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Saddle (hypar) Surfaces

Saddle in a box
The hyperbolic paraboloid or hypar or saddle is one of the nine real quadric surfaces and one of the six which are ruled. In fact it is one of the three doubly ruled surfaces (besides the plane and the hyperboloid), having two distinct independent families of lines generating the surface. There are two distinct grids of lines which are skew but appear parallel when viewed from above. This surface envelops an important point in applied mathematics called the "saddle point" found at its center which appears to be a maximum point in one plane, a minimum point in another plane, and actually is neither.

Construction of the fixed saddle a rectangular box

In rectangular coordinates, the equation kz = x2 - y2, when solved for constant values of x and y, produces the following two curves, which are the projections of the surface:
on the pair of planes x = + a the equation is: kz = (a2-y2) in the yz plane
on the pair of planes y = + b the equation is: kz = (x2-b2) in the xz plane
These provide the patterns for drilling proper holes if two different shaped parabolas are used. If a = b then the parabolas will be the same shape and form a square base.

Construction of the fixed saddle rotated in three-space

Saddle in a wedge
An easier form for the rectangular model of the saddle surface is a saddle rotated pi/4 which results in new axes X and Y where
X = (x+y)/ 2
Y = (-x+y)/ 2

so that the hyperbolic paraboloid kz = y2 - x2 becomes
kz = 2XY

Skew saddle

Thus the surface is generated from curves drawn on two isosceles right triangular planes, ABD and BCD, which are joined orthogonally along their hypotenuse BD. Twelve evenly spaced cuts are sawed into each edge of the sides of two pi/4 isosceles right triangular planes, ABD and BCD. The threading is started by hooking after anchoring one end proceeds from the corner of one plate to the corner of an adjacent plate. Care should be taken to make the initial and final cord ends meet for a common anchoring.

Saddle in a cylinder

Construction of the fixed saddle in a cylinder

Another convenient saddle equation occurs in cylindrical coordinates: z = r2 cos 2t. The intersection of this surface with the cylinder r = ro is the curve
z = ro2 cos 2t or Z = cos 2t,
where Z = kz/ro2 The cosine curve can be plotted on a plane graph using t and Z as rectangular coordinates with the points to be joined indicated. The scale along the t axis has to be adjusted so that the total distance between the extreme positions t and t + pi measures 2 ro, the circumference of the cylinder. The amplitude of the curve ro2 must be chosen so that it fits on the cylinder. The 24 pairs of points, along this cosine curve, which are to be connected by rulings are determined by dividing the t axis into 48 equidistant parts. Then orthogonals to the t axis are drawn at these points and their intersections with the cosine curve are marked. The planar graph can then be wrapped around and pasted to the the cylinder so that the correct holes can be drilled. Threading is begun along the diagonal in the z = 0 plane and proceeding as described in the previous model. Initial and final ends of the cord may be anchored in the same hole.

Construction of the fixed saddle in a sphere

In spherical coordinates these saddle equations become R = cot A csc A sec 2t. When a fixed radius R = R' is chosen a cosine curve drawn on the surface of the sphere will serve as a locus for points which, when connected, will generate the saddle. These points are the projections on the cosine curve taken from the evernly spaced points along the equator.
In spherical coordinates these saddle equations borrowed from the other coordinate systems become: R=cot A csc A sec 2t. When a fixed radius R = R' is chosen 24 pairs of points on the sphere can be generated by computer using as a function of A. That is
t= f(A) = .5 cos-1[1/R'(cot A csc A)].

In fact this is just one cosine curve drawn on the surface of the sphere whose domain takes up the whole equator and whose amplitude can be made conveniently large. If this is drawn carefully it will serve as a locus for pairs of points to be joined by rulings which form the saddle inside the sphere. These points are the projections on the cosine curve taken from the 24 evenly spaced points along the equator.
Threading is done as in the spherical circular hyperboloidal model with a long needle and is more difficult because of the lack of freedom to reach in with fingers and nudge recalcitrant cord. The precautions already mentioned in the cylindrical saddle model about threads out of line because of unevenly spaced rulings.
Four Saddles

Construction of the double/quadruple fixed saddles

Multiple saddles are constructed by using over and over the rectangular model which has been rotated pi/4 described above by grooving around edges instead of hooking. . When two square pieces of acrylic are joined orthogonally along their diagonal it is easy to make four distinct saddles within the four octants just created.

