Processing: Creative Coding and Computational Art

Geometry
Like algebra, geometry dates way back to ancient Egypt and Babylonia (2000 to 500 BC),
where knowledge of geometry was demonstrated through building and development projects.
However, it was Greek influence, beginning with Thales and Pythagoras in around 600
BC, through to Archimedes in around 200 BC, that gave us what we think of as Euclidian
geometry today. Of all the Greeks, it was Euclid of Alexandria, in around 300 BC, who
would leave the most lasting mark on Greek geometry with his famous 13-book treatise,
The Elements of Geometry. This treatise formalized Greek thought on geometry up to that
point, and is even considered by some to be the first textbook ever written. It is certainly
one of the most important and influential books ever written on mathematics, with versions
of it still in print—over 2,500 years after it was written—that would be a lot of royalties
for Euclid!
The word geometry comes from the Greek words for earth and measure, and suggests its
original use in building, surveying, astronomy, and other real-world applications. Geometry
was introduced to Europe during the early Renaissance, and its influence is obvious and
ubiquitous, especially in Renaissance architecture and the visual arts. However, with regard
to computer graphics, the major development in geometry didn’t occur until the 17th century,
with the advent of analytical or coordinate geometry, developed by René Descartes
and Pierre de Fermat. Analytical geometry utilizes the Cartesian coordinate system (also
developed by Descartes) as a system to study geometric curves and shapes plotted utilizing
algebraic equations, which is precisely what is done in computer graphics. (Thankfully,
most of the math is actually done behind the scenes for us.)
Points
The point is the most basic geometric element we deal with. A point has 0 dimensions;
although in Processing, you do see a 1 pixel by 1 pixel output to the screen when you write
the following command (making this technically a 1-pixel-long line—but we’ll let it pass).
point(x, y);
Points, as data structures, are primarily used to store coordinate locations for plotting
curves and shapes. Java has some convenient data structures, or classes, for this very purpose,
such as the aptly named Point class. Remember, in Processing, you have the option
of using Java classes as you see fit.
Lines
COMPUTER GRAPHICS, THE FUN, EASY WAY
Lines occupy one dimension, as they have length, but no width. Lines can be expressed
algebraically with the expression y = mx + b.
This equation for a line is referred to as the slope-intercept form, where m is the slope of
the line and b is the y-intercept (the place where the line intercepts the y-axis). x and y are
the two components of any point on the line. Slope is an important property of a line or
curve in math and computer graphics, and often relates graphically to motion and acceleration.
For example, the graph of the line in Figure 4-5 shows an object moving at a constant
rate. The vertical axis of the graph is distance, and the horizontal axis is time. The
slope of the line can be found by looking at the change in y (∆ y) over the change in x
(∆ x) at any two points on the graph. People commonly refer to the slope as rise over run.
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