matrix transformations 2D - place in world-space but keep size in "object/screen-space"

Question

I have a question concerning matrix transformations. I Think I know how it could be done, but Im not sure its the best/correct way to do it.

I want to apply World translations and rotation to an object but the scale should be consistent. See the image for example. The object closest to origo scales according to World-scale, the other object has the same scale but is placed correctly (World scale is taken into consideration when translating). I know I can do this by applying the inverse scale transform in local space Before translating and applying World-transforms. But is there some better way?

Answer 1

The data visualisation terms for this are
geometric zooming (which literally increases size), and
semantic zooming (scaling underlying positions).

Your initial scaling matrix alone would be a geometric zoom. Everything appears in the correct position, but also becomes larger. Your second operation of inverse-scaling relative to the new position essentially converts it into a semantic zoom, for that marker.

You could implement a semantic zoom more efficiently by just multiplying each marker's translation from the origin by the zoom factor. That way, everything is moved by the correct distance, but the markers' rendered graphics don't change size, so there's no need for the second corrective operation.

Both methods should yield mathematically equivalent results, but the latter is a little more performant (because it doesn't require "undoing" work with an inverse), and easier to read.

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If you do want the markers to be differently sized or rotated, you can of course apply additional transformations once they're translated to their semantic-zoom positions.