Approximations are defined within an object group and they specify how
previously defined polygons, surfaces, and curves should be
tessellated. Within an object group containing free-form surface
geometry the approximation statements are given separately for the
surface itself and for curves used by the surface. The
`surface` approximation statement sets the approximation
technique for the surface itself. If it carries a displacement map
this statement refers to the underlying geometric base surface and
does not take the displacement into account. One may specify the
approximation criteria on the displaced surface with an additional
`displace` approximation statement or even leave out the
`surface` approximation statement altogether.

If the material of the surface does not contain a displacement
shader the `displace` approximation statement is ignored. A
`trim` approximation statement applies to all trimming, hole
and special curves attached to the given surface or surfaces
collectively; it is equivalent to separate `curve`
approximations for each individual curve. When the keyword
`approximate` is directly followed by an approximation
technique it refers to a polygon or a list of polygons. It only has
an effect on displacement mapped polygons. If the `options` statement specifies
approximation statements for base surfaces and/or displacements,
they override the approximation statements in the object. This can
be used for quick previews with low tessellation quality, for
example.

flagsapproximatetechnique[min_{int}max_{int}]flagsapproximate surfacetechnique[min_{int}max_{int}] [ maxmax_{int}] "surface_name" ...flagsapproximate displacetechnique[min_{int}max_{int}] "surface_name" ...flagsapproximate trimtechnique[min_{int}max_{int}] "surface_name" ...flagsapproximate curvetechnique[min_{int}max_{int}] "curve_name" ...flagsapproximate space curvetechnique[min_{int}max_{int}] "spacecurve_name" ...

The dots indicate that there may be more than one
*surface_name* or *curve_name* following the
approximation statement. The given approximation is then used for
all named surfaces or curves. The flags are explained in more
detail in subsection approxflag below.

*technique* stands for one or more of the following:

view offscreen tree grid fine^{3.1}delaunay [ regular ] parametricu_subdiv[v_subdiv] [ regular ] parametricu_subdiv%[v_subdiv%] any sharpsharp^{3.1}lengthedgedistancedistangleanglespatial [ view ]edgecurvature [ view ]dist anglegradingangle

`tree`, `grid`, `fine`, and
`delaunay` are mutually exclusive. `parametric`
cannot be combined with any of the others except `grid`,
which is the default for the parametric case anyway.
`regular` can only be used together with
`parametric`. `view` has no effect unless one of
`length`, `distance`, `spatial`, or
`curvature` is also given. Grading can only be used in
combination with Delaunay triangulation.

**View-dependent approximation** is enabled if the
`view` statement is present. It controls whether the
*edge* argument of the `length` and `spatial`
statements, and the `dist` argument of the `distance`
and `curvature` statements, are in the space the object is
defined in or in raster
space.

Normally geometry outside the viewing frustum is tessellated
much more coarsely because it is not directly visible. However, in
some cases it may become visible due to large flat mirrors, or is
casting shadows on visible geometry. In these cases the
`offscreen` flag can be used to tessellate geometry outside
the viewing frustum at the full accuracy. Note that this can
generate very large numbers of triangles for things like large
displaced ground planes very close to the camera.

**Tree, grid, and Delaunay approximation** algorithms are
available for surface approximation. The grid algorithm tessellates
on a regular grid of isolines in parameter space; the tree
algorithm tessellates in a hierarchy of successive refinements that
produces fewer triangles for the same quality criteria; **Delaunay triangulation** creates a
successive refinement that maximizes triangle equiangularity. By
definition parametric approximations always use the grid algorithm;
all others can use either but `tree` is the default.
`tree`, `grid`, and `delaunay` have no effect
on curve approximations. Delaunay triangulation creates more
regular triangles but takes longer to compute.

**Parametric approximation** subdivides each patch of the
surface into *u_subdiv*· degree equal-sized pieces in the U parameter
direction, and *v_subdiv*· degree equal-sized pieces in the V parameter
direction. If `regular` the number of pieces the whole
surface is subdivided into simply equals the parameter value; the
number of subdivisions may also be specified as the percentage of
*v_subdiv* must be present for `surface` approximations
and must be omitted for `curve` and `trim`
approximations. Note that the factor is a floating point number,
although a patch can only be subdivided an integral number of
times. For example, if a factor of 2.0 is given and the surface is
of degree three, each patch will be subdivided six times in each
parametric direction. If a factor of 0.0 is given, each patch is
approximated by two triangles.

