Free-form surfaces are polynomial patches of any degree up to twenty-one. [11] Supported basis types include Bézier, Taylor, B-spline, cardinal, and basis-matrix form. Any type can be rational or non-rational. Patches can be explicitly or automatically connected to one another, or may be defined to contain explicitly defined points or curves in their approximation. Various approximation types including (regular) parametric, spatial, curvature-dependent, view-dependent, and combinations of these, as well as fine approximation, which can generate microtriangle tessellations very efficiently. Surfaces may be bounded by a trimming curve, and may contain holes.
Surface geometry, like polygonal geometry, is defined by a series of sections. An object containing only surface geometry follows this broad outline:
object "object_name"
[ visible [on|off] ]
[ shadow [on|off] ]
[ shadowmap [on|off] ]
[ trace [on|off] ]
[ reflection [ mode] ]
[ refraction [ mode] ]
[ transparency [ mode] ]
[ select [on|off] ]
[ tagged [on|off] ]
[ caustic [on|off] ]
[ globillum [ on|off ] ]
[ face [front|back|both] ]
[ caustic [mode] ]
[ globillum [mode] ]
[ box [minx miny minz maxx maxy maxz] ]
[ motion box [minx miny minz maxx maxy maxz] ]
[ max displace value ]
[ samples min max ]
[ data null|"data_name" ]
[ tag label_numberint ]
[ basis_list ]
group
vector_list
vertex_list
[ list of curves ]
surface
[ list of surface derivative requests ]
[ list of texture or vector surfaces ]
... # more surfaces
[ list of approximation statements ]
[ list of connection statements ]
end group
end object
Curves, surfaces, approximations, and connections may be interspersed as long as names are defined before they are used. For example, a curve must come before the surface it is trimming, and an approximation must come after the surface to be approximated. Texture and vector texture surfaces must always directly follow the surface they apply to. The individual sections are:
For a description of vector lists and vertex lists, refer to page vector.
When surfaces and curves are present within an object group, it is mandatory that at least one basis has been defined within the object. Bases define the degree and type of polynomials (denoted by Ni,n below) to be used in the description of curves or surfaces. Curves and surfaces reference bases by name. Every surface needs two bases, one for the U and one for the V parameter direction. Both can have a different degree, but must have the same type (for example, rational Bézier in U and Cardinal in V is not allowed). There are five basis types:
basis "basis_name" [ rational ] taylor degreeint
basis "basis_name" [ rational ] bezier degreeint
basis "basis_name" [ rational ] cardinal
basis "basis_name" [ rational ] bspline degreeint
basis "basis_name" [ rational ] matrix degreeint stepsizeint basis_matrix
A parametric representation may be either non-rational or rational as indicated by the rational flag. Rational curves and surfaces specify additional weights at each control point. This flag is optional; it can also be specified in the curves and surfaces that reference the basis.
The degree specifies the degree of the polynomials used in the description of curves or surfaces. Recall that the degree of a polynomial is the highest power of the parameter occurring in its definition. When bases of degree 1 are used control points are connected with straight lines. Cardinal bases always have degree 3. The degree and the type combined determine the length of the parameter vector and the number of control points needed for the surface. The meaning of the parameter vector differs for the different basis types. This is described in detail below.
The supported polynomial types for curves and surfaces are bezier, bspline, taylor, cardinal and matrix.
When a curve or surface is being evaluated and a transition from one segment or patch to the next occurs, the set of control points (the `evaluation window') used is incremented by the stepsize. The appropriate stepsize depends on the representation type expressed through the basis matrix and on the degree.
Consider a curve with k control points {v1 , … , vk}. If the curve is of degree n, then n+1 control points are needed for each polynomial segment. If the stepsize is given as s, then the (1+i)th polynomial segment will use the control points {vis+1 , … , vis+n+1}. For example, for Bézier curves s=n, whereas for Cardinal curves s=1. For surfaces, the above description applies independently to each parametric dimension.
