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Quadratics in design

In this video Norman Wildberger shows how quadratic curves figure in design by explaining how to parametrize curves.
Now we would like to introduce some applications which are very modern for quadratic relations– applications that were discovered by two French car engineers around 1960. They give us a new approach to design, a problem of how to specify curves. So in order to understand this, we have a little bit of a discussion about parameterising curves. This is a point of view that’s shifted a little bit from the idea of an equation describing a curve to the idea of having a curve described by a parameter t as it’s moving. It’s an interesting alternate way of thinking about what a curve is. So let’s start with the parametric equation for a line.
So here’s a line through three points a, b, and c. And we can write down its equation pretty easily. So the equation of the line– it’s called l– is x plus 2y equals 7. And we can check that, because the point 1, 3 satisfies the equation. So does the point 3, 2, so does the point 5, 1, and so on. Now, here’s an alternate form for the same equation. I’m going to make a certain expression. I’m going to write 1 minus t times [1, 3] plus t times [3, 2]. And you’ll see in a minute why this is a good thing to do.
So if we take that combination and we combine it– 1 minus t times 1 plus 3t is 1 plus 2t, and 3 times 1 minus t plus 2t will be a total of 3 minus t. So let’s have a look at this particular expression. t is a parameter here that’s allowed to vary. When t equals 0, then we get the point 1, 3. When t equals 1, then we get 3 here and 2 here. We get point b. If t is some intermediate value between 0 and 1, then we’re going to get some point that’s between a and b.
Those of you who know a little bit about vectors, there’s really a vector formulation here. We’re really taking the point a and adding t times the vector ab to get at this formula. Just another way of thinking about it. So there’s a general formula here. So if we have two arbitrary points– let’s call them p0 and p1– and we’re interested in the line through these two points, so then the line p0 p1 can be described by the coordinates 1 minus t times the coordinate of p0 plus t times the coordinate of p1. That’s basically exactly what I did over here where I used the points a and b to create this thing here.
So this is the general form for what I did in the specific case up here. So this is called an Affine linear combination of these points p0 and p1. With the property that this coefficient and this coefficient always add up to 1, that will guarantee that we’re always getting a point on this line p0 p1. So around 1960, Paul de Casteljau and Pierre Bezier, who were two French engineers working for Renault and Citroen, French car companies, both were tackling the same problem of how to describe mathematically a general kind of curve in a way that the information could easily be translated to somebody else.
They came up with this remarkable idea, which was to extend the parameterised line segment that we were talking about to the quadratic case. And many of you will be familiar with this in some form, because Photoshop, CorelDRAW, Illustrator, a lot of CAD programmes will be using Bezier curves and control points to describe curves. So what’s the basic idea? We start with three points now, p0, p1, and p2. And we imagine these segments here being traversed by parameters. So imagine one person walking from here to here at a steady rate, say time 0 to time 1.
And at the same time, somebody else is walking from here to here at a steady rate, starting at time t equals 0 at time t equals 1. And at all points, we’re connecting the point on this segment with the point on this segment with a green bar. So that green bar is moving. And at the same time as these two fellows are walking, there’s a third person which is walking along the green segment, also starting zero at 1 and finishing at time 1 at the other end.
So if we look at the cumulative effect of all three of these motions– this red point moving, this red point moving, and at the meantime, this green bar moving and this intermediate point moving– then the trace of this point 1 is a curve. And it’s a quadratic curve. It’s a conic section. In fact, it’s a parabola. It’s a parabola which is tangent to these bounding lines and goes through these two control points. This control point is not actually on the curve, but it controls the position of this de Casteljau Bezier curve. It’s a very useful kind of thing. So for example, in the theory of fonts, you have some interesting font that you want to describe.
One approach to that is to cut it up into manageable pieces. And on the manageable pieces, you create some Bezier control points. For example, you might draw tangents there. So you have three control points which would describe that little segment of the curve. So if you do this for a couple of segments, then you can describe the entire curve with just a finite number of control points. It’s a very powerful tool, which is actually used in true type fonts made by Apple and used by Microsoft– just using, basically, these quadratic de Casteljau Bezier curves. Here is the actual Affine combination equivalent of the formula that I wrote down for a line.
So it’s a quadratic version of the formula that actually gives you very specific control over the curve. If you know what p0, p1, p2 are, this is the formula that allows you to describe exactly what’s happening to the general point on the curve. It’s a very important and useful application of quadratic curves in modern design.
Lines and other curves can also be described by parameters. We see how to apply this idea to introduce the remarkable approach to curves of de Casteljau and Bezier introduced in the early 1960s.

Parametrising lines with convex linear combinations

While lines can be described by linear equations, there is also another way, involving parametrising points on the line.
If \(\normalsize{A}\) and \(\normalsize{B}\) are two points, we can traverse the line segment \(\normalsize{AB}\) linearly as time \(\normalsize{t}\) goes from \(\normalsize{0}\) to \(\normalsize{1}\), so that at time \(\normalsize{t=0}\) we are at \(\normalsize{A}\), and at time \(\normalsize{t=1}\) we are at \(\normalsize{B}\). For example, if \(\normalsize{A=[2,3]}\) and \(\normalsize{B=[5,1]}\), then such a path is given by
This is also called a convex linear combination of points \(\normalsize A\) and \(\normalsize B\), because we can write it as
and the two coefficients have the property that they sum always to \(\normalsize 1\).
A nice physical model of this would be to imagine a very light (but strong) wooden rod between the points \(\normalsize A\) and \(\normalsize B\). If we were to take \(\normalsize 1\)kg and then hung \(\normalsize t\) kilograms off point \(\normalsize B\) and the remaining \(\normalsize (1-t)\) kilograms off point \(\normalsize A\) then the center of mass (which is the fulcrum, or the point of balance) would be the same as the point \(\normalsize P\) given by the convex linear combination.
pic of a static labelled convex combinations as box weightsUNSW Australia

De Casteljau Bezier curves

Around 1960, two French car company engineers independently discovered a new way of specifying curves, using parameters. The simplest of these curves are quadratic de Casteljau Bezier curves, and they always give parabolas, but rather general ones, not necessarily of the form \(\normalsize y=ax^2+bx+c\). These curves have gone on to revolutionise design theory, are important in architecture, and figure prominently in the design of modern fonts.
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Maths for Humans: Linear, Quadratic & Inverse Relations

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