![]() ![]() The number of points is used to define what is called the degree of the curve. Instead, the pointsįlow of the curve from the initial point to the last. Point, ends at the last, but does not necessarily go through all the others. Unlike a straight line, it does not pass through all of the points. In a similar fashion a Bézier curve is defined by any number of points, called control points. We all know that between any two points there can be only one straight line hence, weĬan define a specific line using only two points. In the 1960s, engineer Pierre Bézier used special curves in order to specify how he wanted car The answers to our questions originates from a surprising field: that of automobile design. It affect our curve? Is there any way in which we can make amends for our simplifications andĪssumptions that we had to perform earlier? An unlikely answer Is there a reason for this simplicity? Why should it be parabolic, and not some otherĬurve? If we changed the shape of the bridge - say, to make the tower more leaning - how would ![]() Satisfying, for the parabola is such a simple and elegant shape. The result we got - that the outline of the cables is essentially parabolic- is certainly Bottom right: Zoom on the square defined by x and y ranging from 0 to 1 in the tilted parabola, which is the region that represents the bridge. Bottom left: The same parabola tilted by 45 degrees. As we can see, the parabola equation is all the same.įigure 3: Top: The parabola R(S)=S 2/2+1/2. But this new coordinate system needn't frighten us. By replacing our variables and with and we have actually rotated our coordinate system by 45 degrees. However, with a little work it can be shown that if we define and we can rewrite our unfamiliar equation asĪnd this indeed conforms to our well-known parabola equation. Which differs vastly from our result, and you would be correct. Is the shape going to remain an unnamed mathematical relation? In fact, no! While it is not easy to see at first, this is actually the equation for a parabola! To this you might reply that the equation for a parabola is: ![]() So the smooth curve that is hinted at by the chords, the intersection envelope, is the outline you would get from infinitely many chords.įorgoing the detailed calculations (which you can find here), we find that all the points on this curve have coordinates of the type The more chords there are, the smoother the outline becomes. The outline formed by the chords is essentially made out of the intersections of one cable and the one adjacent to it: you connect each intersection point with the one after it by a straight line. Let's look at a coordinate system, The axis corresponds to the base of the bridge, and the axis to the tower from which it hangs.įigure 2: Our axes with evenly spaced chords. This is the core of modelling - taking only the important features from the real world, and We gain in mathematical simplicity, and still capture the beautiful essence of the bridge's form.Īfterwards we will be able to generalise our simple description and apply it to the real bridge structure. While we lose a little accuracy and precision, As the building itself is quite complex, featuring a curved deck and a two-part leaning tower, we will have to simplify things. In order to find out the shape the edges make, we are going to have to devise a mathematical Despite the fact that they draw out discrete, straight lines, we notice a remarkableįeature: the outline of the cables' edges seems strikingly smooth. ![]()
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