The basis_matrix specifies the basis functions used to evaluate a parametric representation. For a basis of degree n the matrix must be of size (n+1)×(n+1). The matrix is laid out in the order b00 , b01 , … , b0n , … , bnn. Note that the generalization to the rational case for all representations is admitted in all cases.
As an example, an object containing a nonrational Bézier surface of degree 3 in one parameter direction and degree 1 in the other parameter direction needs two bases defined at the beginning of the object like this:
object "mysurface"
visible
basis "bez1" bezier 1
basis "bez3" bezier 3
group
...
The surface definition will reference the two bases by their names, bez1 and bez3.
A surface specifies a name and a list of control points. For both parametric dimensions it specifies a basis, a global parameter range, and a parameter list. Optionally, it specifies surface derivative requests, texture surfaces, trimming curves, hole curves, special curves and special points. Special curves and points are included as edges and vertices in the approximation (triangulation) of the surface.
surface "surface_name" "material_name"
"u_basis_name"range u_param_list
"v_basis_name"range v_param_list
hom_vertex_ref_list
[ derivative_request ]
[ texture_surface_list ]
[ surface_specials_list ]
If the enclosing object has the tagged flag set, a label integer must be given instead of a material name (see page tagged). This changes the first line of the preceding syntax block to:
surface "surface_name"label_numberint
The bases used in the definition of the surface must have been defined in the basis list of the object. They are referenced by their basis_names. Their ranges consist of two floating-point numbers specifying the minimum and maximum parameter values used in the respective direction.
The parameter_lists in the basis specifications define the number of patches of the surface and the number of control points. For bases of the types taylor, bezier, cardinal and matrix such a parameter_list consists of a strictly increasing list of at least two floating-point numbers. For bspline bases the parameter_lists specify the knot vector. If the B-spline basis to be used is of degree n, the knot vector ( x0 , … , xq) must have at least q+1=2(n+1) elements. Knot values represent a monotone sequence of floating-point numbers but are not necessarily strictly increasing, i.e. xi ≤ xi+1. Moreover, they must satisfy the following conditions:
(1) x0 < xn+1
(2) xq−n−1 < xq
(3) xi < xi+n for 0 < i < q−n−1
(4) xn ≤ tmin < tmax ≤ xq−n
where [tmin , tmax] is the range over which the B-spline is to be evaluated. Equation (1) demands that no more than n + 1 parameters at the beginning of the parameter list may have the same value. Equation (2) is the same restriction for the end of the parameter list. Equation (3) says that in the middle of the parameter list, at most n consecutive parameters may have the same value. To generate closed B-spline curves, it is often necessary to write a parameter list where the first n and last n parameters in the list produce initial and final curve segments that should not become part of the curve; in this case equation (4) allows choosing a start and end parameter in the range bounded by the first and last parameter of the parameter list.
The number of control points per direction can be derived from the number of parameters p, the degree of the basis n, and the step size s. Their total number can be calculated by multiplying the numbers taken from the following table for each of the U and V directions.
| type | min # of parameters | # of control points |
|---|---|---|
| Taylor | 2 | (p−1) · (n+1) |
| Bézier | 2 | (p−1) · n + 1 |
| Cardinal | 2 | p + 2 |
| Basis matrix | 2 | (p−2) · s + n + 1 |
| B-spline | 2(n+1) | p−n−1 |
Note that only certain numbers of control points are possible; for example, if the U basis is a degree-3 Bézier, the number of control points in the U direction can be 4, 7, 10, 13, and so on, but not 3 or 5. For B-spline bases of degree 3 the minimum number of parameters is 8 corresponding to 4 control points.
Each vertex reference in the hom_vertex_ref_list is an integer index into the vertex list of the current group in the object (index 0 is the first vertex). When the surface is rational, homogeneous coordinates must be given with the control points, by appending a floating-point weight to every vertex reference integer in the hom_vertex_ref_list. There are two methods for specifying weights: either a simple floating-point number that must contain a decimal point to distinguish it from an integer index, or the keyword w followed by a weight value that need not contain a decimal point. The w keyword method is recommended because it eliminates the requirement that numbers contain decimal points, so translators can use %g format specifiers. Weights are used only if the surface is rational and ignored otherwise. If a weight in a rational surface is missing, it defaults to 1.0.
