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.

vector fields : the start

oh boy… this is going to be a frequent topic. i *hate* how vector fields are drawn. i don’t believe i’ve ever seen a vector field drawn well. ever. why? [hello matlab, mathematica, et al…]

  1. square grid sampling: yuk.
  2. non-adaptive sampling: too few samples near equilibria & too many where the vectors are large
  3. long vectors intersect: yuk.
  4. the arrows are drawn poorly. also, yuk.
  5. no/poor use of color.
  6. static pics — vector fields should induce flow (in your mind)

can i fix all these problems? hardly. not easily. but i’m trying. here’s an early attempt. it, too, is flawed. but notice the difference that a hex grid, a bit of style on the arrows, a little 3-d layering, and some ambient occlusion make.

BLUE 4 PROLOGUE : an early attempt at a vector field. ahhh…

soft shadows

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.

BLUE vol 2 chapter 17 : extrema, contour lines, and a soft shadow

a simple surface…

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.

BLUE vol 2 chapter 1