HYPERBOLIC PARABOLOID: DEFINITION, PROPERTIES AND EXAMPLES - MATHS - 2025

A hyperbolic paraboloid is a surface whose general equation in Cartesian coordinates (x, y, z) satisfies the following equation:

(x / a) ² - (y / b) ² - z = 0.

The name "paraboloid" comes from the fact that the variable z depends on the squares of the variables x and y. While the adjective "hyperbolic" is due to the fact that at fixed values ​​of z we have the equation of a hyperbola. The shape of this surface is similar to that of a horse saddle.

Figure 1. Hyperbolic paraboloid z = x ² - y ². Source: F. Zapata using Wolfram Mathematica.

Description of the hyperbolic paraboloid

To understand the nature of the hyperbolic paraboloid, the following analysis will be made:

1.- We will take the particular case a = 1, b = 1, that is to say that the Cartesian equation of the paraboloid remains as z = x ² - y ².

2.- Planes are considered parallel to the ZX plane, that is, y = ctte.

3.- With y = ctte it remains z = x ² - C, which represent parabolas with the branches upwards and vertex below the XY plane.

Figure 2. Family of curves z = x ² - C. Source: F. Zapata using Geogebra.

4.- With x = ctte it remains z = C - y ², which represent parabolas with the branches down and vertex above the XY plane.

Figure 3. Family of curves z = C - y ². Source: F. Zapata through Geogebra.

5.- With z = ctte it remains C = x ² - y ², which represent hyperbolas in planes parallel to the XY plane. When C = 0 there are two lines (at + 45º and -45º with respect to the X axis) that intersect at the origin on the XY plane.

Figure 4. Family of curves x ² - y ² = C. Source: F. Zapata using Geogebra..

Properties of the hyperbolic paraboloid

1.- Four different points in three-dimensional space define one and only one hyperbolic paraboloid.

2.- The hyperbolic paraboloid is a doubly ruled surface. This means that despite being a curved surface, two different lines pass through each point of a hyperbolic paraboloid that totally belong to the hyperbolic paraboloid. The other surface that is not a plane and is doubly ruled is the hyperboloid of revolution.

It is precisely the second property of the hyperbolic paraboloid that has allowed its wide use in architecture since the surface can be generated from straight beams or strings.

The second property of the hyperbolic paraboloid allows an alternative definition of it: it is the surface that can be generated by a moving straight line parallel to a fixed plane and cuts two fixed lines that serve as a guide. The following figure clarifies this alternate definition of the hyperbolic paraboloid:

Figure 5. The hyperbolic paraboloid is a doubly ruled surface. Source: F. Zapata.

Worked Examples

- Example 1

Show that the equation: z = xy, corresponds to a hyperbolic paraboloid.

Solution

A transformation will be applied to the x and y variables corresponding to a rotation of the Cartesian axes with respect to the Z axis of + 45º. The old x and y coordinates are transformed to the new x 'and y' according to the following relationships:

x = x '- y'

y = x '+ y'

while the z coordinate remains the same, that is, z = z '.

By substituting in the equation z = xy we have:

z '= (x' - y ') (x' + y ')

By applying the notable product of the difference by the sum equal to the difference of squares, we have:

z '= x' ² - y ' ²

which clearly corresponds to the initially given definition of hyperbolic paraboloid.

The interception of the planes parallel to the XY axis with the hyperbolic paraboloid z = xy determines equilateral hyperbolas that have as asymptotes the planes x = 0 and y = 0.

- Example 2

Determine the parameters a and b of the hyperbolic paraboloid that passes through the points A (0, 0, 0); B (1, 1, 5/9); C (-2, 1, 32/9) and D (2, -1, 32/9).

Solution

According to its properties, four points in three-dimensional space determine a single hyperbolic paraboloid. The general equation is:

z = (x / a) ² - (y / b) ²

We substitute the given values:

For point A we have 0 = (0 / a) ² - (0 / b) ², an equation that is satisfied whatever the values ​​of the parameters a and b are.

