### 4.6 Hierarchical control

Many 3D applications define a complex scenegraph hierarchy. An important function of the scenegraph is to enforce the geometric and spatial relationships that the scenegraph defines. Just as when the F1 car was moved its constituent parts were also moved. This principle is central to applications that require hierarchical control.

At the scenegraph level, the key to specifying relative positions for Nodes within the scenegraph is the TransformGroup Node. A TransformGroup encapsulates a Transform3D instance, which in turn encodes a 4 ×4 scaling, rotation, and translation matrix. The important principle is that a scenegraph Node’s rotation, scaling, and translation is always specified relative to its parent Node’s rotation, scaling, and translation.

To illustrate these principles, in this section I’ll show you a Java 3D scenegraph to animate a model of a human arm (figure 4.13). Requirements of the model are:

Shoulder joint can be rotated

Elbow joint can be rotated

Wrist joint can be rotated

Upper finger joints can be rotated

Figure 4.13 The human arm—a hierarchical model

This model is obviously hierarchical. It would be most usual if when the elbow joint was rotated the lower arm and the fingers were not also displaced. An important principle of the scenegraph is that the position of a child Node only depends upon the positions of its parent Nodes. In other words, the position of the end of the little finger depends upon

Length of little finger (defines the offset from the center of rotation)

Rotation of little finger joint

Length of wrist

Rotation of the wrist joint

Length of lower arm

Rotation of the elbow joint

Length of upper arm

Rotation of the should joint

Converting the rotational requirements into a scenegraph hierarchical model produces a structure such as that in figure 4.14. Note that we have switched to a left−to−right tree representation of the scenegraph hierarchy to save space.

Figure 4.14 The scenegraph for our conceptual human arm model

As with most things in life, things are not quite that simple. There are a few implementation−related issues that must also be resolved through the scenegraph structure. The scenegraph in figure 4.14 would be fine if you required just a static model of the human arm, but it would be nice to be able to illustrate the example by rotating the various joints to animate the model (as shown in figure 4.15). To simplify rotating the joints you must introduce another TransformGroup into the hierarchy for each limb to store the current joint rotation value. Two TransformGroups are thus required, one to store the length of the limb (i.e., the offset of the coordinate system of the next limb relative to the current limb’s coordinate system) and one to store the joint rotation.

Figure 4.15 Four frames from the ScenegraphTest example to illustrate the simple arm model in action

Unfortunately another minor implementation issue arises. The Cylinder geometric primitive that you are using to create the limbs in the model is created with 0,0,0 at the center of the cylinder. In other words, if you create a cylinder of length L, it will stretch from –L/2 to L/2 in the y direction, with a given radius. We would like our cylinders to stretch from 0 to L in the y direction, so an additional TransformGroup is required to shift the cylinder upward by L/2 in the y direction.

When these refinements have been made, the scenegraph for the arm looks a little more complicated than our initial design but it still fulfills our rotation requirements for each joint. The completed scenegraph is illustrated in figure 4.16.

Figure 4.16 The completed scenegraph for the arm model. TG Joint stores the rotation of each joint. TG Trans shifts the geometry for the cylinder upward by L/2. The RotationInterpolator modifies its parent TG Joint to rotate the joints of the model. TG Offset contains the length of the limb, and hence shifts the coordinate system of the next limb (its child)

To verify the rotational requirements, you can walk up the scenegraph hierarchy from the end of Finger 1. Walking up the hierarchy will tell you what the position of the end of Finger 1 relies upon:

Finger 1: TG Offset (the length of finger 1)

Finger 1: TG Joint (the rotation of finger 1)

Wrist: TG Offset (the length of the wrist)

Wrist: TG Joint (the rotation of the wrist)

Lower Arm: TG Offset (the length of the lower arm)

Lower Arm: TG Joint (the rotation of the lower arm)

Upper Arm: TG Offset (the length of the upper arm)

Upper Arm: TG Joint (the rotation of the upper arm)

Shoulder: TG Shoulder (position/rotation of the shoulder)

The rotational requirements have been satisfied, the scenegraph models the hierarchical structure of the human arm.

There are two slightly undesirable implications of the scenegraph as we have designed it. First, the length of the cylinder that we used for the geometry of the arm is unrelated to the length of the limb, inasmuch as the length of the Cylinder effects the next limb in the hierarchy. In this application, TG Offset and the length of the cylinder we created just happen to correspond. Second, the wrist and fingers are poorly modeled, as the fingers should be offset relative to one another. As modeled here the fingers all attach to the same location on the wrist.

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