6.1 Basic Shape Interface
The scenes that pbrt renders often contain objects that are computationally expensive to process. For many operations, it is useful to have a 3D bounding volume that encloses an object. For example, if a ray does not pass through a particular bounding volume, pbrt can avoid processing all the objects inside of it for that ray.
Axis-aligned bounding boxes are a convenient bounding volume, as they require only six floating-point values to store. They fit many shapes well and it is fairly inexpensive to test for the intersection of a ray with an axis-aligned bounding box. Each Shape implementation must therefore be capable of bounding itself with an axis-aligned bounding box represented by a Bounds3f. The returned bounding box should be in the rendering coordinate system (recall the discussion of coordinate systems in Section 5.1.1).
In addition to bounding their spatial extent, shapes must also be able to bound their range of surface normals. The NormalBounds() method should return such a bound using a DirectionCone, which was defined in Section 3.8.4. Normal bounds are specifically useful in lighting calculations: when a shape is emissive, they sometimes make it possible to efficiently determine that the shape does not illuminate a particular point in the scene.
6.1.2 Ray–Bounds Intersections
Given the use of Bounds3f instances to bound shapes, we will add a Bounds3 method, Bounds3::IntersectP(), that checks for a ray–box intersection and returns the two parametric values of the intersection, if any.
One way to think of bounding boxes is as the intersection of three slabs, where a slab is the region of space between two parallel planes. To intersect a ray with a box, we intersect the ray with each of the box’s three slabs in turn. Because the slabs are aligned with the three coordinate axes, a number of optimizations can be made in the ray–slab tests.
The basic ray–bounding box intersection algorithm works as follows: we start with a parametric interval that covers that range of positions along the ray where we are interested in finding intersections; typically, this is . We will then successively compute the two parametric positions where the ray intersects each axis-aligned slab. We compute the set intersection of the per-slab intersection interval with the current intersection interval, returning failure if we find that the resulting interval is degenerate. If, after checking all three slabs, the interval is nondegenerate, we have the parametric range of the ray that is inside the box. Figure 6.1 illustrates this process, and Figure 6.2 shows the basic geometry of a ray intersecting a slab.
If the Bounds3::IntersectP() method returns true, the intersection’s parametric range is returned in the optional arguments hitt0 and hitt1. Intersections outside of the (0, tMax) range of the ray are ignored. If the ray’s origin is inside the box, is returned for hitt0.
For each pair of planes, this routine needs to compute two ray–plane intersections. For example, the slab described by two planes perpendicular to the axis can be described by planes through points and , each with normal . Consider the first value for a plane intersection, . The parametric value for the intersection between a ray with origin and direction and a plane can be found by substituting the ray equation into the plane equation:
Solving for gives
Because the and components of the plane’s normal are zero, and are zero, and is one. The plane’s coefficient is . We can use this information and the definition of the dot product to simplify the calculation substantially:
The code to compute the values of the slab intersections starts by computing the reciprocal of the corresponding component of the ray direction so that it can multiply by this factor instead of performing multiple divisions. Note that, although it divides by this component, it is not necessary to verify that it is nonzero. If it is zero, then invRayDir will hold an infinite value, either or , and the rest of the algorithm still works correctly.
The two distances are reordered so that tNear holds the closer intersection and tFar the farther one. This gives a parametric range , which is used to compute the set intersection with the current range to compute a new range. If this new range is empty (i.e., ), then the code can immediately return failure.
There is another floating-point-related subtlety here: in the case where the ray origin is in the plane of one of the bounding box slabs and the ray lies in the plane of the slab, it is possible that tNear or tFar will be computed by an expression of the form , which results in a floating-point “not a number” (NaN) value. Like infinity values, NaNs have well-specified semantics: for example, any logical comparison involving a NaN always evaluates to false. Therefore, the code that updates the values of t0 and t1 is carefully written so that if tNear or tFar is NaN, then t0 or t1 will not ever take on a NaN value but will always remain unchanged.
