13.1 Introduction

13.1.1 Name

The name of this component is "Geometry3D". This name shall be used when referring to this component in the COMPONENT statement (see 7.2.5.4 Component statement).

13.1.2 Overview

This clause describes the Geometry3D component of this part of ISO/IEC 19775. This includes how 3D geometry is specified and what shapes are available. Table 13.1 provides links to the major topics in this clause.

13.2 Concepts

13.2.1 Overview of geometry

The geometry component consists of four types of nodes:  shape, geometry, geometry property, and appearance. Together, these node types are used to describe the visual elements of a X3D world.

13.2.2 Shape and geometry nodes

The Shape node associates a geometry node with nodes that define that geometry's appearance. Shape nodes must be part of the transformation hierarchy to have any visible result, and the transformation hierarchy must contain Shape nodes for any geometry to be visible (the only nodes that render visible results are Shape nodes and the background nodes in 24 Environmental effects). A Shape node contains exactly one geometry node in its geometry field, which is of type X3DGeometryNode. For more on the Shape node, see 12 Shape component.

Other components may define additional geometry node types.

13.2.3 Geometric property nodes

Several geometry nodes contain geometric property nodes such as Coordinate, Color, ColorRGBA, and/or Normal. These nodes are specified in 11 Rendering component. The X3DTextureCoordinate nodes specified in 18 Texturing component are also geometry property nodes.

13.2.4 Appearance nodes

Shape nodes may specify an Appearance node that describes the appearance properties (material and texture) to be applied to the Shape's geometry. Appearance is described in 12 Shape component.

13.2.5 Common geometry fields

Several 3D geometry nodes share common fields to describe attributes. These fields specify the vertex ordering, if the shape is solid, if the shape contains convex faces, and at what angle a crease appears between faces, and are named ccw, solid, convex and creaseAngle, respectively. Common 3D geometry fields are described in 11 Rendering component.

13.3 Node reference

13.3.1 Box

Box : X3DGeometryNode { 
SFNode  [in,out] metadata NULL  [X3DMetadataObject]
SFVec3f []       size     2 2 2 (0,∞)
SFBool  []       solid    TRUE
}

The Box node specifies a rectangular parallelepiped box centred at (0, 0, 0) in the local coordinate system and aligned with the local coordinate axes. By default, the box measures 2 units in each dimension, from -1 to +1. The size field specifies the extents of the box along the X-, Y-, and Z-axes respectively and each component value shall be greater than zero. Figure 13.1 illustrates the Box node.

Figure 13.1 — Box node

Textures are applied individually to each face of the box. On the front (+Z), back (-Z), right (+X), and left (-X) faces of the box, when viewed from the outside with the +Y-axis up, the texture is mapped onto each face with the same orientation as if the image were displayed normally in 2D. On the top face of the box (+Y), when viewed from above and looking down the Y-axis toward the origin with the -Z-axis as the view up direction, the texture is mapped onto the face with the same orientation as if the image were displayed normally in 2D. On the bottom face of the box (-Y), when viewed from below looking up the Y-axis toward the origin with the +Z-axis as the view up direction, the texture is mapped onto the face with the same orientation as if the image were displayed normally in 2D. TextureTransform affects the texture coordinates of the Box (see 18.4.8 TextureTransform).

The solid field determines whether the box is visible when viewed from the inside. 11.2.3 Common geometry fields provides a complete description of the solid field.

13.3.2 Cone

Cone : X3DGeometryNode { 
SFNode  [in,out] metadata     NULL [X3DMetadataObject]
SFBool  [in out] bottom       TRUE
SFFloat []       bottomRadius 1    (0,∞)
SFFloat []       height       2    (0,∞)
SFBool  [in out] side         TRUE
SFBool  []       solid        TRUE
}

The Cone node specifies a cone which is centred in the local coordinate system and whose central axis is aligned with the local Y-axis. The bottomRadius field specifies the radius of the cone's base, and the height field specifies the height of the cone from the centre of the base to the apex. By default, the cone has a radius of 1.0 at the bottom and a height of 2.0, with its apex at y = height/2 and its bottom at y = - height/2. Both bottomRadius and height shall be greater than zero. Figure 13.2 illustrates the Cone node.

