Note how we would just get the points in the circle repeated four times from the second part, were it not for the addition. Adding the two circles together therefore we get a parametric formula: This equation allow a computer to draw a circle as we put in values of x and plot the point in 2d. Now recall the parametric formula for a circle:, for x between 0 and. The path given by these cogs can therefore be encoded as (1,5) + (4,1). We also need to know the sizes of the two circles, in this case 5 and 1 (I will not give units as only the relative sizes effect the shape). If we let the big circle go round once, the small circle will go round four times (once for the large circle rotating and three going round the large circle). To plot the curve we need to consider both circles moving round. The final position is the position on the two circles added together. The black circle on the small cog then shows how the red dot moves. The centre of the small cog moves round this circle. We have the circles for the two cogs, but these are not very useful. How do we model this?įirstly lets consider circles. We now look at the red point on the smaller cog and watch it move along the green line. The smaller cog will therefore go round 3 times every time it goes once round the big cog. In this case, we have a cog with 30 pegs and a cog with 10. The sizes of the two cogs show how fast they go round each other. We need a more algebraic form that we can give to the computer, and some way of simply describing the circles. The first step is to get tools we can play with more easily than simply describing the geometry. Now we want to go further, to try to make similar figures in 3d. Luckily humanity was up to the challenge and produced the geometric chuck. ![]() To make images with more than two circle you obviously need a more complicated device as the circles might bump into each other (just think of three cogs). Unfortunately if you add enough circles you can actually get any curve you want, so the method could never be disproved, though it was eventually replaced starting with the brilliance of Copernicus who put forward a model of the solar system with the sun at the center. Adding circles like this gave one of the first predictive models of the planets as they move in their strange paths across the sky. It is easy to find lots of examples of these curves online. In mathematics there is a mess of names to describe the curves produced, I shall just list them, understanding the differences is a good way of learning the subject: epicycloid, hypocycloid, epitrochoid, hypotrochoid. You can make the circles as cogs and then you get a classic toy. There is a great deal that can be done with just these, but what if you want something more complex? Spirographs are a very simple idea, let one circle run around a second. The basic geometric ideas are straight lines and circles.
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