Substituting point B, we obtain:

5/9 = 1 / a ² - 1 / b ²

While for point C it remains:

32/9 = 4 / a ² - 1 / b ²

Finally, for point D we obtain:

32/9 = 4 / a ² - 1 / b ²

Which is identical to the previous equation. Ultimately, the system of equations must be solved:

5/9 = 1 / a ² - 1 / b ²

32/9 = 4 / a ² - 1 / b ²

Subtracting the second equation from the first gives:

27/9 = 3 / a ² which implies that a ² = 1.

In a similar way, the second equation is subtracted from the quadruple of the first, obtaining:

(32-20) / 9 = 4 / a ² - 4 / a ² -1 / b ² + 4 / b ²

Which is simplified as:

12/9 = 3 / b ² ⇒ b ² = 9/4.

In short, the hyperbolic paraboloid that passes through the given points A, B, C and D has a Cartesian equation given by:

z = x ² - (4/9) y ²

- Example 3

According to the properties of the hyperbolic paraboloid, two lines pass through each point that are completely contained in it. For the case z = x ^ 2 - y ^ 2 find the equation of the two lines that pass through the point P (0, 1, -1) clearly belonging to the hyperbolic paraboloid, such that all the points of these lines also belong to the same.

Solution

Using the remarkable product of the difference of squares the equation for the hyperbolic paraboloid can be written like this:

(x + y) (x - y) = cz (1 / c)

Where c is a nonzero constant.

The equation x + y = cz, and the equation x - y = 1 / c correspond to two planes with normal vectors n = <1,1, -c> and m = <1, -1,0>. The vector product mxn = <- c, -c, -2> gives us the direction of the intersection line of the two planes. Then one of the lines that passes through the point P and belongs to the hyperbolic paraboloid has a parametric equation:

= <0, 1, -1> + t <-c, -c, -2>

To determine c we substitute the point P in the equation x + y = cz, obtaining:

c = -1

In a similar way, but considering the equations (x - y = kz) and (x + y = 1 / k) we have the parametric equation of the line:

= <0, 1, -1> + s with k = 1.

In summary, the two lines:

= <0, 1, -1> + t <1, 1, -2> and = <0, 1, -1> + s <1, -1, 2>

They are completely contained in the hyperbolic paraboloid z = x ² - y ² passing through the point (0, 1, -1).

As a check, suppose t = 1 which gives us the point (1,2, -3) on the first line. Check if it is also on the paraboloid z = x ² - y ²:

-3 = 1 ² - 2 ² = 1 - 4 = -3

Which confirms that it does indeed belong to the surface of the hyperbolic paraboloid.

The hyperbolic paraboloid in architecture

Figure 6. Oceanographic of Valencia (Spain). Source: Wikimedia Commons.

The hyperbolic paraboloid has been used in architecture by the great avant-garde architects, among which the names of the Spanish architect Antoni Gaudí (1852-1926) and very particularly the also Spanish Félix Candela (1910-1997) stand out.

Below are some works based on the hyperbolic paraboloid:

-Chapel of the city of Cuernavaca (Mexico) work of the architect Félix Candela.

-The Oceanographic of Valencia (Spain), also by Félix Candela.

References

1. Encyclopedia of mathematics. Ruled Surface. Recovered from: encyclopediaofmath.org
2. Llera Rubén. Hyperbolic paraboloid. Recovered from: rubenllera.wordpress.com
3. Weisstein, Eric W. "Hyperbolic Paraboloid." From MathWorld – A Wolfram Web Resource. Recovered from: mathworld.wolfram.com
4. Wikipedia. Paraboloid. Recovered from: en.wikipedia.com
5. Wikipedia. Paraboloid. Recovered from: es.wikipedia.com
6. Wikipedia. Ruled surface. Recovered from: en.wikipedia.com