Figure 13.2 — Cone node

The side field specifies whether sides of the cone are created and the bottom field specifies whether the bottom cap of the cone is created. A value of TRUE specifies that this part of the cone exists, while a value of FALSE specifies that this part does not exist (not rendered or eligible for collision or sensor intersection tests).

When a texture is applied to the sides of the cone, the texture wraps counterclockwise (from above) starting at the back of the cone. The texture has a vertical seam at the back in the X=0 plane, from the apex (0,  height/2, 0) to the point (0, - height/2, - bottomRadius). For the bottom cap, a circle is cut out of the texture square centred at (0, - height/2, 0) with dimensions (2 ×  bottomRadius) by (2 ×  bottomRadius). The bottom cap texture appears right side up when the top of the cone is rotated towards the -Z-axis. TextureTransform affects the texture coordinates of the Cone (see 18.4.8 TextureTransform).

The solid field determines whether the cone is visible when viewed from the inside. 11.2.3 Common geometry fields provides a complete description of the solid field.

This geometry node is fundamentally a mathematical representation. Displayed geometry shall have sufficient rendering quality that surface and silhouette edges appear smooth, including when textures are applied.

13.3.3 Cylinder

Cylinder : X3DGeometryNode { 
SFNode  [in,out] metadata NULL [X3DMetadataObject]
SFBool  [in out] bottom   TRUE
SFFloat []       height   2    (0,∞)
SFFloat []       radius   1    (0,∞)
SFBool  [in out] side     TRUE
SFBool  []       solid    TRUE
SFBool  [in out] top      TRUE
}

The Cylinder node specifies a capped cylinder centred at (0,0,0) in the local coordinate system and with a central axis oriented along the local Y-axis. By default, the cylinder is sized at "-1" to "+1" in all three dimensions. The radius field specifies the radius of the cylinder and the height field specifies the height of the cylinder along the central axis. Both radius and height shall be greater than zero. Figure 13.3 illustrates the Cylinder node.

The cylinder has three parts: the side, the top (Y = +height/2) and the bottom (Y = −height/2). Each part has an associated SFBool field that indicates whether the part exists (TRUE) or does not exist (FALSE). Parts which do not exist are not rendered and not eligible for intersection tests (EXAMPLE  collision detection or sensor activation).

Figure 13.3 — Cylinder node

When a texture is applied to a cylinder, it is applied differently to the sides, top, and bottom. On the sides, the texture wraps counterclockwise (from above) starting at the back of the cylinder. The texture has a vertical seam at the back, intersecting the X=0 plane. For the top and bottom caps, a circle is cut out of the unit texture squares centred at (0, ±height/2, 0) with dimensions 2 ×  radius by 2 ×  radius. The top texture appears right side up when the top of the cylinder is tilted toward the +Z-axis, and the bottom texture appears right side up when the top of the cylinder is tilted toward the −Z-axis. TextureTransform affects the texture coordinates of the Cylinder node (see 18.4.8 TextureTransform).

The solid field determines whether the cylinder is visible when viewed from the inside. 11.2.3 Common geometry fields provides a complete description of the solid field.

This geometry node is fundamentally a mathematical representation. Displayed geometry shall have sufficient rendering quality that surface and silhouette edges appear smooth, including when textures are applied.

