Stewart Precalc 6e Section 11.FM: Focus on Modeling: Conics in Architecture

Conics in Architecture The photographs on page 776 show six examples of buildings that contain conic sections. Search the Internet to find other examples of structures that employ parabolas, ellipses, or hyperbolas in their design. Find at least one example for each type of conic.

Constructing a Hyperbola In this problem we construct a hyperbola. The wooden bar in the figure can pivot at \(F_1\). A string that is shorter than the bar is anchored at \(F_2\) and at \(A\), the other end of the bar. A pencil at \(P\) holds the string taut against the bar as it moves counterclockwise around \(F_1\).

A Parabola in a Rectangle The following method can be used to construct a parabola that fits in a given rectangle. The parabola will be approximated by many short line segments. First, draw a rectangle. Divide the rectangle in half by a vertical line segment, and label the top endpoint \(V\). Next, divide the length and width of each half rectangle into an equal number of parts to form grid lines, as shown in the figure below. Draw lines from \(V\) to the endpoints of horizontal grid line 1, and mark the points where these lines cross the vertical grid lines labeled 1. Next, draw lines from \(V\) to the endpoints of horizontal grid line 2, and mark the points where these lines cross the vertical grid lines labeled 2. Continue in this way until you have used all the horizontal grid lines. Now use line segments to connect the points you have marked to obtain an approximation to the desired parabola. Apply this procedure to draw a parabola that fits into a 6 ft by 10 ft rectangle on a lawn.

Hyperbolas from Straight Lines In this problem we construct hyperbolic shapes using straight lines. Punch equally spaced holes into the edges of two large plastic lids. Connect corresponding holes with strings of equal lengths as shown in the figure on the next page. Holding the strings taut, twist one lid against the other. An imaginary surface passing through the strings has hyperbolic cross sections. (An architectural example of this is the Kobe Tower in Japan, shown in the photograph.) What happens to the vertices of the hyperbolic cross sections as the lids are twisted more?

Tangent Lines to a Parabola In this problem we show that the line tangent to the parabola \(y = x^2\) at the point \((a, a^2)\) has the equation \(y = 2 a x - a^2\).

[The construction uses Dandelin spheres: two spheres inscribed in a cone and tangent to a cutting plane. Let \(F_1\) and \(F_2\) be the points where the upper and lower spheres touch the plane, and for a point \(P\) on the conic curve let \(Q_1, Q_2\) be the points where the cone's generator through \(P\) touches the upper and lower spheres.]