Re-Learning Mathematics

This collection grows with time, so come back to check it out sometime in the future. If it stays quite for too long, ping me and let me know!

New 3D animation techniques have me revisiting mathematics. In my math book research, I found this information below to be particularly interesting and/or useful.

Perspective! by David Chelsea
Why not start with a bit of inspiration first? Would you be inspired to know that you can learn to draw environments like a professional comic book artist… while also forming questions as to why all these proportions line up almost magically.

Precalculus Mathemetics in a Nutshell by George F. Simmons
AH! It looks like a text book!! Don’t worry, it’s actually tiny, at about 100 pages. The author teaches calculus and states that he teaches precalculus (algebra, geometry and trigonometry) in a single day. It will take most of us longer than that… but this information gives a great grounding position to pounce off from.

The Secrets of Triangles by Alfred S. Posamentier
I’ve only read a few sections, but this appears to be a great place to start for those that want to understand points in space and their relations to each other.

Measurement by Paul Lockhart
It’s all about the relationships. Want to play with some math without writing letters or numbers? Start here.

Euclid’s Window by Leonard Mlodinow  (haven’t read yet, but I kept encountering it while looking at the books above).

For the 3D animation programmer out there, you have quite a few to select from and I will have to take some time in the future listening those out, as well.

For the person who hates math, I recommend checking this list out, as it may help you understand where math becomes beautiful.

Vector Math for 3D Graphics and Animation

2014-01-27 09.32.05First, I’m not a “graphics programmer”. I do quite a bit of scripting (Maya’s MEL scripting language) and enter dozens of expressions (Houdini), but have never compiled a single line of C++.

I was never a big fan of math, especially algebra (grammar = yuck), though I did breeze through my Geometry class. I survived a decade working in the 3D animation industry working as a rigger, technical artist and production technology lead and only implemented a little triangle math only a handful of times. *note: This does NOT mean you should skip out on getting a proper understanding of Euler rotation orders and gimbal lock, if you want to be a good character TD.

Though no one needs to understand 3D math in order to become a 3D artist, even some digital sculptors realize the importance of a little trigonometry. Having an elementary 3D math base gives an artist a better understanding of the ‘why”s.

So, since I don’t remember too much of my high school math and never took a single trigonometry class (I’ll blame the education system), I eventually realized that if I was going to produce anything jaw-dropping, I was going to need to study up a bit…

…and luckily the internet has the answers!

Where to start?
There is a boatload of free online courses available to those willing to learn from text and videos, but mathematics is a massive world and it’s difficult to figure out where to start. For those that have completed highschool (maybe even just 10th grade) and are interested playing with some bit of 3D code, here’s some items I found quite useful. Keep in mind that most 3D animation software packages will include tools (commands, expressions, functions, etc.) that will do the math for you. Remember to search the documentation!!

If you have a line, what direction is it pointing? If you have an object in space, what direction is it moving? This is your vector. The length (aka “magnitude”) of a vector can tell you it’s speed. A normalized vector is always 1, and is useful to apply to other operations. You can subtract two vectors and you’ll get another vector which tells you the direction from one vector to another. More examples with images on MathIsFun.com.

cross product (aka vector product)
a × b = |a| |b| sin(θ) n
I have two vectors and I want to find another vector that is perpendicular (90 degress) to these two vectors. The cross product of the two vectors will help. It’s useful if I have a plane and I need to know the surface normal to the plane. For those using particles, this tells us the direction that particles would travel if we are emitting from the middle of a surface. BUT, be careful, the direction of this newly calculated vector depends on what order you specified a & b. This is where the dot product comes in very handy! More cross product details on MathIsFun.com.

dot product
a · b = |a| × |b| × cos(θ)
If you have two vectors, the dot product will tell you if the vectors are moving in the same general direction (greater than 0) or moving in two completely different directions and will never intersect the same plane (less than 0). If you have just used the cross product to find a perpendicular vector (like a surface normal), you can use the dot product to see if the if it’s facing the right direction (up?). More info on MathIsFun.com.

For video explanations, Khan Academy has a great section on vector dot and cross products or trigonometry and precalculus if you already have a basic understanding of geometry.

Here’s a list links to more information on vector math:

So what’s next?  Perhaps some matrix calculations? Maybe linear algebra via MIT’s online course or N.J. Wildberger (known for “rational trigonometry”).

Finally, checkout this list of free technical online courses, including programming.

Thanks to Les for the inspiration to catalog my findings.

Numbers and Patterns in Nature

Was recently introduce to Vi Hart’s blog. Tons of inspirational mathematical games can be found on her blog, including a practical introduction to the Fibonacci spiral.

Finally, if you have not seen “Nature by Numbers”, well here is a real feast for your eyes… and soul.

Nature by Numbers from Cristóbal Vila on Vimeo.

Don’t Be Afraid, Math Doesn’t Always Hurt

Nature by Numbers by ETERIA.

If only this became the new definition of “infographics”.

And here’s a Youtube link for those of you on smartphones.

I saw the beginnings of Drew’s visualization of DNA replication about 8 years ago in L.A. Have things improved since then? Well, my cheeks start to hurt when I think of how happy I am for him.  Watch this.

Vi Hart’s blog is another visual explanation of math in nature.

Constructing the Universe w/ Michael Schneider

Spirit Science’s presentation on sacred geometry creates an intense ride through the many patterns that can be found in nature and describes it’s effects on our consiousness… and our consciousness on it!

What is Pi? Simple animations can help see where the 3.14 comes from.

Finally, if you are super geek, Professor Norman J. Wildberger has gone back in time to explain how trigonometry was supposed to be taught. I swear, my dread would have been turned to love, had I been exposed to trig like this.

Nature By Numbers – The Magic Spiral.

Some are familiar with the Fibonacci sequence, but if you’re not, I can’t think of a better example of it’s existence in nature.

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