Constructing Ellipses

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.

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.

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

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.

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.

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.

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.

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.

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 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.