The surface specials list is used to define trimming curves, hole curves, special curves, and special points (vertex references). A surface may be further modified by approximation and connection statements, as described below.
For example, an object with a simple degree-3 Bézier surface can be written as:
object "mysurface"
visible
basis "bez3" bezier 3
group
0.314772 -3.204608 -7.744229 # vector 0
0.314772 -2.146943 -6.932366
0.314772 -1.089277 -6.120503
0.314772 -0.031611 -5.308641
-0.660089 -2.650739 -8.465791 # vector 4
-0.660089 -1.593073 -7.653928
-0.660089 -0.535407 -6.842065
-0.660089 0.522259 -6.030203
-1.634951 -2.096869 -9.187352 # vector 8
-1.634951 -1.039203 -8.375489
-1.634951 0.018462 -7.563627
-1.634951 1.076128 -6.751764
-2.609813 -1.543000 -9.908914 # vector 12
-2.609813 -0.485334 -9.097052
-2.609813 0.572332 -8.285189
-2.609813 1.629998 -7.473326
v 0 v 1 v 2 v 3 # vertices
v 4 v 5 v 6 v 7
v 8 v 9 v 10 v 11
v 12 v 13 v 14 v 15
surface "surf1" "material"
"bez3" 0.0 1.0 0.0 1.0
"bez3" 0.0 1.0 0.0 1.0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
end group
end object
First, 16 vectors are defined, each of which is used to build one vertex (control point). Next, a surface is defined that uses basis bez3 for both the U and V parameter directions. Since the surface is built from only one 4 × 4 Bézier patch, the parameter vector after the basis range has only length 2. If there had been two patches in the U direction and three in the V direction, the bases would have been referenced as
"bez3" 0.0 1.0 0.0 0.5 1.0
"bez3" 0.0 1.0 0.0 0.33333 0.66667 1.0
Alternatively, the parameter vector may be given as
"bez3" 0.0 2.0 0.0 1.0 2.0
"bez3" 0.0 3.0 0.0 1.0 2.0 3.0
by changing the parameter range of the basis. This has no influence on the geometry of the surface, but generates UV texture coordinates in a different range (here, [0.0, 2.0] × [0.0, 3.0]). However, a different parameterization does affect the texture surface range (see below), and the range of trimming, hole, and special curves (which do not define their own ranges but borrow the range from the surface they apply to).
The optional surface_specials_list that completes the surface definition is a sequence of trimming curves, hole curves, special curves, and special points as described in the next section.
mental ray can automatically generate surface derivative vectors if requested. First derivatives describe the UV parametric gradient of a surface; second derivatives describe the curvature. They are computed and stored only if requested by derivative_request statements in the surface definition:
derivative numberint [ numberint ]
There can be one or more derivative statements that request first and/or second derivatives. Valid values for number are 1 and 2, for first and second derivatives, respectively.
mental ray does not use derivative vectors but makes them available to shaders. First derivatives are presented as two vectors (dS du and dS dv, with S being the surface and the derivatives evaluated at the current point in parameter space); second derivatives are presented as three vectors ( d2S du2, d2S dv2, and d2S du dv). This is the same format that can be explicitly given for polygonal data using the d keyword in vertices. Surfaces always compute the vertex derivatives analytically, explicit vertex derivatives given by d keywords are ignored.
A plain surface statement defines the geometry of the surface. If a texture is to be mapped on the surface, it is necessary to include texture surfaces. A texture surface defines a mapping from raw UV coordinates to texture coordinates as used by shaders. A vector texture is a variation of a texture surface that additionally defines a pair of basis vectors; it is used for bump mapping.
The texture or vector texture directly following a surface defines texture space number 0, the next defines texture space number 1, and so on, exactly like the first t statement after the v statement in a vertex used for building polygonal geometry defines texture space number 0, the next t defines texture space number 1, and so on. Basically, texture and vector texture surfaces replace the t statements used by polygonal geometry, because attaching textures to control points that usually are not part of the surface is not useful.