13.3.4 ElevationGrid

ElevationGrid : X3DGeometryNode {
MFFloat [in]     set_height
MFNode  [in,out] attrib          []   [X3DVertexAttributeNode]
SFNode  [in,out] color           NULL [X3DColorNode]
SFNode  [in,out] fogCoord        NULL [FogCoordinate]
SFNode  [in,out] metadata        NULL [X3DMetadataObject]
SFNode  [in,out] normal          NULL [X3DNormalNode]
SFNode  [in,out] texCoord        NULL [X3DTextureCoordinateNode]
SFBool  []       ccw             TRUE  
SFBool  []       colorPerVertex  TRUE
SFFloat []       creaseAngle     0    [0,∞)
MFFloat []       height          []   (-∞,∞)
SFBool  []       normalPerVertex TRUE
SFBool  []       solid           TRUE
SFInt32 []       xDimension      0    [0,∞)
SFFloat []       xSpacing        1.0  (0,∞)
SFInt32 []       zDimension      0    [0,∞)
SFFloat []       zSpacing        1.0  (0,∞)
}

The ElevationGrid node specifies a uniform rectangular grid of varying height in the Y=0 plane of the local coordinate system. The geometry is described by a scalar array of height values that specify the height of a surface above each point of the grid.

The xDimension and zDimension fields indicate the number of elements of the grid height array in the X and Z directions. Both xDimension and zDimension shall be greater than or equal to zero. If either the xDimension or the zDimension is less than two, the ElevationGrid contains no quadrilaterals. The vertex locations for the rectangles are defined by the height field and the xSpacing and zSpacing fields:

• The height field is an xDimension by zDimension array of scalar values representing the height above the grid for each vertex.
• The xSpacing and zSpacing fields indicate the distance between vertices in the X and Z directions respectively, and shall be greater than zero.

Thus, the vertex corresponding to the point P[i, j] on the grid is placed at:

  P[i,j].x = xSpacing × i

P[i,j].y = height[ i + j × xDimension]

P[i,j].z = zSpacing × j

where 0 ≤ i < xDimension and 0 ≤ j < zDimension,
and P[0,0] is height[0] units above/below the origin of the local
coordinate system

If the rendering algorithm being used requires tessellation, the quadrilaterals are split into triangles along the seam starting at the initial vertex of the quadrilateral and proceeding to the opposite vertex. The positive direction for the normal of each triangle shall be on the same side of the quadrilateral. The triangles are defined either counterclockwise or clockwise depending on the value of the ccw field.

EXAMPLE  In Figure 13.4 with the ccw field set to TRUE, the first polygon is split into triangles with vertices [0, 5, 6] and vertices [0, 6, 1].

The set_height inputOnly field allows the height MFFloat field to be changed to support animated ElevationGrid nodes.

The color field specifies per-vertex or per-quadrilateral colours for the ElevationGrid node depending on the value of colorPerVertex. If the color field is NULL, the ElevationGrid node is rendered with the overall attributes of the Shape node enclosing the ElevationGrid node (see 12 Shape component).

The colorPerVertex field determines whether colours specified in the color field are applied to each vertex or each quadrilateral of the ElevationGrid node. If colorPerVertex is FALSE and the color field is not NULL, the color field shall specify a node derived from X3DColorNode containing at least ( xDimension-1)×( zDimension-1) colours; one for each quadrilateral, ordered as follows:

    QuadColor[i,j] = Color[ i + j × (xDimension-1)]

where 0 ≤ i < xDimension-1 and 0 ≤ j < zDimension-1,
and QuadColor[i,j] is the colour for the quadrilateral defined
by height[i+j × xDimension], height[(i+1)+j × xDimension],
height[(i+1)+(j+1) × xDimension] and height[i+(j+1) × xDimension]

If colorPerVertex is TRUE and the color field is not NULL, the color field shall specify a node derived from X3DColorNode containing at least xDimension ×  zDimension colours, one for each vertex, ordered as follows:

    VertexColor[i,j] = Color[ i + j × xDimension ]

where 0 ≤ i < xDimension and 0 ≤ j < zDimension,
and VertexColor[i,j] is the colour for the vertex defined by
height[ i+j × xDimension ]

The normal field specifies per-vertex or per-quadrilateral normals for the ElevationGrid node. If the normal field is NULL, the browser shall automatically generate normals, using the creaseAngle field to determine if and how normals are smoothed across the surface (see 11.2.3 Common geometry fields).