Curves are subdivided in subdiv ·
degree equal pieces by the `parametric` approximation
and into *subdiv* equal pieces by the `regular
parametric` approximation.

For displacement mapped polygons and displacement mapped
surfaces with a `displace` statement `regular
parametric` has the same meaning as `parametric` in the
approximation. For displacement mapped polygons the *u_subdiv*
constant specifies that each edge in the triangulation of the
original polygon is subdivided for the displacement 2^{u_subdiv} times. If a `displace`
approximation is given for a displacement mapped surface, the
initial tessellation of the underlying geometric surface is
subdivided in the same way as for polygons. For example, a value of
2 leads to a fourfold subdivision of each edge. Non-integer values
for the subdivision constant are admissible. Nothing is done if the
expression above is smaller than 2 (if *u_subdiv* *<
1*). The *v_subdiv* constant is ignored for the parametric
approximation of displacement maps.

**Length/distance/angle (LDA) approximation** specifies
curvature-dependent approximation according to the criteria
specified by the `length`, `distance`, and
`angle` statements. These statements can be given in any
combination and order, but cannot be combined with parametric
approximation in the same `approximate` statement. If they
are preceded by the `any` keyword the approximation stops as
soon as any of the criteria is satisfied.

The `length` statement subdivides the surface or curve
such that no edge length of the tessellation exceeds the
*edge* parameter. *edge* is given as a distance in the
space the object is defined in, or as a fraction of a pixel
diagonal in raster space if the
`view` keyword is present. Small values such as 1.0 are
recommended. For tree and grid approximation the *min* and
*max* parameters, if present, specify the minimum and maximum
number of recursion levels of the adaptive subdivision. The
*min* parameter is a means to enforce a minimal triangulation
fineness without any tests. Edges are further subdivided until they
satisfy the given criterion is fulfilled or the *max*
subdivision level is reached. The defaults are 0 and 5,
respectively; 5 is a very high number. mental ray imposes a hard
maximum of 7; mental ray 3.3 and higher have no hard limit for
displacement. Good results can often be achieved with a maximum of
3 subdivisions. For Delaunay approximation, the number *max*
following the keyword `max` specifies the maximum number of
triangles of the surface tessellation. This number will be exceeded
only if required by trimming, hole, and special curves because
every curve vertex must become part of the tessellation regardless
of the specified maximum.

For displacement mapped polygons and displacement mapped
surfaces with a `displace` approximation statement the
length criterion in the approximation limits the size of the edges
of the displaced triangles and ensures that at least all features
of this size are resolved. Subdivision stops as soon as an edge
satisfies the criterion or when the maximum subdivision level is
reached. It cannot be ruled out that at an even finer scale new
details may show up which would lead again to longer edges. This
caveat about the potential miss of high-frequency detail applies
also to the distance and angle criteria.

The `distance` statement specifies the maximum distance
*dist* between the tessellation and the actual curve or
surface. The value of *dist* is a distance in the space the
object is defined in, or a fraction of a pixel diagonal in raster space if the `view`
statement is present. As a starting point, a small distance such as
0.1 is recommended. For tree and grid approximation the *min*
and *max* parameters, if present, specify the minimum and
maximum number of recursion levels of the adaptive subdivision. For
Delaunay approximation, the number *max* following the keyword
`max` specifies the maximum number of triangles of the
surface tessellation.

For displacement mapped polygons and displacement mapped
surfaces with a `displace` approximation statement the
distance criterion cannot be used in the same way because the
displaced surface is not known analytically. Instead, the
displacements of the vertices of a triangle in the tessellation are
compared. The criterion is fulfilled only if they differ by less
than the given threshold. Subdivision is finest in areas where the
displacement changes. For example, if a black-and-white picture is
used for the displacement map the triangulation will be finest
along the borders between black and white areas but the resolution
will be lower away from them in the uniformly colored areas. In
such a case one could choose a moderately dense parametric
`surface` approximation that samples the displacement map at
sufficient density to catch small features, and use the
curvature-dependent `displace` approximation to resolve the
curvature introduced by the displacement map. Even if the base
surface is triangulated without adding interior points, as if its
trim curve defined a polygon in parameter space, it is still
possible to guarantee a certain resolution by increasing the
*min* subdivision level. Only the consecutive subdivisions are
then performed adaptively.

