# Quadratics in design

## 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 \[\Large{P(t)=[2(1-t)+5t,3(1-t)+t]}.\] This is also called a*convex linear combination*of points \(\normalsize A\) and \(\normalsize B\), because we can write it as \[\Large{P(t)=(1-t)[2,3]+t[5,1]}\] 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*.

^{UNSW 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.#### Maths for Humans: Linear, Quadratic & Inverse Relations

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