Constructing Ellipses

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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Foci and Axes

Given the major and minor axes of an ellipse, you can always find the foci. You need the foci for some construction methods. Just draw radii of length a from the ends of the minor axis.

Given the foci, however, you can't uniquely determine the axes. You need additional information such as the length of one axis. However, the major axis is always along the line through the foci and the minor axis always perpendicularly bisects the line between the foci.

Pin and String

Mostly useful as a heuristic tool to help students visualize ellipses, but it can be useful for constructing large ellipses.

Put a pin in each focus and tie a string to each pin leaving slack with length 2a. Pull the string taut with a pencil point and slide the pencil to draw the ellipse.

This makes use of the fact that an ellipse is the locus of points whose distances from two fixed points have a constant sum.

http://www.sahraid.com/ has a spiffy tool that makes this method much handier to employ.

Trammel Method

In my view the best method. With the major and minor axes constructed (and extended) mark a piece of paper with points O, A and B so that OA = a and AB = b. Slide O along the minor axis and B along the major axis. Point A traces out the ellipse.

With any reasonable care this method is quite accurate and it is very fast.

For cases where the axes are similar in size, the method above may be inaccurate. It is also possible to draw an ellipse using an external trammel.

Envelope Method

Given the focus and a circle whose diameter equals the major axis, draw a radius from the focus to the circle, then a line perpendicular to that. The perpendiculars sweep out the ellipse.

This is fast and can be quite accurate. If you use an index card to draw the perpendiculars you can dispense with drawing the radii. Just let one edge of the card serve as a radius and use the other to draw the perpendicular.

For a quick and dirty way to sketch an ellipse this rivals the trammel method. It roughs out the area enclosed by the ellipse a bit faster.

The Draftsman's Method

This method is given in a lot of drafting texts. For extreme accuracy it's probably the best method. It's convenient for use on a drafting board with T-square and triangles.

Construct the major and minor axes and draw circles with each axis as diameter. Also construct radii as shown. Angles aren't critical so radii can be closer in areas where greater accuracy is needed.

Draw vertical lines from the intersection of each radius with the outer circle and horizontal lines from the intersection of each radius with the inner circle. The intersection of each pair of corresponding lines is a point on the ellipse.

It's easy to see this is simply an implementation of the parametric equation of an ellipse: x = a cos t, y = b sin t.

When the desired number of points are drawn, construct the ellipse.

Paralellogram Method

Sometimes you know an ellipse can be enclosed within a parallelogram, for example, a foreshortened view of a circle or a spherical object sheared out of shape.

Construct the parallelogram and divide it into quarters. Divide all the lines into equal numbers of segments and number them as shown.

From the midpoints of opposite sides, draw straight lines through the tick marks as shown.

Continue the construction for all quadrants of the parallelogram.
Connect all the 1-1, 2-2 intersections, etc., to construct the ellipse.

This construction will work perfectly well if the parallelogram is a rectangle, so it will work to construct an ellipse if the major and minor axes are known.

If the parallelogram is a square, the resulting ellipse is a circle. The intersecting lines are perpendicular, and the construction is the famous one of constructing a right angle inside a semicircle. The general construction here simply works by deforming the construction so an ellipse results.

Five-Center Method

This was long a standard in drafting texts, and pretty much obsolete today. Any CAD program will draw true ellipses, and the trammel and envelope methods are faster. It's here mostly out of respect for tradition.

Draw the major and minor axes of the ellipse and draw a rectangle around them. Construct XY and draw P-C1 perpendicular to XY.

Let X-C2 = r and locate Q such that YQ = 2r. Draw a circle of radius r centered on C2. By symmetry, everything on the right side of the diagram is repeated exactly on the left.

Using C1 as center and C1-Q as radius, draw an arc through Q. Call the intersection of this arc with the circle X-C2 C4.

Construct line C1-C4 and extend it to T. Using C1 as center, draw arc YU.

Construct line C4-C2 and extend it to L. Using C4 as a center, construct arc UV. The successive arcs YUVX are an approximation to the ellipse.

The true ellipse is shown in red, the approximation in purple. The approximation is quite good for slightly or moderately eccentric ellipses but becomes obviously incorrect for very elongated ellipses.

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Created 17 June 2005, Last Update 30 January 2012

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