## Points of an Ellipse

This is about an image that is intended

• to be decorative,
• to establish the mathematical construction, with ruler and compass, of points of an ellipse.

The image is above as a `jpg` file, created by means of the `gimp` program from a `pdf` file, in turn created by means of `latex` and packages including `pstricks`, `pst-eucl`, and `a0poster`. The page size is actually A1, and Ayşe and I had the `pdf` file printed in this size on glossy paper; the result is now tacked to our bathroom wall. (The walls are painted a light blue that the firm calls yağmur damlası, that is, raindrop.)

The ideas behind the image come out of the paper, “Conic Diagrams,” discussed near the end of the recent post “Loneliness (Iliad Book IX).” The paper raises the question of whether a mathematical diagram can be art, at least in the sense of Immanuel Kant, even though, for him, “All stiff regularity (such as borders on mathematical regularity) is inherently repugnant to taste.”

The mathematical diagram in the image above is accompanied by text

• that explains it, and
• that it illustrates.

The text and diagram constitute a problem in the Greek mathematical sense: something to be done, as distinct from something to be seen, which would be called a theorem. The mathematical content of the diagram can be reconstructed from the text, which is given below. The idea is that a segment VW is divided at random at a point M, which is an endpoint of another segment, whose other endpoint is D. We know from Apollonius that there is a unique ellipse of which VW is a diameter and MD a corresponding ordinate. It is often overlooked today that ordinates of a conic section need not be at right angles to the corresponding diameter. Thus, when we start out, three parameters have been chosen:

• the ratio of VM to MW;
• the angle VMD;
• the ratio of VM to MD.

In the ensuing construction, we first draw a right angle DMB. We can let MB have any ratio to MD that we like, with one exception, to be described below.

There will be ordinates XP of our ellipse that are parallel to MD, with X lying on VW, so that P lies on the ellipse itself. This means the proportion

(XP : MD)² :: (VX : VM)(XW : MW)

is satisfied. Here VX and XW are the two abscissas corresponding to the ordinate XP. Thus, in words, the characteristic property of the ellipse is that the square on the ordinate varies jointly as the abscissas.

The circle might then be described as an ellipse in which the ordinates corresponding to any diameter are at right angles to it. In particular, MD will be an ordinate corresponding to the diameter BC of a circle. This means the triangle BCD has a right angle at D.

Now we should have chosen B so that BV and CW are not parallel, but intersect at a point A.

Here then is the text in the image; the sections are labelled with the parts of a proposition that Proclus lists in his Commentary on the First Book of Euclid’s Elements.

Construction of Points of an Ellipse

Enunciation. Of an ellipse, given a diameter, an ordinate, and the foot of another ordinate, to find the head of that ordinate.

Exposition. Given are

• a segment VW of a straight line,
• points M and X on the segment,
• a point D not on the segment.

Specification. Of the ellipse of which VW is a diameter and MD an ordinate, to find a point P so that XP is an ordinate, that is,

XPMD,
(XP : MD)² :: (VX : VM)(XW : MW).

Construction.

• Draw right angle DMB.
• Draw right angle BDC.
• Let DC meet BM at C.
• Draw the circle with diameter BC.
• Let VB and WC intersect at A.
• Let AX intersect BC at N.
• Let the parallel to MD through N meet the circle with diameter BC at J.
• Let the parallel to MD through X meet AJ at P.
• Let the parallel to BC through X meet AB at Q and AC at R.

Demonstration. We use Thales’s Theorem, and that NJ and MD are themselves ordinates of an ellipse, namely the circle with diameter BC.

 (XP : MD)² :: ((XP : NJ)(NJ : MD))² :: (XP : NJ)² (NJ : MD)² :: (XA : NA)² (NJ : MD)² :: (QX : BN)(XR : NC)(NJ : MD)² :: (QX : BN)(XR : NC)(BN : BM)(NC : MC) :: (QX : BN)(BN : BM)(XR : NC)(NC : MC) :: (QX : BM)(XR : MC) :: (VX : VM)(XW : MW).

That’s the text in the image. Proclus lists a sixth part, a conclusion, asserting that the problem stated in the enunciation has been accomplished.

All of the points can be in one plane, but they need not be. The point B can be out of the plane of V, W, and D. In this case, if the colors in the diagram are of any mathematical help, this is to show that:

• A is the apex of a cone whose base is the circle BCD, which has diameter BD and whose color is cyan, except for the green right triangle BCD;
• the plane that is parallel to the base and that contains X cuts the cone in a circle that has diameter QR and corresponding ordinate XP.

Thus MD and XP are ordinates of circles with parallel diameters and lying in parallel planes. This allows us to show that they are also parallel ordinates of the same ellipse.

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