3D Theory - Transform Group

Prerequisites

How to combine rotations and translations is scenegraph shown here:

Order of transforms

In addition to the translation and rotation, described here, the Transform group also allows a (possibly) non-uniform scale about an arbitrary point.

To do these 3 things we have the following parameter:

Given a 3-dimensional point P and Transform node, P is transformed into point P' in its parent's coordinate system by a series of intermediate transformations. In matrix transformation notation.

P' = T × C × R × SR × S × -SR × -C × P
The following Transform node:

Transform {
centre C
rotation R
scale S
scaleOrientation SR
translation T
children [...]
}
is equivalent to the nested sequence of:

Transform {
	translation T 
	children Transform {
		translation C
		children Transform {
			rotation R
			children Transform {
				rotation SR 
				children Transform {
					scale S 
					children Transform {
						rotation -SR 
						children Transform {
							translation -C
							children [...]
						}
					}
				}
			}
		}
	}
}
 

Code

Java code see sftransform class in source code.

public void calcTransform(sfvec3f translation, sfrotation rotation, sfvec3f center, sfvec3f sc, sfrotation scaleOrientation){
   setIdentity();
   if (translation != null) translate(translation);
   if (rotation != null) rotate(rotation,center);
   if (scaleOrientation != null) rotate(scaleOrientation,center);
   if (sc != null) { // if scale is (0,0,0), such as when scale first
   // enabled this will generate a non-afine error
   // so do following check
   if ((sc.x != 0) & (sc.y != 0) & (sc.z != 0)) scale(sc);
   if (scaleOrientation != null) rotate(scaleOrientation.minus(),center);
}
 

Example: To implement a trackball control

Assume we have a shape in a scenegraph under one or more Transform Groups, we want a trackball control to apply rotation to the Transform adjacent to the Shape3D node. The problem is that the cumulative rotations above it will modify it, so when we move the mouse right we want the object to rotate about the y-axis but the cumulative rotation will effect this and cause it to rotate about some other axis.

What we want is the equivalent of the following where a transform is added at the root of the tree, this will ensure that moving the mouse right will make the object to rotate about the y-axis.

The problem with this solution is that any other shapes in the scenegraph will also be transformed. What we want to work out is an equivalent rotation in the local Transform which will have the same effect as this.

So in the required rotation is:

[Rcumulative] * [Rlocal]

where:

We want this to be equivalent to:

[trackball] * [Rcumulative] * [Rlocalb]

where:

Since we want these to be equivalent then:

[Rcumulative] * [Rlocal] = [trackball] * [Rcumulative] * [Rlocalb]

so if we pre-multiply both sides by the conjugate of Rcumulative we get:

[Rcumulative]' * [Rcumulative] * [Rlocal] = [Rcumulative]' * [trackball] * [Rcumulative] * [Rlocalb]

cancelling out [Rcumulative]' * [Rcumulative] gives

[Rlocal] = [Rcumulative]' * [trackball] * [Rcumulative] * [Rlocalb]

Further Reading

Or you may be interested in how these quantities are used to simulate physical objects:


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