The `angle` statement specifies the maximum angle
*angle* in degrees between normals of adjacent tiles of a
displaced polygon or the tessellation of a surface or its
displacement or between tangents of adjacent segments of the curve
approximation. Large angles such as 45.0 are recommended. For tree
and grid approximation the *min* and *max* parameters, if
present, specify the minimum and maximum number of recursion levels
of the adaptive subdivision. For Delaunay approximation, the number
*max* following the keyword `max` specifies the maximum
number of triangles of the surface tessellation.

**Spatial approximation** as specified by a `spatial`
statement is a special case of an LDA approximation that specifies
only the `length` statement. For backwards compatibility,
the `spatial` statement has been retained; it is equivalent
to the `length` statement plus an optional `view`
statement.

**Curvature-dependent approximation** as specified by the
`curvature` statement is also a special case of LDA
approximation, equivalent to a `distance` statement, an
`angle` statement, and an optional `view` statement.
The `spatial` and `curvature` statements can be
combined, but future designs should use `length`,
`distance`, and `angle` directly.

**Grading** applies only to Delaunay triangulation controls the
density of triangles around the border of the surface. It allows
the density of triangles to vary quickly in a smooth transition
between a finer curve approximation and a coarser surface
approximation. The *angle* constant specifies a lower bound
related to the degree of the minimum angle of a triangle. Values
from 0.0 to 30.0 can be specified. Small values up to 20.0 are
recommended. The default is 0.0. When using high grading values it
is recommended to specify a maximum number of triangles because
otherwise high grading values might result in a huge number of
triangles or endless mesh refinement. The purpose of this option is
to prevent a large number of tiny triangles at the trimming or
hole curve to abruptly join very
large triangles in the interior of the surface.

The `sharp`^{3.1} keyword controls the **sharp approximation** of normal vectors. If set to 0.0, mental
ray uses the interpolated normal as specified by the base surface,
modified by displacement if available. If the argument *sharp*
is set to 1.0, mental ray will use the geometric normal for a
faceted look. This is primarily useful in `fine` mode.
mental ray allows any *sharp* value between 0.0 and 1.0.

If no approximation statement is given the parametric technique
is used by default with *u_subdiv = v_subdiv* = 0 for
surfaces, or *u_subdiv* = 0 in the case of curves and
polygons.

The following exceptions apply to subdivision surfaces:

- The
`displace`approximation is ignored, and the regular approximation is used. Subdivision surfaces are refined during displacement in a single integrated pass, so the separation between base surface and displaced surface does not apply. - There are no restrictions on the approximation techniques in fine mode. Any LDA mode may be used.

Standard approximations as described in the
previous section work under the assumption that as few triangles as
possible should be used to approximate a surface to achieve a
user-defined quality. mental ray also supports an approximation
mode called *fine approximation*, which addresses the problem
from a different angle: it is capable of efficiently expending very
large numbers of triangles to faithfully approximate even very
complex surfaces, especially displaced surfaces, without excessive
memory consumption.

This is done by reducing the granularity of mental ray's cache manager. In mental ray 3.0, it operated on entire objects, which could become very large when tessellated. mental ray 3.1 applies cache management to smaller units formed by splitting objects into smaller sets, which can be individually tessellated without excessive memory requirements. This is especially useful for extremely detailed displacement maps.

