The Geometry of Active and Passive Transformations

The missing geometric story behind the standard slogan on active and passive transforms.

Take a change of basis to be \(\mathcal{B}\)

Lorem ipsum Table 1.

Table 1: High-level contrast between active and passive transforms

	Active transform	Passive transform
	Single frame, \(B\)	Spans two frames, \(B\) and \(G\)
Forward transform	\(T\)	\({}^G T_B\)
Reverse/Inverse transform	N/A	\({}^B T_G = \left({}^G T_B\right)^{-1}\)

The commonly computed passive case is just the incomplete visible form of the fuller passive geometric story.

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Lorem ipsum Figure 1.

Figure 1: Left: Anti-pattern: Active transformation of ego vehicle’s sensor frame origin and body frame origin all within the body frame. Needless to say, this transformation doesn’t make sense when these origins correspond to physical points that can’t move relative to the ego vehicle. Right: Correct: Passive transformation of ego vehicle’s sensor frame origin and body frame origin from body frame to global frame. In practical calculations, one normally only needs to transform the coordinates of p_S and p_B from their body frame representation to their global frame representation without giving any thought to the physical reference-frame basis-vectors/axes. But it is the transform of the physical reference-frame basis-vectors/axes shown in this figure that people are talking about when they say “passive is opposite of active”.

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