The best way to begin is probably to construct a hyperboloid and describe its properties.

In the diagram at the right, let AC be the axis of rotation, and let OP be any line not parallel to or intersecting the axis. Let CO be the shortest distance R between the two lines, which will be normal to both. Let α be the angle between OP and OB, parallel to AC. Let (x,y,z) be rectangular coordinates with origin at C, the z-axis along AC and the x-axis along CO. Take any point P on OP, which is a distance [R^{2} + (z tan α)^{2}]^{1/2} from the axis of rotation. P will be a point on the hyperboloid, and will describe a circle about the axis as OP is rotated. The equation of the surface is then x^{2} + y^{2} = R^{2} + z^{2}tan^{2}α. Rotation of a line making an angle -α with OB would give the same surface. The surface is then obviously doubly ruled. The smallest horizontal section is described by point O, a circle of radius R. This is called the *throat* or *gorge* of the hyperboloid.

It is easy to demonstrate generating a hyperboloid by using an electric hand drill to rotate the generating line. I drilled a hole in one end of the plastic T-handle of a small gimlet and attached a length of stainless steel wire with a small loop at its centre with a 3/8 x 4 round head wood screw. The wire, about 150 mm long overall, was positioned at about a 45° angle. The effect is best seen against a dark background with a good light on the bright wire. It is remarkable to see the hyperboloid magically appear when the drill rotates at no great speed.

The equation of the hyperboloid can be expressed in the standard form (x/a)^{2} + (y/b)^{2} - (z/c)^{2} = 1 if a = b = R, c = R/tan α. The exponents 2 make it a *quadric* surface (like the ellipsoid and sphere). If a ≠ b, the cross-sections normal to z are ellipses instead of circles. Although the hyperboloid is a (doubly) ruled surface, it cannot be laid out flat, a property shared with the sphere: it is said to be nondevelopable. It is even more difficult to map a hyperboloid than to map a sphere. When z is very large compared to R, we can neglect R and then the hyperboloid is asymptotic to a cone of angle α. The equation also shows that the sections with a plane passing through the z-axis are hyperbolas with foci in the z = 0 plane, so the hyperboloid can also be formed by rotating a hyperbola about the axis, a property not as interesting as the one we used to find the equation. If the hyperbolas have foci on the z-axis, rotation about the z-axis produces a different surface, the hyperboloid of two sheets, which also has asymptotic cones. The equation of this surface is the same, but with +1 replaced by -1.

The cylinder and the cone are degenerate cases of the hyperboloid. When α = 0 we have a cylinder of radius R, and when R = 0 we have a cone of angle α. The cylinder is generated by a generator parallel to the axis, while a cone is generated by a generator that intersects the axis. The hyperboloid is closely related to the sphere, since it is obtained by changing the sign of z^{2}, which gives a hyperboloid with α = 45° and a gorge radius of r. This reminds one of the relativistic metric, where s^{2} = x^{2} - c^{2}t^{2}. The radius of curvature of any normal section of a sphere at a point on the sphere is the same in any direction, the radius of the sphere. The radius of curvature of a normal section of a hyperboloid changes sign--the centre of curvature of a vertical section outside the surface, while that of a horizontal section is inside. The hyperboloid is negatively curved, the sphere positively. The curvature is zero for sections in which the generators lie.

A. N. Tolstoy (a distant relative of L. N. Tolstoy of *War and Peace*) wrote the science-fiction novel *Engineer Garin's Hyperboloid* in 1926-27. It wasn't a surface, but a ray gun, and Tolstoy really wanted a paraboloid, not a hyperboloid, to focus the rays.

The pioneer of hyperboloidal structures is the remarkable Russian engineer V. Shukhov (1853-1939) who, among other accomplishments, built a hyperboloidal water tower for the 1896 industrial exhibition in Nizhny Novgorod. Hyperboloidal towers can be built from reinforced concrete or as a steel lattice, and is the most economical such structure for a given diameter and height. The roof of the McDonnell Plantarium in St. Louis, the Brasilia Cathedral and the Kobe Port tower are a few recent examples of hyperboloidal structures. The most familiar use, however, is in cooling towers used to cool the water used for the condensers of a steam power plant, whether fuel burning or nuclear. The bottom of the tower is open, while the hot water to be cooled is sprayed on wooden baffles inside the tower. Potentially, the water can be cooled to the wet bulb temperature of the admitted air. Natural convection is established due to the heat added to the tower by the hot water. If the air is already of moderate humidity when admitted, a vapor plume is usually emitted from the top of the tower. The ignorant often call this plume "smoke" but it is nothing but water. Smokestacks are the high, thin columns emitting at most a slight haze. The hyperboloidal cooling towers have nothing to do with combustion or nuclear materials. Two such towers can be seen at the Springfield Nuclear Plant on The Simpsons. The large coal facility at Didcot, UK also has hyperboloidal cooling towers easily visible to the north of the railway west of the station. Hyperboloidal towers of lattice construction have the great advantage that the steel columns are straight.