The normalPerVertex field determines whether normals are applied to each vertex or each quadrilateral of the ElevationGrid node depending on the value of normalPerVertex. If normalPerVertex is FALSE and the normal node is not NULL, the normal field shall specify a node derived from X3DNormalNode containing at least ( xDimension−1)×( zDimension−1) normals; one for each quadrilateral, ordered as follows:

    QuadNormal[i,j] = Normal[ i + j × (xDimension-1)]

where 0 ≤ i < xDimension-1 and 0 ≤ j < zDimension-1,
and QuadNormal[i,j] is the normal for the quadrilateral defined
by height[i+j × xDimension], height[(i+1)+j × xDimension],
height[(i+1)+(j+1) × xDimension] and height[i+(j+1) × xDimension]

If normalPerVertex is TRUE and the normal field is not NULL, the normal field shall specify a node derived from X3DNormalNode containing at least xDimension ×  zDimension normals; one for each vertex, ordered as follows:

    VertexNormal[i,j] = Normal[ i + j × xDimension]

where 0 ≤ i < xDimension and 0 ≤ j < zDimension,
and VertexNormal[i,j] is the normal for the vertex defined
by height[i+j × xDimension]

The texCoord field specifies per-vertex texture coordinates for the ElevationGrid node. If texCoord is NULL, default texture coordinates are applied to the geometry. The default texture coordinates range from (0,0) at the first vertex to (1,1) at the last vertex. The S texture coordinate is aligned with the positive X-axis, and the T texture coordinate with positive Z-axis. If texCoord is not NULL, it shall specify a node derived from X3DTextureCoordinateNode containing at least ( xDimension)×( zDimension) texture coordinates; one for each vertex, ordered as follows:

    VertexTexCoord[i,j] = TextureCoordinate[ i + j × xDimension]

where 0 ≤ i < xDimension and 0 ≤ j < zDimension,
and VertexTexCoord[i,j] is the texture coordinate for the vertex
defined by height[i+j × xDimension]

The ccw, solid, and creaseAngle fields are described in 11.2.3 Common geometry fields.

By default, the quadrilaterals are defined with a counterclockwise ordering. Hence, the Y-component of the normal is positive. Setting the ccw field to FALSE reverses the normal direction. Backface culling is enabled when the solid field is TRUE.

See Figure 13.4 for a depiction of the ElevationGrid node.

Figure 13.4 — ElevationGrid node

13.3.5 Extrusion

13.3.5.1 Syntax

Extrusion : X3DGeometryNode {
MFVec2f    [in]     set_crossSection
MFRotation [in]     set_orientation
MFVec2f    [in]     set_scale
MFVec3f    [in]     set_spine
SFNode     [in,out] metadata         NULL                      [X3DMetadataObject]
SFBool     []       beginCap         TRUE
SFBool     []       ccw              TRUE
SFBool     []       convex           TRUE
SFFloat    []       creaseAngle      0                         [0,∞)
MFVec2f    []       crossSection     [1 1 1 -1 -1 -1 -1 1 1 1] (-∞,∞)
SFBool     []       endCap           TRUE
MFRotation []       orientation      0 0 1 0                   [-1,1] or (-∞,∞)
MFVec2f    []       scale            1 1                       (0,∞)
SFBool     []       solid            TRUE
MFVec3f    []       spine            [0 0 0 0 1 0]             (-∞,∞)
}   

13.3.5.2 Overview

The Extrusion node specifies geometric shapes based on a two dimensional cross-section extruded along a three dimensional spine in the local coordinate system. The cross-section can be scaled and rotated at each spine point to produce a wide variety of shapes.

An Extrusion node is defined by:

1. a 2D crossSection piecewise linear curve (described as a series of connected vertices);
2. a 3D spine piecewise linear curve (also described as a series of connected vertices);
3. a list of 2D scale parameters;
4. a list of 3D orientation parameters.

