The topic of always-online live-service games shutting down and ultimately becoming unplayable has been a popular topic of internet discourse for many years. Through my work on Towerborne I've seen first-hand just how challenging and time-consuming it can be to make a game originally designed like this work offline. Every game has its own unique challenges in both design and technical architecture, making offlining a uniquely complex undertaking that is hard to understand for many of its players. However, the Native AOT approach we leveraged shows that it is possible and I hope other developers finding themselves in a similar position find it useful in showcasing one possible path.
Museum saves Tommy Cooper's 'gag file'
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Another way to approach the linear combination is to look at it geometrically. This is where the idea of barycentric coordinates can help. A barycentric coordinate system describes the location of a point as the weighted sum of the regular coordinates of the vertices forming a simplex. In other words, it describes a linear combination with respect to a set of points, where in -dimensional space.