“What are these photo’s you brought?” The interviewer asked during my senior year at the university. I was applying for my first engineering job and I brought pictures of the devices I had made for my dad’s cabinet shop. I think it was the photo’s that convinced the interviewer that I had good practical hand’s on engineering experience. I got the job. We packed up our little household and moved to California, or should I say ‘returned to California,’ the place of my birth.
My assignment? Design and test supporting structures for satellite payloads. Requirements? High strength to weight ratio’s with high reliability. I learned about space environments, material properties, computer aided design systems, and how aerospace companies work.
After a couple years I became a test engineer, then system safety analyst, then electronics packaging designer, then… Each assignment gave me experience in new areas of engineering. I tried diligently to learn company goals and objectives and participate in process improvement initiatives. This focus brought trust and new opportunities to serve and grow. But I wasn’t finished with school yet. I wanted a masters degree and I wanted to continue my education; I loved to learn.
I applied for graduate school, was accepted and returned to the university. I was in tears as I left the math building that warm June day. Summer on college campus was relatively quiet, but my mind was clamoring with the noise of theorems and derivations, and proofs drumming away at my confidence. “What have I gotten myself into?” I thought. I had left a good paying job and returned to college after nine years to pursue a master of science degree in mechanical engineering. The very first class I had was Linear Algebra. It was a lot of work relearning matrices and vectors, moving into linear transformations, determinants, eigenvalues and all their applications. There were times when I didn’t know if I could do it, but I kept at it and looked for ways each day to apply my new knowledge so that it would be interesting and meaningful. How meaningful could linear algebra be?
Do you remember the other day when you played a video game, or went to an animated digital movie. How did the movie makers make those complex graphic images look so convincing and real? Perhaps just today you swiped your fingers on a touch pad or touch screen and the photograph you just captured moved or rotated or zoomed at your command. Chances are pretty good that the people who programmed your device used linear algebra or matrix arithmetic to pan, zoom, scale, rotate, or even give depth perspective and reflection to the scene making it look real, like you were really there.
If you think of each point or pixel on the screen as a member of a large array or matrix of vectors (↗: lines with magnitude and direction), then using the rules or theorems of linear algebra you can program all these points to move or change color or take on different shades of grey or even reflect light coming from another point (a.k.a., ray tracing). You can even make one object appear to disappear behind the one in front of it (It’s called “hidden line or object removal”). It’s pretty amazing actually. Suppose one matrix represents an object on your screen, say the eight corners of a cube, and let’s say you throw the cube off a tall building, or at least you want your audience to think it is really being thrown off a building, you can use the laws of physics (classical mechanics) to calculate what a real cube would do as it falls (speed, rotation, trajectory or arc), and then multiply the cube matrix by the speed, rotation, and trajectory pipeline of matrices to get the next frame of the movie, update the pixels on the screen and then repeat the process over and over again forty times per second until the cube hits the ground. But wait, don’t stop there, you can continue the scene as the cube bounces or crushes, or gets stepped on…
Linear Algebra is used in computer graphics, games, chemistry, flying real airplanes, economics, forecasting the weather, data compression (e.g. JPEG), sociology, traffic flow, electrical circuits, and many many other applications.
During my graduate work, and after, I have used Linear Algebra to write my own computer graphics software, develop mechanical systems to reshape complex surfaces, and many other things. When I create using a computer aided design (CAD) application, I understand what the software is doing when I click the mouse or drag a feature from one point to another. When I sit down to a digitally animated movie, I’m a little distracted from the story because “I know how they do it!” I know how they made all those characters move around and do what they do. Linear Algebra is a powerful tool. No tears anymore, just determination. I still don’t have all the rules memorized; but that’s okay, they’re not hidden. I know where to find them.
In graduate school I took linear algebra, finite element method, CAD software development, and utilized these tools in my thesis to research numerical-to-physical surface shape manipulation. I wanted to morph surfaces; surfaces that could be used as forms to shape other objects. I applied for funding and built numerically controlled surfaces (See Engineering Story “The Ribbon Cutting” for a fictionalized story based on my thesis). While the academics were fresh in mind, I took the professional engineering exam to make sure doors of opportunity were always open.
Following graduate school, I worked in industrial automation designing new methods to handle printed circuit boards during production (See fictionalized Engineering Story “Get A Grip”). I also worked as an Engineering Manager during those years. I think the master of science degree was an advantage. Eventually I returned to the aerospace world with modeling and simulation work on guidance, navigation, and control instruments. Along the way I developed my writing skills, a plus that opened more and more opportunities.
I have also looked for opportunities to serve as a mentor to other young engineers; another investment with definite returns. Although not my motive, I was always improved, when I sought to improve and help others. Several stories in this book, for example, are based on experiences gained while coaching and mentoring engineering college seniors.
Such has been my satisfying journey to and through engineering. What will your journey be like?