Bounds3 also provides a specialized IntersectP() method that takes the reciprocal of the ray’s direction as an additional parameter, so that the three reciprocals do not need to be computed each time IntersectP() is called.
This version of the method also takes precomputed values that indicate whether each direction component is negative, which makes it possible to eliminate the comparisons of the computed tNear and tFar values in the original routine and to directly compute the respective near and far values. Because the comparisons that order these values from low to high in the original code are dependent on computed values, they can be inefficient for processors to execute, since the computation of their values must be finished before the comparison can be made. Because many ray–bounds intersection tests may be performed during rendering, this small optimization is worth using.
This routine returns true if the ray segment is entirely inside the bounding box, even if the intersections are not within the ray’s range.
If the ray direction vector is negative, the “near” parametric intersection will be found with the slab with the larger of the two bounding values, and the far intersection will be found with the slab with the smaller of them. The implementation can use this observation to compute the near and far parametric values in each direction directly.
The fragment <<Check for ray intersection against slab>> is analogous and is not included here.
This intersection test is at the heart of traversing the BVHAggregate acceleration structure, which is introduced in Section 7.3. Because so many ray–bounding box intersection tests are performed while traversing the BVH tree, we found that this optimized method provided approximately a 15% performance improvement in overall rendering time compared to using the Bounds3::IntersectP() variant that did not take the precomputed direction reciprocals and signs.
6.1.3 Intersection Tests
Shape implementations must provide an implementation of two methods that test for ray intersections with their shape. The first, Intersect(), returns geometric information about a single ray–shape intersection corresponding to the first intersection, if any, in the (0, tMax) parametric range along the given ray.
In the event that an intersection is found, a SurfaceInteraction corresponding to the intersection point and the parametric distance along the ray where the intersection occurred are returned via a ShapeIntersection instance.
There are a few important things to keep in mind when reading (and writing) intersection routines:
- The provided tMax value defines the endpoint of the ray. Intersection routines must ignore any intersections that occur after this point.
- If there are multiple intersections with a shape along the ray, the closest one should be reported.
- The rays passed into intersection routines are in rendering space, so shapes are responsible for transforming them to object space if needed for intersection tests. The intersection information returned should be in rendering space.
The second intersection test method, Shape::IntersectP(), is a predicate function that determines whether or not an intersection occurs without returning any details about the intersection itself. That test is often more efficient than a full intersection test. This method is used in particular for shadow rays that are testing the visibility of a light source from a point in the scene.
6.1.4 Intersection Coordinate Spaces
For some shapes, intersection routines are most naturally expressed in their object space. For example, the following Sphere shape computes the intersection with a sphere of a given radius positioned at the origin. The sphere being at the origin allows various simplifications to the intersection algorithm. Other shapes, like the Triangle, transform their representation to rendering space and perform intersection tests there.
Shapes like Sphere that operate in object space must transform the specified ray to object space and then transform any intersection results back to rendering space. Most of this is handled easily using associated methods in the Transform class that were introduced in Section 3.10, though a natural question to ask is, “What effect does the object-from-rendering-space transformation have on the correct parametric distance to return?” The intersection method has found a parametric distance to the intersection for the object-space ray, which may have been translated, rotated, scaled, or worse when it was transformed from rendering space.
Using the properties of transformations, it is possible to show that the distance to the intersection is unaffected by the transformation. Consider a rendering-space ray with associated origin and direction . Given an object-from-rendering-space transformation matrix , we can then find the object-space ray with origin and direction .
If the ray–shape intersection algorithm finds an object-space intersection at a distance along the ray, then the object-space intersection point is
Now consider the rendering-space intersection point that is found by applying ’s inverse to both sides of that equation:
Therefore, the value that was computed in object space is the correct value for the intersection point in rendering space as well. Note that if the object-space ray’s direction had been normalized after the transformation, then this would no longer be the case and a correction factor related to the unnormalized ray’s length would be needed. This is one reason that pbrt does not normalize object-space rays’ directions after transformation.
