the rest of the assembly. Starting from the root of the interaction graph, we perform a breadth first traversal. At each traversal step, we compute a set of nodes S that includes the frontier of newly visited nodes, as well as any previously visited nodes that are in contact with this frontier. We then generate a frame that highlights S by rendering all other parts in a desaturated manner. To emphasize the motion of highlighted parts, each frame includes any non-contact arrow whose parent part is highlighted, as well as any contact-based arrow whose two associated parts are both highlighted. If a highlighted part only has contact arrows and none of them are included based on this rule, we add the part’s longest contact arrow to the frame to ensure that every highlighted part has at least one arrow. In addition, arrows associated with previously visited parts are rendered in a desaturated manner. For animated visualizations, we allow the user to interactively step through the causal chain while the animation plays; at each step, we highlight parts and arrows as described above.

5. 3. Highlighting keyframes of motion

As explained in Section 2, some assemblies contain parts that move in complex ways (e.g., the direction of motion changes periodically). Thus, static illustrations often include keyframes that help clarify such motions. We automatically

figure 9. exploded view results. our system automatically generates exploded views that separate the assembly at coaxial part interactions to reduce occlusions. these two illustrations show two different configurations for the planetary gearbox: one with fixed outer rings (a), and one with free outer rings (b). the driver part is in blue, while fixed parts are in dark gray.

(a)

Planetary gearbox w/ fixed outer rings (b)

Planetary gearbox w/ free outer rings

compute keyframes of motion by examining each translational part in the model: if the part changes direction, we add keyframes at the critical times when the part is at its extremal positions. However, since the instantaneous direction of motion for a part is undefined exactly at these critical times, we canonically freeze time d after the critical time instances to determine which direction the part is moving in (see Figure 8). Additionally, for each part, we also add middle frames between extrema-based keyframes to help the viewer easily establish correspondence between moving parts. However, if such frames already exist as the extrema-based keyframes of other parts, we do not add the additional frames (see Figure 8).

We also generate a single frame sequence that highlights both the causal chain and important keyframes of motion. As we traverse the interaction graph to construct the causal chain frame sequence, we check whether any newly highlighted part exhibits complex motion. If so, we insert keyframes to convey the motion of the part and then continue traversing the graph (see Figure 10c).

5. 4. exploded views

In some cases, occlusions between parts in the assembly make it difficult to see motion arrows and internal parts. To reduce occlusions, our system generates exploded views that separate portions of the assembly (see Figure 9). Typical exploded views separate all touching parts from one another to ensure that each part is visually isolated. However, using this approach in how-things-work illustrations can make it difficult for viewers to see which parts interact and how they move in relation to one another.

To address this problem, we only separate parts that are connected via a coaxial interaction; since such parts rotate rigidly around the same axis, we believe it is easier for viewers to understand their relative motion even when they are separated from one another. To implement this approach, our system first analyzes the interaction graph and then cuts coaxial edges. The connected components of the resulting graph correspond to sub-assemblies that can be separated from one another. We use the technique of Li et al. 16 to compute explosion directions and distances for these sub-assemblies.

6. ResuLts

We used our system to generate both static and animated how-things-work visualizations for ten different input models, each of which contains from 7 to 27 parts, collected from various sources. Figures 2, 7–10 show static illustrations of all ten models. Other than specifying the driver part and its direction of motion, no additional user assistance was required to compute the interaction graph for seven of the models. For the drum, hammer, and chain driver models, we manually specified lever, cam, rod and belt parts, respectively. We also specified the crank and piston parts in the piston model. In all of our results, we colored the driver blue, fixed support structures dark gray, and all other parts light gray. We render translation arrows in green, and side and cap arrows in red. In all the examples, analysis takes 1–2 s, while visualization runs at interactive rates.