Texture spaces is what ends up in the
state→tex_list array where it can be accessed by
texture shaders to decide which texture is mapped which way.
Texture space 0 is the first entry in that array, which is used by
the shader for the first texture listed in the texture list in the
material definition. In general, there is one texture space per
texture on a material, although shaders making nonstandard use of
texture spaces could be written.
The syntax for texture surfaces is a simplified version of geometric surfaces. The texture_surface_list in the grammar summary at the beginning of the "Surfaces" section above expands to zero or more copies of the following block:
[ volume ] [ vector ] texture
"u_basis_name"u_param_list
"v_basis_name"v_param_list
vertex_ref_list
Unlike geometric surfaces, no surface name and material name is given. Bases are given like in geometric surfaces. Texture surfaces use the ranges of the geometric surface they are attached to, they are not repeated in the texture surface basis statements. The vertex_ref_list follows the same rules as the geometric surface's vertex_ref_list. Texture surfaces have no specials such as trimming curves or holes.
The optional volume keyword in the texture surface definition disables seam compensation. It should be used for 3D textures where each texture vector should be used verbatim. If the volume flag is missing, the tessellator detects textures that span the geometric seam on closed surfaces, and prevents rewinding. Consider a sphere with a 2D texture that is shifted slightly in the U parameter direction: a triangle might have u0 = 0.0 on one side and u1 = 0.1 on the other side. If the texture is shifted towards higher u coordinates by 0.05, u0 and u1 will map to texture coordinates t0 = 0.95 and t1 = 0.05, assuming an otherwise normal UV mapping. Even though u0 < u1, t0 >> t1, causing a fast "rewind" of the texture. Seam compensation corrects t1 to 1.05. This is undesirable for 3D textures, which should have the volume keyword set. Most problems with strangely shifted textures are caused by inappropriately used or missing volume keywords.
The optional vector keyword in the texture surface definition is a flag that causes bump basis vectors to be calculated during tessellation. This flag must be used if the texture surface is used for a bump map that expects to find bump basis vectors in the geometry. However, this is not a frequently used requirement - the standard mental ray shaders don't depend on them, for example.
This is an example for the simplest of all texture surfaces, a bilinear mapping:
object "mysurface"
visible
basis "bez1" bezier 1
basis "bez3" bezier 3
group
# ... 16 vectors used for the surface
0.0 0.0 0.0 # vector number 16
0.0 1.0 0.0 # vector number 17
1.0 0.0 0.0 # vector number 18
1.0 1.0 0.0 # vector number 19
# ... 16 vertices used for the surface
v 16 # vertex number 16
v 17 # vertex number 17
v 18 # vertex number 18
v 19 # vertex number 19
surface "surf1" "material"
"bez3" 0.0 1.0 0.0 1.0
"bez3" 0.0 1.0 0.0 1.0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
texture
"bez1" 0.0 1.0
"bez1" 0.0 1.0
16 17 18 19
end group
end object
This texture surface defines a bilinear mapping from the UV
coordinates computed during surface tessellation to the texture coordinates. To define other
than bilinear mappings, the texture surface needs to have more
control points than just one at every corner of the surface.
Whenever the surface tessellator generates a triangle vertex, it
uses the UV coordinate of that vertex to look up the texture
surface and interpolate the texture
coordinate from the nearest four points of the texture surface.
The resulting texture
coordinate is stored with the vertex and becomes available in
state→tex_list when the material shader is called
because a ray has hit the surface.
If more than one texture surface is given, one texture coordinate is computed for each
texture surface and stored in sequence in the generated triangle
vertices. Each texture surface is said to define a "texture space". They are
available in the state→tex_list array in the same
order. The number and order of texture surfaces should agree with
the number and order of textures given in the texture list in the
material definition. (Note that not all material shaders support
multiple textures.)
