In a previous post I discussed the relationship between derivatives, logarithms, and transforms. This blog post adds to that and explains how to find the second and higher order derivatives of a transform.
The Concept of Derivatives of Transforms
Let’s quickly recap my earlier blog post. Suppose we have a transform \(T\) and a point at the initial position of \(x(0)\). The position of the point at time t and the first derivative of that point are
\(x(t) = T^t * x(0)\)
\(x'(t) = log(T) * x(t)\).
where \(log(T)\) is the logarithm of the transform. Both \(T\) and \(log(T)\) are constant with respect to t.
What’s the second derivative?
We know the first derivative (\(x'(t) = log(T) * x(t)\)), so to find the second derivative, we start with
\(\dfrac{d}{dt}x'(t) = \dfrac{d}{dt}log(T) * x(t)\)
Since \(log(T)\) is constant with respect to time, we then have
\(x''(t) = log(T) * \dfrac{d}{dt}x(t)\)
and earlier we stated that \(x'(t) = log(T) * x(t)\), so we just plug that in, giving us our answer:
\(x''(t) = log(T) * log(T) * x(t)\).
Neat! In other words, if we have the first derivative, then the second derivative is just another multiplication by \(log(T)\).
In the earlier blog post, we found the tangent vector of a point by multiplying the point by \(log(T)\). Similarly, we can now find the acceleration vector of a point by multiplying it by \(log(T) * log(T)\).
What’s the nth derivative?
The beauty of this approach is how simple it is. For the nth derivative, we can follow the same pattern. More generally, the nth derivative is
\(x^{n}(t) = log(T)^n * x(t)\)
Interactive Example
Here’s an interactive example where you can see the tangent vector (green arrow) and acceleration vector (red arrow) of a point as it is interpolated from the origin to the gizmo’s transform. You can manipulate the gizmo to translate and rotate the transform. You can also enable a visualization of the velocity vector field or the acceleration vector field.
The source code for the applet can be found here.
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