illustrating & animating mathematics
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 😉
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…
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…
this overhead still from BLUE 2 chapter 17 is a decent effort: the soft shadow maps on the base plane are warm & undistracting. the contrast between the surface shader and the contour lines is also a win. when animated, this is not bad.
here’s an early picture that i drew for BLUE vol 2 chapter 1: it’s a simple surface, with cel shading and a nice (though primitive) color palette. some things work. i like how the base plane accepts shadows & helps frame the figure in 3-d. in retrospect, i should have used soft shadows instead of hard & the specular on the cel shading is distracting. it looks a bit better animated.
but — it’s the hidden lines and countours that tell the real story. see that cusp singularity? there’s a lot of mathematics hiding in a single figure.