13.3.5.3 Algorithmic description

Shapes are constructed as follows. The cross-section curve, which starts as a curve in the Y=0 plane, is first scaled about the origin by the first scale parameter (first value scales in X, second value scales in Z). It is then translated by the first spine point and oriented using the first orientation parameter (as explained later). The same procedure is followed to place a cross-section at the second spine point, using the second scale and orientation values. Corresponding vertices of the first and second cross-sections are then connected, forming a quadrilateral polygon between each pair of vertices. This same procedure is then repeated for the rest of the spine points, resulting in a surface extrusion along the spine.

The final orientation of each cross-section is computed by first orienting it relative to the spine segments on either side of point at which the cross-section is placed. This is known as the spine-aligned cross-section plane (SCP), and is designed to provide a smooth transition from one spine segment to the next (see Figure 13.5).

Figure 13.5 — Spine-aligned cross-section plane (SCP) at a spine point.

The SCP for each point is determined by first computing its Y-axis and Z-axis, then taking the cross product of these to determine the X-axis. These three axes are then used to determine the rotation value needed to rotate the Y=0 plane to the SCP. This results in a normal to the plane that is the approximate tangent of the spine at the point, as shown in Figure 13.5. First the Y-axis is determined, as follows:

Let n be the number of spines and let i be the index variable satisfying
0 ≤ i < n:

1. For all points other than the first or last: The Y-axis for spine[i] is found by normalizing the vector defined by
   ( spine[i+1] − spine[i−1]).
2. If the spine curve is closed: The SCP for the first and last points is the same and is found using ( spine[1] − spine[n−2]) to compute the Y-axis.
3. If the spine curve is not closed: The Y-axis used for the first point is the vector from spine[0] to spine[1], and for the last it is the vector from spine[ n−2] to spine[ n−1].

The Z-axis is determined as follows:

4. For all points other than the first or last: Take the following cross-product:
   

   	Z = (spine[i+1] − spine[i]) × (spine[i-1] − spine[i])
   

5. If the spine curve is closed: The SCP for the first and last points is the same and is found by taking the following cross-product:

   	Z = (spine[1] − spine[0]) × (spine[n-2] − spine[0])
   

6. If the spine curve is not closed: The Z-axis used for the first spine point is the same as the Z-axis for spine[ 1]. The Z-axis used for the last spine point is the same as the Z-axis for spine[ n−2].
7. After determining the Z-axis, its dot product with the Z-axis of the previous spine point is computed. If this value is negative, the Z-axis is flipped (multiplied by −1). In most cases, this prevents small changes in the spine segment angles from flipping the cross-section 180 degrees.

Once the Y- and Z-axes have been computed, the X-axis can be calculated as their cross-product.

Each SCP is then rotated by the corresponding orientation value. This rotation is performed relative to the SCP itself. For example, to impart twist in the cross-section, a rotation about the local Y-axis (0 1 0) would be used. Other orientation values are valid and may rotate the cross-section out of the plane of the original SCP.

13.3.5.4 Special cases

13.3.5.4.1 Overview

There are a number of special cases require specific handling. These concern the numbers of values or points, and collinear or coincident spine points.

13.3.5.4.2 Number of scale or orientation values

If the number of scale or orientation values is greater than the number of spine points, the excess values are ignored. If they contain one value, it is applied at all spine points. The results are undefined if the number of scale or orientation values is greater than one but less than the number of spine points.

13.3.5.4.3 Collinear spine points

If the three points used in computing the Z-axis are collinear, the cross-product is zero so the value from the previous point is used instead.

If the Z-axis of the first point is undefined (because the spine is not closed and the first two spine segments are collinear) then the Z-axis for the first spine point with a defined Z-axis is used.

If the entire spine is collinear, the SCP for all the spine points is computed by finding the rotation of a vector along the positive Y-axis ( v1 ) to the vector ( v2 ) defined by (spine[n] - spine [0]), where spine[n] is the first spine point not coincident with spine [0].If v2 is parallel to and in the direction of the negative-Y axis, the rotation will be a 180 degree rotation about the Z-axis. The Y=0 plane is then rotated by this value.

13.3.5.4.4 Coincident spine points

If two or more sequential points in a spine array are coincident, they are each treated as a single point when computing the corresponding SCP, and each will have an identical SCP.

