, [eq]#p~b~#, and [eq]#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, <<triangle_perspective_interpolation,
perspective interpolation>> is performed as described above.
When the code:NoPerspective decoration is used,
<<triangle_linear_interpolation, linear interpolation>> is performed in the
same fashion as for depth values, as described above.
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.

ifdef::VK_AMD_shader_explicit_vertex_parameter[]
When the +VK_AMD_shader_explicit_vertex_parameter+ device extension is
enabled the code:CustomInterpAMD <<shaders-interpolation-decorations,
interpolation decoration>> can: also be used with fragment shader inputs
which indicate that the decorated inputs can: only be accessed by the
extended instruction code:InterpolateAtVertexAMD and allows accessing the
value of the inputs for individual vertices of the primitive.
endif::VK_AMD_shader_explicit_vertex_parameter[]

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

// refBegin VkPolygonMode Control polygon rasterization mode

The method of rasterization for polygons is determined by the
slink:VkPipelineRasterizationStateCreateInfo::pname:polygonMode property of
the currently active pipeline, which takes the following values:

include::../api/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.

// refEnd VkPolygonMode


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

// refBegin vkCmdSetDepthBias Set the depth bias dynamic state

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::../api/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.

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 [eq]#m# of a triangle is

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

where [eq]#(x~f~, y~f~, z~f~)# is a point on the triangle.
[eq]#m# may: be approximated as

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

The minimum resolvable difference [eq]#r# is an implementation-dependent
parameter that depends on the depth buffer representation.
It is the smallest difference in framebuffer coordinate [eq]#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 [eq]#pname:z~f~# values that differ by
$r$, will have distinct depth values.

For fixed-point depth buffer representations, [eq]#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, [eq]#e#, in the range of [eq]#z# values
spanned by the primitive.
If [eq]#n# is the number of bits in the floating-point mantissa, the minimum
resolvable difference, [eq]#r#, for the given primitive is defined as

  :: [eq]#r = 2^e-n^#

If no depth buffer is present, [eq]#r# is undefined.

The bias value [eq]#o# for a polygon is

[latexmath]
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
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}
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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

For fixed-point depth buffers, fragment depth values are always limited to
the range [eq]#[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.

.Valid Usage
****
  * The currently bound graphics pipeline must: have been created with the
    ename:VK_DYNAMIC_STATE_DEPTH_BIAS dynamic state enabled
  * If the <<features-features-depthBiasClamp,depth bias clamping>> feature
    is not enabled, pname:depthBiasClamp must: be code:0.0
****

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