Many rendering systems, particularly those based on scanline or -buffer algorithms, support the concept of shapes being “one-sided”—the shape is visible if seen from the front but disappears when viewed from behind. In particular, if a geometric object is closed and always viewed from the outside, then the backfacing parts of it can be discarded without changing the resulting image. This optimization can substantially improve the speed of these types of hidden surface removal algorithms. The potential for improved performance is reduced when using this technique with ray tracing, however, since it is often necessary to perform the ray–object intersection before determining the surface normal to do the backfacing test. Furthermore, this feature can lead to a physically inconsistent scene description if one-sided objects are not in fact closed. For example, a surface might block light when a shadow ray is traced from a light source to a point on another surface, but not if the shadow ray is traced in the other direction. For all of these reasons, pbrt does not support this feature.
In pbrt, area lights are defined by attaching an emission profile to a Shape. To use Shapes as area lights, it is necessary that shapes be able to return their surface area of a shape in rendering space.
A few methods are necessary to sample points on the surface of shapes in order to use them as emitters. Additional Shape methods make this possible.
There are two shape sampling methods, both named Sample(). The first chooses points on the surface of the shape using a sampling distribution with respect to surface area and returns the local geometric information about the sampled point in a ShapeSample. The provided sample value u, a uniform sample in , should be used to determine the point on the shape.
The ShapeSample structure that is returned stores an Interaction corresponding to a sampled point on the surface as well as the probability density with respect to surface area on the shape for sampling that point.
Shapes must also provide an associated PDF() method that returns probability density for sampling the specified point on the shape that corresponds to the given Interaction. This method should only be called with interactions that are on the shape’s surface. Although Sample() already returns the probability density for the point it samples, this method is useful when using multiple importance sampling, in which case it is necessary to compute the probability density for samples generated using other sampling techniques. An important detail is that implementations are allowed to assume that the provided point is on their surface; callers are responsible for ensuring that this is the case.
The second shape sampling method takes a reference point from which the shape is being viewed. This method is particularly useful for lighting, since the caller can pass in the point to be lit and allow shape implementations to ensure that they only sample the portion of the shape that is potentially visible from that point.
Unlike the first Shape sampling method, which generates points on the shape according to a probability density with respect to surface area on the shape, the second one uses a density with respect to solid angle from the reference point. This difference stems from the fact that the area light sampling routines evaluate the direct lighting integral as an integral over directions from the reference point—expressing these sampling densities with respect to solid angle at the point is more convenient.
Information about the reference point and its geometric and shading normals is provided by the ShapeSampleContext structure. The reference point position is specified using the Point3fi class, which can represent the numerical uncertainty in a ray intersection point computed using floating-point arithmetic. Discussion of related topics is in Section 6.8. For points in participating media that are not associated with a surface, the normal and shading normal are left with their default values of .
ShapeSampleContext provides a variety of convenience constructors that allow specifying the member variable values directly or from various types of Interaction.
For code that does not need to be aware of numeric error in the intersection point, a method provides it as a regular Point3.
A second PDF() method comes along with this sampling approach. It returns the shape’s probability of sampling a point on the light such that the incident direction at the reference point is wi. As with the corresponding sampling method, this density should be with respect to solid angle at the reference point. As with the other Shape PDF() method, this should only be called for a direction that is known to intersect the shape from the reference point; as such, implementations are not responsible for checking that case.
Some of the PDF() method implementations will need to trace a ray from the reference point in the direction to see if it intersects the shape. The following ShapeSampleContext methods should be used to find the origin or the ray itself rather than using the point returned by ShapeSampleContext::p(). This, too, stems from a subtlety related to handling numeric error in intersection points. The implementation of these methods and discussion of the underlying issues can be found in Section 6.8.6.