Tactile Math Explorations

Although I took math in high school and college, my math skills have atrophied over time due to lack of usage.  How can I possibly do 3D graphics if I can’t remember the math?  It takes a lot of referencing back to books, and code I wrote years ago.  So, perhaps there’s a better way of teaching some math skills which will make the lessons stick?

Here we see 3 of the quadrics.  The cone, cylinder, and hyperboloid.  Each of them has their own bi-parametric equation that describes them.  In this particular case though, I’ve used only one formula to describe three different shapes.  I have used the Hyperboloid, because by slightly changing various parameters, you can get all three shapes.  The Banate CAD code looks like this:

local usteps = 180
local wsteps = 80
local radius = 5
local height = 10

local con = {{radius, 0, 0}, {0, 0, height}}
local cyl = {{radius, 0, 0}, {radius, 0, height}}
local hyp = {{radius, 0, 0}, {0, radius, height}}

function hyper(params)
USteps = usteps,
WSteps = wsteps,
StartPoint = params[1],
EndPoint = params[2],
--ColorSampler = checkerboard:new({Columns=usteps, Rows=wsteps})


translate({12, 0, 0})

translate({24, 0, 0})

In the beginning there, I setup the radius, height, and resolution I’m going to generate the curves.  Now, the hyperboloid takes some explanation.  Of course you can look it up online, but the thing to realize is that you take a line, any line, and sweep it around in a circle.  for the simplest case, the cone, just take a stick, stand it straight up on your desk, some distance from the center of your circle, and move it around that center.  You have just traced out a cylinder in space.

Now, let’s say you want to make a cone.  Take that stick again, and instead of it standing straight up and down, lean it in towards the center.  Now as you sweep it around, the top part stays fixed, as the bottom traces out a circle.

And lastly, for the hyperboloid, tiltl the stick slightly back, and to the left, and again, trace it around.

This is kind of hard to describe in words, and some visual aids would sure make it easier.

I’ve done exactly the same thing here, execept I’ve changed the resolution to make the facets more obvious.  Now if you go back and read the explanation again, it might make a little more sense.

In each case, all I’ve done is change the starting and ending points for the stick to be traced around the circle.

local con = {{radius, 0, 0}, {0, 0, height}}
local cyl = {{radius, 0, 0}, {radius, 0, height}}
local hyp = {{radius, 0, 0}, {0, radius, height}}

Each of these little arrays defines a starting and ending point for the hyperboloid function.  The con (cone), has a bottom radius of ‘radius’ and a top radius of 0.  The cylinder maintains the same radius at the bottom and top.  The hyperboloid maintains the same radius, but with a twist as it shifts from the x-axis to the y-axis.

This is a nice revelation.  For the code itself, it means that I don’t actually have to write three different functors to represent these things.  I can write just one, the hyperboloid, and just change the parameters I feed it in order to get the others.  Same for ellipse/sphere.  A sphere is just an ellipse where the two axes are the same length.

How does this help me learn maths?  Well, I can write the hyperboloid equation on paper, or better yet, plug it into a graphing calculator, and play around with the results.  Better still, I can plug it into Banate CAD, and see it on the screen in real time.  And even bestest of all, I can lower the facet count, and print these out so I can play with them in my hands.  It’s this latter case that’s the most exciting to me.  If I can make maths a tangible physical thing, then I’m more likely to remember it.  A set of these hyperboloid relatives, printed out as play things, would really help me retain and understand the math behind them.


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