Fine approximations support a small subset of approximation techniques since the remainder exists only to trade off triangle counts vs. quality, which is no longer a problem for fine approximations:

fine [ nosmoothing ] [ sharpsharp] [ view ] lengthedgeparametricu_subdivv_subdiv

The `fine` keyword enables fine approximation. It can be
used for polygon displacement, free-form surface displacement,
subdivision surface displacement and free-form surface
approximations, but not for curves. `fine nosmoothing` can
be used for polygon displacement for turning off a smoothing
procedure which is used by default during fine polygon
displacement. As with standard approximations, the `sharp`
keyword controls normal-vector calculations. If set to 0.0, mental
ray uses the interpolated normal as specified by the base surface,
modified by displacement if available; if set to 1.0, mental ray
will use the geometric normal to achieve a sharp faceted look.
[12]

Fine approximation requires the choice of one of three techniques:

`view length`specifies that all triangles should be subdivided until they are smaller than*edge*pixel diagonals. The*edge*value is typically around 0.5, or 0.25 or even 0.1 for very high quality. This technique is recommended for all fine approximations.`length`specifies that the triangle edge length should stay under*edge*units in the object's object space. The*edge*parameter needs more careful tuning than in the view-dependent case, and very high values defeat the purpose of fine approximation.`parametric`[13] tessellates a free-form surface such that all microtriangles have the same size. This results in very regular meshes but is harder to optimize, and may use significantly more memory.

This simplicity makes it very easy to control fine displacement, without the risk of accidentally creating billions of triangles until memory runs out, and without juggling a large number of temperamental displacement-mapping parameters.

However, fine displacement critically depends on the
specification of a cache size limit, because otherwise the fine
tessellation results would not flow through the cache but
accumulate until memory runs out. The default cache limit
(unlimited) can be overridden with the `-memory`
command-line option. A good choice is 100 MB less than the maximum
allocatable memory on 32-bit machines. If the number is too large,
the operating system may run out of virtual address space; if it is
too small, mental ray will perform too many cache flush
operations.

If *fine* is used to approximate displaced geometry, it is
also important to specify a correct *max displace* value. This
parameter specifies the maximum absolute scalar value a
displacement shader may return and serves to give mental ray a hint
of the maximum extension of the displaced object. It is measured in
object space. This parameter is
somewhat "sensible": if chosen too large, it may affect rendering
performance; if chosen too small, any larger values returned by a
displacement shader will be clipped. mental ray issues a warning
message if *max displace* is chosen too small and a greater
value is returned by the shader during rendering, thus providing a
way to adjust *max displace* optimally. mental ray relies on
max displace exclusively; if accidentally left at the default of
zero, all displacement will disappear (with a warning message).

Fine approximation cannot be used together with merging and connections.

mental ray 3.2 allows flagged approximations. An
object or instance may optionally prefix `approximate`
statements with one or more of the flag keywords `visible`,
`trace`, `shadow`, `caustic`, and
`globillum`. The approximation then applies only if the
object is hit by the corresponding ray or photon. If no flags are
given, the approximation applies to all rays and photons, which is
equivalent to specifying all five flags. For example,

visible approximate fine view length 0.5 ... trace shadow approximate regular parametric 3 3 ... visible approximate displace fine view length 0.25 ...

specifies that where the object is visible to primary rays, it
will be fairly detailed (fine view length 0.5), but its
reflections, refractions, and shadows are very coarse (regular
parametric 3 3). Visible displacements are even finer (fine view
length 0.25). In particular, final gathering falls under
`trace`, and will also be coarsely approximated. In this
example, mental ray will tessellate the object twice, once for
`visible` and once for `trace shadow`. Missing
approximations, in this example for caustic and globillum, use the
default parametric 0 0. The trace flag also applies to final gathering and probe rays
(*mi_trace_probe*).

The intended use is for simple stand-ins. If the scene contains
many detailed visible objects, it is undesirable to use the same
level of detail for trace objects to compute shadows or global
illumination because those rays are far less coherent, and pull in
far more objects into the geometry cache simultaneously. Hence,
simpler stand-ins should be used. This can be done by specifying
entire separate objects with `visible` and `trace`
object flags, but it is simpler to use the same objects and
applying the flags only to the approximations.

Since low-resolution shadow and trace stand-in objects are
usually a small distance offset from the high-resolution visible
object, it is often useful to offset rays at least that distance
away from the visible object, using *mi_ray_offset* in the shader.

[12] mental ray supports any value between 0.0 and 1.0.

[13] Parametric approximation is supported only for free-form surfaces and not for displaced geometry.

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