NOTE  This case is useful when animating the spine array without needing to simultaneously modify the corresponding orientation and scale arrays.

If each coincident point has a different orientation value, the surface is constructed by connecting edges of the cross-sections as normal. This is useful in creating revolved surfaces.

NOTE  Combining coincident and non-coincident spine segments, as well as other combinations, can lead to interpenetrating surfaces which the extrusion algorithm makes no attempt to avoid.

13.3.5.4.5 Number of distinct spine points

If only 2 distinct, non-coincident, spine points are provided, the corresponding SCP planes for each are perpendicular to the vector defined by these two points.

If fewer than 2 non-coincident spine points are provided, the extrusion is not well defined and no results are rendered.

13.3.5.5 Common cases

The following common cases are among the effects which are supported by the Extrusion node:

Surfaces of revolution:

If the cross-section is an approximation of a circle and the spine is straight, the Extrusion is equivalent to a surface of revolution, where the scale parameters define the size of the cross-section along the spine.

Uniform extrusions:

If the scale is (1, 1) and the spine is straight, the cross-section is extruded uniformly without twisting or scaling along the spine. The result forms a parallelepiped with a uniform cross section.

Bend/twist/taper objects:

These shapes are the result of using all fields. The spine curve bends the extruded shape defined by the cross-section, the orientation parameters (given as rotations about the Y-axis) twist it around the spine, and the scale parameters taper it (by scaling about the spine).

13.3.5.6 Other fields

Extrusion has three parts: the sides, the beginCap (the surface at the initial end of the spine) and the endCap (the surface at the final end of the spine). The caps have an associated SFBool field that indicates whether each exists (TRUE) or doesn't exist (FALSE).

When the beginCap or endCap fields are specified as TRUE, planar cap surfaces will be generated regardless of whether the crossSection is a closed curve. If crossSection is not a closed curve, the caps are generated by adding a final point to crossSection that is equal to the initial point. An open surface can still have a cap, resulting (for a simple case) in a shape analogous to a soda can sliced in half vertically. These surfaces are generated even if spine is also a closed curve. If a field value is FALSE, the corresponding cap is not generated.

Texture coordinates are automatically generated by Extrusion nodes. Textures are mapped so that the coordinates range in the U direction from 0 to 1 along the crossSection curve (with 0 corresponding to the first point in crossSection and 1 to the last) and in the V direction from 0 to 1 along the spine curve (with 0 corresponding to the first listed spine point and 1 to the last). If either the endCap or beginCap exists, the crossSection curve is uniformly scaled and translated so that the larger dimension of the cross-section (X or Z) produces texture coordinates that range from 0.0 to 1.0. The beginCap and endCap textures' S and T directions correspond to the X and Z directions in which the crossSection coordinates are defined.

The browser shall automatically generate normals for the Extrusion node, using the creaseAngle field to determine if and how normals are smoothed across the surface. Normals for the caps are generated along the Y-axis of the SCP, with the ordering determined by viewing the cross-section from above (looking along the negative Y-axis of the SCP). By default, a beginCap with a counterclockwise ordering shall have a normal along the negative Y-axis. An endCap with a counterclockwise ordering shall have a normal along the positive Y-axis.

Each quadrilateral making up the sides of the extrusion are ordered from the bottom cross-section (the one at the earlier spine point) to the top. So, one quadrilateral has the points:

    spine[0](crossSection[0], crossSection[1])
spine[1](crossSection[1], crossSection[0])

in that order. By default, normals for the sides are generated as described in 13.2.2 Shape and geometry nodes.

For instance, a circular crossSection with counter-clockwise ordering and the default spine form a cylinder. With solidTRUE and ccwTRUE, the cylinder is visible from the outside. Changing ccw to FALSE makes it visible from the inside.

The ccw, solid, convex, and creaseAngle fields are described in 11.2.3 Common geometry fields.