Suppose two nonparallel shafts that do not intersect are to be connected by gearing. Let C be the shortest distance between the two shafts. This line is perpendicular to both shafts. At some point on the line, let there be a straight line. When this line is rotated around the shafts, two hyperboloids are generated which will be tangent along this line. If the successive positions of this line on the hyperboloids are provided with teeth, the rotation of one hyperboloid will compel the rotation of the other. There will be sliding along the tooth, as we shall see. If the two shafts intersect, the hyperboloids degenerate to cones, and the gears are bevel gears, with no sliding along the teeth. In the general case, the gears are called *skew bevel* or *hypoid* gears. The gears will have line contact, like spur gears, which allows them to handle large powers and run smoothly. Helical gears can serve the same purpose, but they have only point contact.

Let the angle between the two shafts be θ as projected on a plane normal to the shortest distance, and let α and β be the angles between the projections of the shafts and the generator of the hyperboloids. For the moment, let the shafts intersect at O so that C = 0, as shown in the figure. Now we have cones rolling on each other, which would be pitch surfaces for bevel gears. Let the gear whose pitch radius is AQ have N teeth, and that with pitch radius BQ have N' teeth. The velocity ratio is N/N'. The teeth must have the same pitch on each gear, so the radius AQ = PN/2, where P is the diametral pitch, and BQ = PN'/2. The distance OQ is common to both triangles, so PN/2 = OQ sin α and PN'/2 = OQ sin β. Therefore, N/N' = sin α/sin β. If N/N' and θ are given, we can calculate α and β as follows: eliminate β using β = θ - α and solve for α. The result is shown, tan α = sin θ/(N/N' + cos θ). Then β = θ - α.

For general hyperboloidal gears, let us now calculate α and β in the same way in terms of the velocity ratio. Let us focus our attention on the position of closest approach, which will give us the gorge radii R, R' of the hyperboloids, such that R + R' = C. However, the ratio R/R' is not the same as the velocity ratio. In order for the gears to mesh, they must have the same normal pitch. However, the radii are determined by the pitch in the plane of rotation, which differs from the normal pitch by a factor cos α or cos β. This means that R/R' = (N/cos α)/(N'/cos β) = N cos β/N' cos α = tan α/tan β. The velocity ratio is the ratio of the sines, but the gorge radius ratio is the ratio of the tangents. Then, the equations of the two hyperboloids are of the form x^{2} + y^{2} = R^{2}[1 + λz^{2}], where λ is the same for both, and only the gorge radius is different.

It may be helpful to review what is meant by pitch. The linear or circumferential pitch P is the distance between corresponding points on successive teeth, expressed, say, in inches per tooth. Its reciprocal is the number of teeth per inch of circumference on the pitch circle (since we are generally talking about circular gears). Multiplying by π gives the number of teeth per inch of *diameter*. This is the diametral pitch, D = π/P, which is the most commonly used expression of pitch, since it is very convenient in specifying gears. A gear, of course, must have an integral number of teeth. For example, a 4-pitch gear has 4 teeth for each inch of pitch diameter, so that a 32-tooth gear has a pitch diameter of 8 inches. The circumferential pitch is P = 0.7854". When a tooth is inclined at an angle of θ to the axis, the diametral pitch is reduced by a factor cos θ. The normal pitch is important because the teeth may be generated by a cutter that would produce spur gear teeth of this pitch. The pitch diameter is the diameter of the equivalent cylinder for friction gearing.

The figure at the left should give a better idea of how hyperboloids roll on one another. The axis of the hyperboloid of gorge radius R_{1} is a-a, the axis of the hyperboloid of gorge radius R_{2} is b-b, and the common generator of the two is t-t. The hyperboloids are in contact at the pitch point P on the shortest distance between the two axes, and at all points along t-t, such as Q. Note that the lines AQ and BQ in the top view are not shown in true length.
Let the angle between the shafts in a plane passing through the pitch point and normal to the shortest distance be θ. The line t-t also passes through the pitch point and lies in this plane, making angles α and β with the two shafts, so that α + β = θ. These angles are, of course, the angle parameters of the two hyperboloids.

If ω is the angular velocity and R the gorge radius of the hyperboloid whose axis makes an angle α with the line of contact, and ω', R' similar quantities for the other hyperboloid, then considering the velocities at the pitch point, the equality of the components of the velocities normal to the line of contact gives us ωR cos α = ω'R' cos β. From this we easily can find the angular velocity ratio: ω/ω' = (R'/R)(cos β/cos α) = sin β/sin α. Starting from the desired angular velocity ratio, we can find the angles α and β, and from them the ratio R/R'. Since the distance between the shafts is R + R', we can find the individual gorge radii. Usually only a small part of each hyperboloid is used in a pair of gears, and these parts are not usually near the gorges. They look a lot like bevel gears, but the teeth are quite different.

The pitch of hyperboloidal gear teeth increases with distance from the gorge, analogously to the increase of pitch of bevel gear teeth with distance from the point of intersection. This makes the teeth difficult to design and manufacture. For gears of limited thickness, the change in pitch can be neglected and the teeth generated as parallel teeth.

Use Wikipedia Search for "hyperboloid" and visit the interesting links displayed there. There are many illustrations of hyperboloids, including an excellent one made by strings stretched between two circular discs.

P. Schwamb, A. L. Merrill, W. H. James and V. L. Doughtie, *Elements of Mechanism*, 6th ed. (New York: John Wiley & Sons, 1947). Sec. 9-17, pp. 219-224.

E. Buckingham, *Analytical Mechanics of Gears* (New York: Dover, 1988). Chapter 17, pp. 352-382.

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Composed by J. B. Calvert

Created 6 March 2007

Last revised 13 March 2007