[HN Gopher] Visualizing Complex Functions
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Visualizing Complex Functions
Author : hyperific
Score : 59 points
Date : 2024-08-10 07:05 UTC (3 days ago)
(HTM) web link (vankessel.io)
(TXT) w3m dump (vankessel.io)
| bmitc wrote:
| A good book is _Visual Complex Functions: An Introduction with
| Phase Portraits_ : https://www.amazon.com/Visual-Complex-
| Functions-Introduction...
| azeemba wrote:
| The animation looks awesome! Looks like the author used
| matplotlib (as they mention in a comment on the website):
| https://github.com/vankessel/sandbox/blob/master/graph/inter...
|
| In the past, I have used manim to make mathematical animations:
| https://www.manim.community/ Manim is more flexible but that
| comes with some overhead of complexity and learning. Example of
| some animations using manim:
|
| - List of videos using manim:
| https://www.manim.community/awesome/
|
| - A blog post I made: https://azeemba.com/posts/degenerate-
| matter.html
| unconed wrote:
| I couldn't disagree more to be honest, about this post.
| Animations are good when they provide object permanence, and
| let you track what's changing and how.
|
| This post linearly interpolates complex functions blindly,
| which doesn't tell you anything useful, unless the thing being
| interpolated is an affine or projective transform where that
| makes sense.
|
| e.g. For complex powers, the most natural animation is to
| animate the exponent, which will show a continuous folding or
| unfolding. Here the squaring just looks like the extra 360deg
| appears out of nowhere.
|
| For mobius-like transforms, interpolating the inverse might be
| better.
|
| One particularly good example is e.g. visualizing equally
| spaced points on a circle, and their various combinations as
| roots and poles of complex functions.
|
| The goal of math animation should be to highlight and travel
| the natural geodesics of the concept space, with natural starts
| and stops too.
|
| The rest is cargo culting.
| seanhunter wrote:
| Not sure what you mean by this.
|
| > The goal of math animation should be to highlight and
| travel the natural geodesics of the concept space, with
| natural starts and stops too.
|
| > The rest is cargo culting.
|
| A geodesic as I understand it is the curve representing the
| shortest path between two points in some manifold.
|
| So take one thing that I have found math animations useful
| for: showing the path of travel of some parametric system. Is
| that a geodesic? Not necessarily in the cartesian space of
| the system. I don't know what it would mean for it to be a
| natural geodesic of the concept space.
|
| For me the goal of math animation is the same as the goal of
| any math visualisation: to improve understanding and
| intuition. When I animate something (Which I only ever do for
| myself) that is why I do it. Am I cargo culting in your
| estimation?
|
| Let's take another example: Say I do an animation of some
| sort of force problem in mechanics. I can show the paths of
| some particles in the simulation and the magnitude and
| direction of the various vectors vs time. Is that cargo
| culting? It's definitely not any kind of geodesic. Does it
| help my understanding? Quite possibly.
|
| In that sense in the blog post you are addressing, in my
| opinion the position vs momentum distribution animation is
| really great because it really helps my intuition of how
| those probability distributions are related and how one would
| change as the other changes.
| ttoinou wrote:
| I animated the power in this video after 18sec :
| https://www.shadertoy.com/view/Ms2Bz3
|
| Please note that complex powers involve the complex
| logarithm, which is multivalued, it should be a surface in 3D
| to really see the whole function. The animation I made is
| only taking one value of the power
| fauria wrote:
| One of the best explanations on imaginary numbers I've seen is
| Kalid Azad's from Better Explained:
| https://betterexplained.com/articles/a-visual-intuitive-guid...
| liminal wrote:
| This is very cool, but would greatly benefit from a perceptually
| linear color space. I think the author is using HSL -- HCL would
| be a better choice.
| hyperific wrote:
| The author's code is available in an archived repo so it should
| be possible to experiment with different colormaps.
|
| https://github.com/vankessel/sandbox/tree/master
| hyperific wrote:
| This was also posted previously in 2019.
|
| https://news.ycombinator.com/item?id=19423278
| mkaic wrote:
| If you want to mess around with these sorts of visualizations
| yourself, I recommend checking out David Bau's little web app for
| it: http://davidbau.com/conformal
| ttoinou wrote:
| Plug of my own work on the topic :
|
| Using images as input to show conformal deformation
| https://www.youtube.com/watch?v=CMMrEDIFPZY
|
| Better phase portraits with a grid, zeroes, poles
| https://www.shadertoy.com/view/Ms2Bz3
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