Approximations are defined within an object group. 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.

flags approximate
    technique [ minint maxint ]

flags approximate surface
    technique [ minint maxint ] [ max maxint ] "surface_name" ...
flags approximate displace
    technique [ minint maxint ] "surface_name" ...

flags approximate trim
    technique [ minint maxint ] "surface_name" ...

flags approximate curve
    technique [ minint maxint ] "curve_name" ...

flags approximate space curve
    technique [ minint maxint ] "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:

[ regular ] parametric u_subdiv [ v_subdiv ]
[ regular ] parametric u_subdiv% [ v_subdiv% ]
sharp sharp
length edge
distance dist
angle angle
spatial [ view ] edge
curvature [ view ] dist angle
grading angle

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 2u_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 sharp3.1 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:

Fine Approximations

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 ]
[ sharp sharp ]
[ view ] length edge
parametric u_subdiv v_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:

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.

Flagged Approximations

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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