Derivatives and Logarithms of 3D Transforms

Date Tue 07 February 2023 Tags Programming Given a transform \(T\) and a point x, we can find the transformed point with \(T * x\). But what if we want to smoothly interpolate \(T\) so it moves \(x\) along the path from its initial position to its position transformed by \(T\)? What we want to find is the point \(x\) at time \(t\): \(x(t) = T(t) * x(0)\) where \(x(0)\) is the point’s initial position, and \(T(t)\) is the transform at time \(t\). Since we have only a single transform \(T\), we need to find a way to interpolate it over time. One way to accomplish this is to raise \(T\) to the power of \(t\), which can be done using the exponential and logarithm of the transform. Interestingly, the logarithm of a transform can also be used to easily find the velocity of a point \(x\) in space: the velocity vector (also called the tangent vector) is just \(log(T) * x\). This blog post shows the relationship between the logarithm and velocity. Example Check out this interactive examp ... Read full article.