If the material name of a surface is empty (two consecutive double quotes), the surface uses the material from the closest instance (this is called material inheritance).
Curves are two-dimensional parametric curves when they are referenced by surfaces. They are used as trimming curves, hole curves, and special curves. They must be defined before the surface which references them. Curves are three-dimensional parametric curves when referenced by space curves. A curve is defined as:
curve "curve_name" "basis_name"
parameter_list
hom_vertex_ref_list
[ special special_point_list ]
The parameter_list of a curve is a list of monotonically increasing floating-point numbers that define the number of segments of the curve and the number of control points. Curve parameter lists work very much the same way as surface parameter lists except that no range needs to be provided, because they are supplied by the surfaces that reference the curve under consideration as explained in the next section. For details on parameter lists, see the sections on bases and surfaces above.
Each vertex reference in the hom_vertex_ref_list is an integer index into the vertex list of the current group in the object (index 0 is the first vertex), optionally followed by the keyword w and a weight value. (For backwards compatibility, the w keyword may be omitted if the weight is a floating-point value containing a decimal point.) Weights are used only if the curve is rational, they are ignored otherwise. If a weight in a rational curve is missing, it defaults to 1.0. The vertices indexed by the integers in the hom_vertex_ref_list should have no normals or textures (no n and t statements), and the third component of the vector ( v statement) should be 0.0 because curves are defined in UV space, not 3D space.
The optional special_point_list specifies points that are included in the approximation of the curve. After the special keyword, a sequence of integers follows that index into the vertex list, just like the integers in the hom_vertex_ref_list. The first component of the vector is used as the t parameter; it forces the point on the curve at parameter value t to become part of the curve approximation. Of course t must be in the range of parameters allowed by the surface definition.
A surface may reference curves to trim the surface, to cut holes into it, and to specify "special curves" that become part of the tessellation of the surface. Special points in surfaces work like special points in curves, except that they provide a point in the parameter range of the surface, that is, a two-dimensional UV coordinate, rather than a one-dimensional curve parameter. They specify single points on the surface that are to be included in the tessellation. As all curves and points are in UV space, the third component of the vectors provided for them is ignored. None of the above types of curves and points may exceed the range of (0.0, 1.0) at any point.
No two curves may intersect each other, and no curve may self-intersect. This is an important point because trimming curves and holes that are not closing or intersecting themselves or other loops can produce unexpected tessellation results.
Trimming, hole, and special curves and special points are defined at the end of the surface definition. The curves are composed of segments from the list of curves of the surface's group. The surface_specials_list given in the previous section is a list of zero or more of the following four items:
trim "curve_name"min max
...
hole "curve_name"min max
...
special "curve_name"min max
...
special vertexint
...
The dots indicate that each trim, hole, and special statement may be followed by more than one curve segment or vertex, respectively. All listed segments are concatenated to form a single curve.
The vertex integers specify vertices from the vertex section of the current group in the current object. Such a vertex specifies the UV coordinate of the special point that is to be included in the tessellation.
Each of the three types of curves references a curve that has been defined earlier with a curve statement. If a single trim, hole, or special statement is followed by more than one curve, the resulting trimming, hole, or special curve is pieced together by concatenating the given curves. The min and max parameters allow using only part of the curve referenced. min and max must be in the range of the parameter vector of the curve which in turn must be mapped into the parameter range of the surface. The min and max parameters of two different curve pieces are independent, they only depend on the curve parameter lists. For example, a trimming curve can be built from two curves, using the first three quarters of the first curve and the last three quarters of the second curve:
curve "trim1"
"bez1" 0.0 1.0 2.0 3.0 4.0
0 1 2 3 4
curve "trim2"
"bez1" 0.0 1.0 2.0
3 5 0
surface "patch1" "mtl"
"bez3" 0.0 1.0 0.0 1.0
"bez3" 0.0 1.0 0.0 1.0
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
trim "trim1" 0.0 3.0
"trim2" 0.5 2.0
Both trimming curves use the basis bez1, which is assumed to be a degree-1 linear curve. Hence, trim1 connects the UV vertices 0, 1, 2, 3, and 4 with straight lines, and trim2 connects the vertices 3, 5, and 0. If these two curves are put together by the trim statement in the surface definition, all parts of the surface that fall outside the polygon formed by the UV vertices 0, 1, 2, 3, and 5 are trimmed off. The trim2 curve includes vertex 0 to close the trimming curve. Holes and special curves are constructed exactly the same way. Trimming curves and holes must form closed loops but special curves are not restricted in this way.