13.3.6 IndexedFaceSet

IndexedFaceSet : X3DComposedGeometryNode {
MFInt32 [in]     set_colorIndex
MFInt32 [in]     set_coordIndex
MFInt32 [in]     set_normalIndex
MFInt32 [in]     set_texCoordIndex
MFNode  [in,out] attrib            []   [X3DVertexAttributeNode]
SFNode  [in,out] color             NULL [X3DColorNode]
SFNode  [in,out] coord             NULL [X3DCoordinateNode]
SFNode  [in,out] fogCoord          NULL [FogCoordinate]
SFNode  [in,out] metadata          NULL [X3DMetadataObject]
SFNode  [in,out] normal            NULL [X3DNormalNode]
SFNode  [in,out] texCoord          NULL [X3DTextureCoordinateNode]
SFBool  []       ccw               TRUE
MFInt32 []       colorIndex        []   [0,∞) or -1
SFBool  []       colorPerVertex    TRUE
SFBool  []       convex            TRUE
MFInt32 []       coordIndex        []   [0,∞) or -1
SFFloat []       creaseAngle       0    [0,∞)
MFInt32 []       normalIndex       []   [0,∞) or -1
SFBool  []       normalPerVertex   TRUE
SFBool  []       solid             TRUE
MFInt32 []       texCoordIndex     []   [-1,∞)
}

The IndexedFaceSet node represents a 3D shape formed by constructing faces (polygons) from vertices listed in the coord field. The coord field contains a Coordinate node that defines the 3D vertices referenced by the coordIndex field. IndexedFaceSet uses the indices in its coordIndex field to specify the polygonal faces by indexing into the coordinates in the Coordinate node. An index of "−1" indicates that the current face has ended and the next one begins. The last face may be (but does not have to be) followed by a "−1" index. If the greatest index in the coordIndex field is N, the Coordinate node shall contain N+1 coordinates (indexed as 0 to N). Each face of the IndexedFaceSet shall have:

1. at least three non-coincident vertices;
2. vertices that define a planar polygon;
3. vertices that define a non-self-intersecting polygon.

Otherwise, The results are undefined.

The IndexedFaceSet node is specified in the local coordinate system and is affected by the transformations of its ancestors.

Descriptions of the coord, normal, and texCoord fields are provided in Coordinate, X3DNormalNode, and X3DTextureCoordinateNode, respectively.

Details on lighting equations and the interaction between color field, normal field, textures, materials, and geometries are provided in 11 Rendering component and 12 Shape component.

If the color field is not NULL, it shall contain a node derived from X3DColorNode whose colours are applied to the vertices or faces of the IndexedFaceSet as follows:

4. If colorPerVertex is FALSE, colours are applied to each face, as follows:
   1. If the colorIndex field is not empty, one colour is used for each face of the IndexedFaceSet. There shall be at least as many indices in the colorIndex field as there are faces in the IndexedFaceSet. If the greatest index in the colorIndex field is N, there shall be N+1 colours in the X3DColorNode. The colorIndex field shall not contain any negative entries.
   2. If the colorIndex field is empty, the colours in the X3DColorNode node are applied to each face of the IndexedFaceSet in order. There shall be at least as many colours in the X3DColorNode node as there are faces.
5. If colorPerVertex is TRUE, colours are applied to each vertex, as follows:
   1. If the colorIndex field is not empty, colours are applied to each vertex of the IndexedFaceSet in exactly the same manner that the coordIndex field is used to choose coordinates for each vertex from the Coordinate node. The colorIndex field shall contain at least as many indices as the coordIndex field, and shall contain end-of-face markers (−1) in exactly the same places as the coordIndex field. If the greatest index in the colorIndex field is N, then there shall be N+1 colours in the X3DColorNode node.
   2. If the colorIndex field is empty, the coordIndex field is used to choose colours from the X3DColorNode node. If the greatest index in the coordIndex field is N, then there shall be N+1 colours in the X3DColorNode node.

If the color field is NULL, the geometry shall be rendered normally using the Material and texture defined in the Appearance node (see 12 Shape component for details).

