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Mathematics Doesn't Have To Be Difficult: How Computer Graphics Helps Student Understanding

by John Vince 21st May 2019

Computer graphics can help students grasp mathematics. John Vince explains how

Let’s be honest: not everyone wants to be a professional mathematician. Nevertheless, many of us are interested in mathematics, and have to use it to solve problems associated with our every-day work. I belong to this second category.

I was reasonably good at mathematics at school, and developed the subject during my electrical engineering studies. However, as my career travelled towards computer science and, later on, computer graphics, I realised that I would have to become more familiar with the subject. In particular, vector analysis, 3D geometry, matrix analysis and the design of algorithms were essential topics. And when I moved across to lecturing, I found myself having to teach mathematics and computer science to artists and graphic designers, which required new skills.

I have a visual brain, where an image is important to understanding a new concept as opposed to people who process symbols and abstract ideas. Artists and designers are interested in form, colour, texture, symmetry, contours, shading, typography, etc., and are not generally mathematically literate. So there were obvious challenges.
Computer graphics provides the perfect vehicle to bring mathematics alive.
Fortunately, computer graphics provides the perfect vehicle to bring mathematics alive. 3D-Cartesian geometry, where any point (x,y,z) is changed into another point simply by adding or subtracting a number to the coordinates is an easy concept to appreciate, which leads to simple equations and eventually matrix notation. Within a short period of time, students are able to write programs incorporating library functions, where objects are rotated in space to create animations. This started for me before interactive computers were invented. In fact, my first main-frame computer required a large air-conditioned room with a massive 24KB of memory!

As my lectures progressed I introduced concepts of perspective, shading algorithms, lighting, shadows, Bézier curves, and B-splines. I always took jars of Jelly-Baby sweets and fudge into my lectures, which established an air of informality. Very often, I asked students to close their eyes to visualise the graphs of 2D and 3D equations. This emphasised the one-to-one relationship between mathematics and the visual arts. Once, I taught a blind student mathematics, and traced curves on his back with my finger to communicate the shape of an equation.

As computer systems improved, students were given access to high-performance PCs and commercial animation systems. They could thus be taught more advanced concepts of computer science and mathematics, and undertook some amazing animation projects.

The final-year mathematics exam paper for my class was challenging, but was passed by most students, which showed that I was managing to communicate a wide range of mathematical concepts using graphical tools. These graduates secured top jobs in computer animation and special-effects companies, because of their multi-disciplinary talents. One of my early (1980) Technical Illustrator students became so engrossed by computer graphics and programming, he wrote his own hidden-surface removal algorithm and shading algorithm, and for a final-year project showed NASA’s space shuttle taking off and landing! Another student constructed the Eiffel tower and Statue of Liberty with an animated sequence.

After reading several books and papers, the penny dropped, and all became clear. I became aware that I was in a unique position, and had acquired some useful information that could be communicated to a wider audience through books. I began by writing some small introductory books on computer graphics, and one on Mathematics for Computer Graphics, which is now in its 5th edition. Writing is harder than lecturing, as one does not know the background of the reader, nor does one know the particular issues that will cause problems for them. Nevertheless, without talking down to the reader, I developed a writing style which introduced potentially difficult mathematical topics to a non-mathematical reader. I always remember coming across quaternions for the first time whilst working for a flight-simulator company, and found them impossible to understand. The same thing happened with geometric algebra: the subject was completely new to me and completely obscure. Then one day, whilst walking with my dog, and some intense thinking, another penny dropped, followed by two more books.

So, if you find mathematics difficult, be reassured that you are not alone. But today, there are some amazing websites and YouTube videos that will increase your understanding of this.
Featured image: 3D cube weurfel by Marcus Herold. Available on Wikimedia Commons, CC BY-SA 4.0