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Spherical Coordinates, Forward and Inverse Maps with Interactive Desmos Link

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Tags: visual-proofspherecoordinates

This project was inspired by 3Blue1Brown’s interactive quaternions video—the moment where you don’t just watch the math, you touch it. I wanted that same feeling for spherical coordinates. The result is a short, highly visual lesson paired with an interactive Desmos scene that mirrors the video one-to-one. Watch the video from start to finish to see the structure unfold cleanly; at the end there’s a quick walkthrough showing exactly how to use the Desmos file to recreate everything you’ve just seen—right down to the final picture. The focus is conceptual clarity you can feel: how the forward map (r,θ,φ)→(x,y,z) paints the sphere, how the inverse map (x,y,z)→(r,θ,φ) “reads” points back into angles, and where the representation gets delicate near the poles. No prior knowledge is required; the visuals do the heavy lifting while the narration keeps the rigor intact. Then the Desmos model turns the ideas into a sandbox: drag a slider, nudge an angle, and watch the geometry breathe. I built this to help the material sink in through doing—first a beautiful, guided pass, then hands-on reconstruction. It’s meant to be enjoyable to look at, satisfying to tinker with, and faithful to the math. Everything is free by design so students and teachers can use, remix, and share. If it resonates, please pass it on; the more people who interact with the ideas, the better the ideas stick. Here is a Desmos link with the clone with everything seen in the video: https://www.desmos.com/3d/og7qio7wgz It is recommended to watch the video from start to end, to then interact with everything that was in the video and also watch the Desmos walkthrough at the end for the perfect Desmos experience: https://www.desmos.com/3d/og7qio7wgz This could be perfect for any classroom interactive visualization from high school to undergraduates as no priors are required to interact.

Analytics

6 Overall score*

77 Rank

17 Votes

8 Comments

Comments

7.4

The interactive Desmo is interesting, though the phi sign go crazy when in some range of phi_1

The explanation was very clear. But I could use some motivation for when you would use this.

5.7

Quite obviously, the videos greatest strength is the extensive use of visuals. The animation clearly brings a lot to the understanding of spherical coordinates. You should be careful not to overload frames, because on one hand, it gets harder to follow your arguments, and on the other hand, most screens arent too large. At some points (eg drawing out all longitudes), the animations are a little overused, even though you make up a lot of it explaining in the meantime. I couldn’t score higher on the novelty side, because unfortunately theres nothing fundamentally new about spherical coordinates. Overall, I think the video was above average.

Some general editing tips first - make sure you line up your audio with what’s being shown on screen. For example at 3:00, make sure when you say the x direction is sin(theta)… that the equation shows on the screen at the same time you say that. Also the audio is a bit too sharp. Try recording with your microphone a bit off-axis (to the side) or you can do it in post depending on your software.

Motivation (2) - No motivation is really introduced for this topic (in the video)

Clarity (7) - Clear explanations

Novelty (8) - desmos!

Memorability (4) - Mainly impacted by the lack of motivation. Nice video, but no clear reason why it’s interesting which negatively impacts the memorability.

8.1

perfect!

I think it was too long and needed to focus on some central question.

6.5

Provides a great tool for people learning spherical coordinates for the first time, or for those who need to review it. I liked that you went through the trouble of creating phi, and theta from plots. I enjoy that it introduces desmos to potentially a new audience.

6.5

I like Desmos and I like spherical coordinates. The explanations and visuals were good, if a little static. I honestly probably enjoyed the Desmos demonstration over the actual video, it was a very cool project.