Construction of eight saddles in a single frame

These eight saddles are accomplished by repeatedly using the rectangular model which has been rotated pi/4 described above by grooving around edges instead of hooking. There is not much choice in threading the previous four-saddle since a corner point must match another corner point of an adjacent side. If, however, the joined planes were of a different shape, made of sharp angled triangles instead of the pi/4 - pi/4 - pi/2 triangle in the previous model, then spectacular results are possible such as the schooner. In fact it is possible to cement several sharp triangles to an irregularly shaped base, so that eight very deep saddles are possible instead of just four shallow saddles. Sixteen are possible as well but would be too busy to observe anything meaningful.
Eight saddles on one frame
The model with eight saddles can be made to resemble a schooner if the two narrow (colored) triangles are cut in the shape of triangular sails and the base is cut in the shape of a ship's bow. Then grooves are saw-cut into the sails and into both ends of the bow so that the sails can be cemented to the bow at right angles. The sides of the bow are not straight but can still be used to house one collection of points for a saddle. The other collection of required generating points sit along the outside of the sail. The side of the horizontal bow is really used twice, once for each sail. The saddle resulting will be very steep and will only accommodate one of the two possible rulings for the saddle. The other side of the bow will also be used in the same way for two other saddles making four saddles above the bow. Under the bow four more saddles are constructed making a total of eight saddles. The shape of the two right triangular sails differ so that four pairs of congruent saddles result. All four saddles on one side are different but are matched by their four partners on the other side.
It is necessary to have the same number of grooves cut into the tops of the sails as in the sides of the bow. The number 40 grooves (the number should be a multiple of four) is a suitable number of cuts depending on how large the pieces are. As for the lower part of the sails just half the number are required and each of these 20 grooves will accommodate two rulings, so the lower saddles are quite crowded.

Constructing the triple point of three intersecting saddles

Triple point of three Saddles
Three saddles (the rotated version) are enclosed in a regular tetrahedron, and opposite edges are connected with line. The three resulting saddles will intersect in pairs leaving three distinctive space curves and these three curves will meet in a point. An equilateral (6"sides) acrylic parallelogram with pi/3 - 2 pi/3 angles has 12 equally spaced holes drilled along the shorter diagonal and has 12 equally spaced grooves saw-cut along each of the four edges (being careful not to get too close to the corners). The acrylic material is heated along the short diagonal and the two equilateral sides are bent to form a pi/3 angle. A 6" rod angle-cut to pi/3 at each end has 12 equally spaced holes drilled along it. The rod is cemented to the two apexes of the two equilateral triangles.
After the bonding is strong enough the six edges are ready for threading (using three colors) - by both methods of sewing through the rod to the drilled joint and hooking along the two other pairs of edges. The sewing is done first, since it is more difficult. Anchoring one end of the white thread start at either end of the joint (so that more than one of the six ends may be anchored together) and connect to either end of the rod and then continue sewing to the end. While it makes no difference which end of the rod is chosen now, it will determine which end of the next two threadings are chosen. Now it merely means that the first saddle rises to the right or to the left, but the next two saddles must follow this same choice. Now the red and then the blue are fastened and hooked into place. It is easy to get confused and thread to the wrong corner. Since hooking is so easy and quick, not much time is lost if a mistake is made. A good hint, though, that the hooking is correct is the fact that all the blue threads will pass between the the third and fourth white threads and all the red thread will pass between the the third and fourth blue threads.
Another version of this model avoids the 6" rod which can be difficult to bond securely. An equilateral trapezoid compose of 4 " sides and top and 8 " base and 12 equally spaced grooves are saw-cut along each 4" side with 24 along the 8" side. Also 12 equally spaced holes are drilled along the two diagonals. It is then heated along the two diagonals and bent at 60o angles to form the required regular tetrahedron. The threading and hooking then accomplish this sturdier version of the model just described.

Construction of the flexing saddle

The saddle as a developable surface can be continuously transformed from a plane two dimensional lamina into a shallow saddle and gradually into a deep saddle. This model demonstrates this fact because it is a flexible saddle. A 6" square piece of acrylic is cut along the diagonal and then 20 evenly spaced cuts are made along the legs of the right triangles. The two pieces are joined along the diagonal using transparent tape or more attractive plastic hinges which are available. A small piece of acrylic is attached along the diagonal of one to prevent the resulting model from closing completely. It should be open about 60o before threading. The threading is done using elastic thread. The resulting figure can be opened flat to display a plane surface or gradually opened to display saddles of varying depth. Struts of different lengths can be used to keep the saddle open at fixed angles.

An assortment of saddles


[1] AH.M. Cundy and A.P. Rollett, "Mathematical Models",
London: Oxford University Press, 1961

[2] J. MacDonnell, S.J., "The mathematics of three straight line constructions," Journal of the International Society LEONARDO, 17 (1984) 104-107

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