The Sydney Opera House in Polar Equations
Designing for practicality and functionality is obviously vital to the success of architecture projects, but one must never forget that it is also an art form. There’s beauty in its curves, textures and edges. When my Honors Precalculus class at Concordia International School Shanghai required us to complete a polar equations project that mapped out a building using polar equations in Desmos, my mind immediately jumped to the iconic Sydney Opera House. The process, though challenging and time consuming, taught me an immense amount about rectangular to polar conversions. Here was my process:
Parabolic Curves
To begin, I recognized that the famous curved roofs on the Sydney Opera House could be broken down into parabolas, typically represented by the rectangular equation y=ax²+bx+c. With that said, my first approach turned out to be a flawed one: constructing the parabolas via trial and error. For the standard equations (x-h)²=4p(y-k) and (y-k)²=4p(x-h), I attempted to adjust the h and k values, which determine the coordinate of the vertex, and the p-value, which determines the distance between the focus and the vertex, until I landed on an equation that matched my desired curve. Although Desmos’ slider feature allows you to efficiently adjust the values of variables with ease, I soon realized that this process took far too long and yielded mediocre results.
Then, I recalled a set of procedures I had learned from previous math courses: finding the equation of a parabola using 3 points. Because any parabola can be represented by y=ax²+bx+c, any 3 points on that parabola provide values for a solvable system of equations that would yield the needed a, b and c values. Thus, all I had to do was plot 3 points along a desired curve on the image and use a calculator to solve the derived system of equations. This yielded almost perfect results and required little time.
As one of the constraints of the project was that all equations had to be in polar, the next step was converting my rectangular y=ax²+bx+c equation into a polar equation with r (directed distance from pole) and θ (directed angle from pole). First, all x and y values can be represented by rcosθ and rsinθ respectively, as proven by the definitions of r, cosθ and sinθ. Second, all terms must be moved to one side of the equation for the sake of combining like terms. In this step, factor out the common variables of r; for example, rcosθ+rsinθ should be written as r(cosθ+sinθ). Essentially, you are rewriting the equation to become a different version of y=ax²+bx+c, where the x² and x are now represented by r² and r. This is critical for the next step: using the quadratic formula. Just as the quadratic formula can be used to solve a quadratic equation for x, it can also be used to solve for r using these newfound a, b and c values from the last step, which are written in terms of cosθ, sinθ and a constant. Plugging these new values for a, b and c into the quadratic formula, you now have a polar equation with r on one side and a complicated amalgamation of cosθ, sinθ and other values on the other side. Though this process seems complex, its procedural nature makes it highly efficient in the long run.
Now that the equation has been converted to polar, the final step is entering a θ range that fits only the desired region of a curve. Here, it is critical to pay attention to the gridlines on the Desmos graph, which indicate θ values in 15º intervals. To find the desired interval, estimate roughly where the curve should start and end by referencing the gridlines and then adjust as necessary.
There were some instances where the shape of the roof did not perfectly align with a parabolic shape. In reality, it’s more of a rounded “shell.” In such cases, I combined a parabola with a straight line, which I will further explain in the next section.
Straight Lines
Now that the parabolic curves are completed, the straight lines are fairly straightforward. First, plot two points that align with a straight line in the image. Next, use these points to find the value of the linear line in the form y=mx+b. To do so, you can solve for the slope using (y1-y2)/(x1-x2), and then use this slope and the coordinate of one point to solve for b, the y-intercept. Finally, convert the rectangular equation into a polar equation by converting x and y into rcosθ and rsinθ respectively. Once again using factoring as described in the last section, you can apply algebraic principles to isolate r on one side and end up with a constant over either the subtraction or addition of sinθ and cosθ values. These procedures should yield a straight line in polar form.
To finish the process, find the correct θ range by referencing the grid lines in Desmos. Throughout this process, I found that horizontal lines and straight lines with a very small slope tended to be difficult to find a perfect θ range for. Because of how polar equations graph lines, flat or almost completely flat lines can experience a dramatic change in appearance from a small change in θ interval. Therefore, finding the correct θ range requires a good deal of trial and error.
In total, the project included 131 polar equations and many laborious hours of work, but the end product seemed to do the beauty of the building justice. Below, I have included a compilation of every polar equation that went into constructing the Sydney Opera House:
