Quaternion rotation shader

After writing yesterday's post I wondered how easy it would be to use the mathematics in a shader. This would be a natural way to extend my cross product shader, replacing the simple product in that with a more general geometric product.

Angle between two vectors

Another use for inverse trigonometric functions is finding the angle between two vectors. Again there is more than one way to do it, and again the most straightforward approach is often not the best.

Variable speed rotation with complex numbers

I wrote only last week that complex numbers are best for mostly fixed or uniform speed rotations. This is true in general, but there are ways to use complex numbers when the speed varies, as long as it varies in a straightforward way. In particular if the speed increases linearly, so with uniform angular acceleration, it can be modelled with complex numbers.

Accuracy

I came across a few issues related to accuracy in my ballistics app, both expected and unexpected. As these have wider application than ballistics simulations and are interesting in their own right I thought them worth their own post.

Varying the rotation speed

One objection to rotating using complex numbers instead of angles is that it works best with a fixed rotation speed. This is correct: if the speed needs to vary from frame to frame then the 'delta' needs to be recalculated each frame, and it may be easier to just use the angle to calculate the rotation each time.

But as long as the speed is mostly fixed complex numbers work well. If for example the rotation speed changes in steps, but between these is constant, then the 'delta' needs only be recalculated at these steps.

Trig-free rotation blending

As mentioned yesterday complex numbers can be used to do rotations in two dimensions, by just multiplying by a suitable (complex) value. One particular application is rotation blending or interpolation, where the direction of something is blended smoothly over time. An example would be aiming a gun, where having it move over time to aim is more realistic and interesting.


Normally this would be done by blending angles, so an angle is updated in steps from one direction to another. But this is expensive as two trigonometric calculations are required each frame to update the direction or to transform whatever is being rotated. It is quicker to use complex numbers and avoid trigonometry altogether, except at the start.

Complex numbers

Complex numbers are a much under-appreciated topic in mathematics, or at least that's how it seems to me looking back on them. Very often they are introduced almost as a mathematical exercise, as a way of solving mathematical problems such as quadratics which are otherwise insoluble. But once this is done all it gives is an impossible solution.


For example in my ballistics application complex number solutions to the quadratic formula in it (given by a negative discriminant) arise when the target is unreachable. And many applications of complex numbers seem like this. Except the more you study mathematics and physics the more useful they become, arising in diverse areas such as dynamics, electro-magnetism, and quantum mechanics.

Angle-free rotation

In my last post I showed how to aim a gun at a target above or below the gun to fire a ballistic missile, i.e. one moving under gravity. The mathematics was entirely angle and trigonometry free, but I noted that the angle can be calculated, in case it's need to e.g. rotate a gun turret. But it's not actually needed for that: it's possible to generate the rotation matrix to line up an object with a direction, without using angles.