If the normal field is not NULL, it shall contain a node derived from X3DNormalNode whose normals are applied to the vertices or faces of the IndexedFaceSet in a manner exactly equivalent to that described above for applying colours to vertices/faces (where normalPerVertex corresponds to colorPerVertex and normalIndex corresponds to colorIndex). If the normal field is NULL, the browser shall automatically generate normals, using creaseAngle to determine if and how normals are smoothed across shared vertices (see 11.2.3 Common geometry fields).

If the texCoord field is not NULL, it shall contain a node derived from X3DTextureCoordinateNode. The texture coordinates in that node are applied to the vertices of the IndexedFaceSet as follows:

6. If the texCoordIndex field is not empty, then it is used to choose texture coordinates for each vertex of the IndexedFaceSet in exactly the same manner that the coordIndex field is used to choose coordinates for each vertex from the Coordinate node. The texCoordIndex field shall contain at least as many indices as the coordIndex field, and shall contain end-of-face markers (−1) in exactly the same places as the coordIndex field. If the greatest index in the texCoordIndex field is N, then there shall be N+1 texture coordinates in the X3DTextureCoordinateNode.
7. If the texCoordIndex field is empty, then the coordIndex array is used to choose texture coordinates from the X3DTextureCoordinateNode node. If the greatest index in the coordIndex field is N, then there shall be N+1 texture coordinates in the X3DTextureCoordinateNode node.

If the texCoord field is NULL, a default texture coordinate mapping is calculated using the local coordinate system bounding box of the shape. The longest dimension of the bounding box defines the S coordinates, and the next longest defines the T coordinates. If two or all three dimensions of the bounding box are equal, ties shall be broken by choosing the X, Y, or Z dimension in that order of preference. The value of the S coordinate ranges from 0 to 1, from one end of the bounding box to the other. The T coordinate ranges between 0 and the ratio of the second greatest dimension of the bounding box to the greatest dimension. Figure 13.6 illustrates the default texture coordinates for a simple box shaped IndexedFaceSet with an X dimension twice as large as the Z dimension and four times as large as the Y dimension. Figure 13.7 illustrates the original texture image used on the IndexedFaceSet used in Figure 13.6.

Figure 13.6 — IndexedFaceSet texture default mapping

Figure 13.7 — ImageTexture for IndexedFaceSet in Figure 13.6

11.2.3 Common geometry fields, provides a description of the ccw, solid, convex, and creaseAngle fields.

13.3.7 Sphere

Sphere : X3DGeometryNode { 
SFNode  [in,out] metadata NULL [X3DMetadataObject]
SFFloat []       radius   1    (0,∞)
SFBool  []       solid    TRUE
}

The Sphere node specifies a sphere centred at (0, 0, 0) in the local coordinate system. The radius field specifies the radius of the sphere and shall be greater than zero. Figure 13.8 depicts the fields of the Sphere node.

Figure 13.8 — Sphere node

When a texture is applied to a sphere, the texture covers the entire surface, wrapping counterclockwise from the back of the sphere ( i.e., longitudinal arc intersecting the -Z-axis) when viewed from the top of the sphere. The texture has a seam at the back where the X=0 plane intersects the sphere and Z values are negative. TextureTransform affects the texture coordinates of the Sphere (see 18.4.8 TextureTransform).

The solid field determines whether the sphere is visible when viewed from the inside. 11.2.3 Common geometry fields provides a complete description of the solid field.

This geometry node is fundamentally a mathematical representation. Displayed geometry shall have sufficient rendering quality that surface and silhouette edges appear smooth, including when textures are applied.

13.4 Geometry3D component support levels.

The Geometry3D component provides three levels of support as specified in Table 13.2. Level 1 provides the basic indexed geometry types with limited support for some fields, as well as the geometric primitives and the Shape node. Level 2 adds support for the IndexedFaceSet node. Level 3 adds support for the ElevationGrid node to enable lightweight terrain and data visualization and supports all fields in all nodes supported at Level 3. Level 4 adds support for the Extrusion node.