mobius strips i have not loved

is there any nice way to generate a mobius strip (from the computer graphics point of view)? i am doubtful. you won’t get it parametrically (normals are messed up; closure is a pain) and don’t try realizing it implicitly in 3-d 😉

BLUE vol 4 ch 9: a mobius strip

i think it helps to have a fattened boundary, as well as to have the surface partially translucent (fresnel shader) with some subtle reflectivity. this definitely looks much better when animated, but you’ll have to wait for BLUE vol 4 to hit youtube for that…


this is day 3 of trying and failing to properly animate stokes’ theorem in 3-d. stills are not hard: it’s the animation that’s rough.

so, as a way to procrastinate (& practice), here is a cartoon drawing of a sheaf of vector spaces over a graph, thought of as an algebraic data structure.

cellular sheaf of vector spaces over a graph

this has a nice greyscale scheme, with good sense of scope. it does not do a very good job of suggesting the heterogeneity of the stalks, nor the linear transformations between them. but you can’t have everything.

on hidden lines, a recurring topic

when i started doing illustration for calculus BLUE, i had insufficient appreciation for the technicalities of hidden lines. it’s taken me a long time to really understand how they can best be used, and i’m still learning. here are some simple examples from the exercises in the e-text version of volume 4, used for induced orientations in stokes’ theorem.

none of these are particularly good or deep examples. yet the combination of the hidden lines and the shading makes such a difference in grasping the surface. when we get to colors, thicknesses, and intersections, there will be much more to discuss…

BLUE volume 3 title screen

i’ve recently released volume 3 of Calculus BLUE on my YouTube channel. you can check out the playlist here. it’s 6.5 hours of animated content & will keep you busy.

let’s focus on an image from the title screen…

calcBLUE vol 3 title screen

what do we have here? as with the others, there are a few hints and foreshadowings… this volume is on multivariate integrals. what do you notice? i will admit to the following:

  • lots of blocks, hinting at riemann sums (but in a hex grid…hmmmm…)
  • a plane sweeping from side-to-side, hinting at the fubini theorem
  • a lot of falling balls, bouncing off the blocks: ahha, a modification of a classic quincunx, since we do a lot of applications to probability
  • is that a gaussian rising from the bottom? hmmmm…
  • and what is with those three circles?

there are a few visual touches that were intentional. the blocks are translucent with a fresnel shader, so they are invisible head-on and you only see them by the absence of balls. the sweeping plane is luminescent and does some nice things to the blocks.