Note that trimming and hole curves must be listed in the correct order, outside in. If there is an outer trimming curve, it must be listed first, followed by the holes. If a hole has a hole, the inner hole must be listed after the outer hole. Since curves may never intersect, there is always an unambiguous order - if a curve A encloses curve B, A must be listed before B. Curves that do not enclose one another can be listed in any order.
This example omits the vector and vertex parts of the group in the object. Here is an example that defines a complete object containing a surface with a trimming curve that precisely follows the outer boundary. A trimming curve that follows the outer surface boundary does not actually clip off any part of the surface, but it is still useful if surfaces are to be connected, because connections work on trimming curves.
object "mysurface"
visible
basis "bez1" bezier 1
basis "bez3" bezier 3
group
# ... 16 vectors used for the surface
0.0 0.0 0.0 # vector number 16
1.0 0.0 0.0 # vector number 17
1.0 1.0 0.0 # vector number 18
0.0 1.0 0.0 # vector number 19
# ... 16 vertices used for the surface
v 16 # vertex number 16
v 17 # vertex number 17
v 18 # vertex number 18
v 19 # vertex number 19
curve "trim1"
"bez1" 0.0 0.25 0.5 0.75 1.0
16 17 18 19 16
surface "surf1" "material"
...
trim "trim1" 0.0 1.0
end group
end object
The trimming curve in the example is linear, using a degree-1 Bézier basis. This means that the parameter vector has five equally-spaced parameters, one for each corner in counter-clockwise order and back to the first corner to close the trimming curve. Trimming and holes always require a closed curve or sequence of curves (they can be pieced together by multiple curves as long as the pieces form a closed loop together). The results are undefined if trimming or hole loops are not closed, or intersect.
If the trimming curve would be a degree-3 Bézier going through four corner points, a parameter vector with 3·5+1=16 parameters would be required (again, the 5 is the number of corners visited including the return to the first to close the curve).
For details on the parameter vector following the basis name in the definition of the curve, refer to section curve. The bases and parameter vectors for curves and surfaces follow the same rules, except that curves have no explicit range; they always use the implicit range given by the parameter list.
Free-form surfaces may be connected to each other using the connect statements, respectively.
The connect statement is "manual" in that it requires an explicit specification of the parts of two surfaces to be connected. These parts refer to intervals of trimming curves or hole curves of these surfaces, see section curve. A connection is defined as:
connect "surface_name1" "curve_name1" min1 max1
"surface_name2" "curve_name2" min2 max2
This statement closes the gap between two surfaces surface_name1 and surface_name2 by connecting their trimming curves curve_name1 and curve_name2.The curves are connected only in the range ( min1...max1) and ( min2...max2), respectively. They share the same points, but normals, textures etc. are evaluated on the individual surfaces. Only surfaces that have trimming curves can be connected by an explicit connect statement. Trimming curves used in connections must satisfy three conditions:
The range values min1,2 and max1,2 must not exceed the range of the trimming curve segment as referenced by a trim statement of the corresponding surface. The minimum value must be less than the maximum value; it is not possible to satisfy the third condition by inverting the range.
Best results are obtained if the curves to be connected are close to each other in world space and have at least approximately the same length. The connect statement is not meant to be a replacement for proper modeling. For carefully modeled surfaces these techniues will not be necessary most of the time. Their purpose is to close small cracks between adjacent surfaces that are already not too far from each other. Topologically complex situations with several connections meeting in a point are beyond its scope.
[11] The algorithms used impose no inherent limit. The limit may be increased in future versions.
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