An implicit surface is a 2-dimensional surface in 3-dimensional space defined as the locus of zeros of a given function. Many useful shapes such as sphere, cylinder or cone can be expressed using this approach, known as a quadric surfaces.

Sub-classes of this node compute and render an implicit surface on the GPU using a GLSL shader function. A screen-aligned quad is drawn, representing the screen space bounding box of the algebraic shape. Then, this quad is ray-casted and a ray/shape intersection is applied per fragment to draw the final shape.

Several predefined sub-classes are provided for convenience, including OIV.Inventor.Nodes.SoAlgebraicCone, OIV.Inventor.Nodes.SoAlgebraicCylinder and OIV.Inventor.Nodes.SoAlgebraicSphere. These nodes can be used in an application scene graph similar to the corresponding classic geometry nodes OIV.Inventor.Nodes.SoCone, OIV.Inventor.Nodes.SoCylinder and OIV.Inventor.Nodes.SoSphere. Use a transform node, e.g. OIV.Inventor.Nodes.SoTransform, to position the shape node in 3D space. Use an OIV.Inventor.Nodes.SoMaterial node to assign material properties. See the notes and limitations section on this page for some important differences between algebraic and geometric shapes.

Extending OIV.Inventor.Nodes.SoAlgebraicShape:

Derived classes must implement the bounding box computation function computeBBox() in C#.And also implement the ray/shape intersection function OivASRayIntersection() in GLSL. This function returns true if there is an intersection between the ray and the shape, false otherwise. Create an OIV.Inventor.Nodes.SoFragmentShader to hold the GLSL function and set this node in the rayIntersection field.

bool
OivASRayIntersection ( in OivASRay ray, inout OivASPoint point )
{
  DO SOMETHING
  return [ true | false ];
}

See the GLSL include file oivAlgebraicShape.h in $OIVHOME/shaders/include/Inventor/AlgebraicShape. It declares ray, a structure that contains ray parameters:

struct OivASRay {
 vec3 rs; // ray start
 vec3 re; // ray end
 vec3 rd; // ray direction
};

struct OivASPoint {
 vec3 position;
 vec3 normal;
 vec4 color;
};

Note that ray parameters and point information are defined in the reference frame specified by the workspace field, an enum of type OIV.Inventor.Nodes.SoAlgebraicShape.ASWorkSpaces. This frame can be the camera space, the world space or the normalized space of the bounding box of the shape. By default, the bounding box space is used.

A GLSL helper function for solving quadratic functions (i.e. a*x^2 + b*x + c = 0) is provided:

bool OivASSolveQuadric ( in vec3 abc, inout vec2 roots );

with abc, a vector containing the coefficients {a, b, c} of the polynomial. A quadratic equation has zero, one or two solutions, called roots. It returns true if there are solutions, false otherwise. Note that only helper function for quadric surfaces are provided but higher order surface such as Torus (i.e. degree 4) may be implemented using user-defined polynomial solver.

All quadric shape equations can be solved using this function. For instance, the equation of a sphere centered at the origin with a radius of 1 is defined by: To find the intersection point between such a sphere with a ray as defined above, we have to solve the quadric sphere equation such as: which leads to, It means solving a quadratic equation with:

• a = 1 (i.e. dot(rd, rd) = 1),

• b = 2 * dot(rs, rd),

• c = dot(rs, rs) - 1.0.

If a solution exists (1 or 2), the OivASSolveQuadric function returns true and roots are stored in the parameter roots. The roots (i.e. t1 and t2) represent the solution for the parameter t such as solutions are:

• p1 = rs + t1*rd

• p2 = rs + t2*rd

The smallest positive root is the first intersection point along the ray direction rd. If there are two positive roots, the larger one is the intersection point with the back face. If a root is negative, it means that there is an intersection in the opposite ray direction.

While this node is designed to address algebraic surfaces, the ray intersection function could be used with other types of surfaces to find the intersection between the ray and the shape (e.g. distance functions).

Note that this node supports instancing using OIV.Inventor.Nodes.SoMultipleInstance to render millions of algebraic shapes in a more efficient way than than using geometric shapes.

The application can also provide custom color shaders to shade the surface or use built-in shading based on light model and material properties (transparency is supported as well).

Notes:

• Shape hints (OIV.Inventor.Nodes.SoShapeHints) do not affect rendering. Algebraic shapes are always rendered as if "two-sided lighting" is enabled.

• Complexity (OIV.Inventor.Nodes.SoComplexity) does not affect rendering. Algebraic shapes are not tessellated, so are always "full resolution".

• Material binding (OIV.Inventor.Nodes.SoMaterialBinding) does not affect rendering. (You can't color the caps differently like you can with OIV.Inventor.Nodes.SoCylinder, etc.)

• Algebraic shapes can be picked, but no OIV.Inventor.Details.SoDetail is available.

• Wireframe rendering is not supported since this node does not generate real geometry.

Limitations:

• Texturing (OIV.Inventor.Nodes.SoTexture2) does not affect rendering.

• Projection (OIV.Inventor.Nodes.SoProjection) does not affect rendering.

• Draw style (OIV.Inventor.Nodes.SoDrawStyle) does affect rendering, but the result is a single line segment, not a wire frame shape.

Reference