Carpentry Math - Drawing an Ellipse

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— Calvin and Hobbes


Figure 1: Ellipses are Common in Carpentry (Source).

Figure 1: Ellipses are Common in Carpentry (Source).

Today, I received an issue of Fine Homebuilding Magazine (Dec/Jan 2011) that contained a bit of math that caught my eye. In this issue's "Tips & Techniques" section, there is a short piece called "A No-Math Method to Draw An Ellipse." Of course, this method does use mathematics, just no equations. It is a nice little geometric construction.

The basic construction approach shown here requires only a compass and square – tools in every carpenter's kit. Figure 2 illustrates the basic approach. First, draw two circles, one having a diameter equal to the ellipse's major axis and the other having a diameter equal to the ellipse's minor axis. You can locate points on the ellipse by drawing a number of lines through the axes' origin and both circles. For each line, draw perpendiculars from the line's intersection points on the outer and inner circles to the x and y-axes, respectively (see Figure 2).  The intersection of the perpendiculars determines the points on the ellipse. Once we have a sufficiently dense set of points on the curve, we can accurately draw an ellipse by interpolating between the points. Carpenters usually interpolate between the points using a thin strip of wood called a spline.

Figure 1: Ellipse Construction Using Compass and Square.

Figure 1: Ellipse Construction Using Compass and Square.

If you want a proof that this construction produces an ellipse, all you need is a bit of algebra.

x = a \cdot \cos \left( \theta \right),{\text{ }}y = b \cdot \cos \left( \theta \right)

\frac{x}{a} = \cos \left( \theta \right),{\text{ }}\frac{y}{b} = \sin \left( \theta \right)

{\left( {\frac{x}{a}} \right)^2} + {\left( {\frac{y}{b}} \right)^2} = 1{\text{, the equation of an ellipse}}

 
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