What is the guideline? (Geometry) - science - 2023

Content

The directive in geometry it consists of a curve, surface or volume that remains fixed and determines the way in which a geometric object is formed. For example, a line establishes other curves such as conics, and surfaces of revolution, such as the right circular cylinder.

The drive curve can also be a circle. A right circular cylinder can be formed by leaving fixed a directive circumference of radius R.

The circumference, which is on the plane drawn in the figure, determines the shape of the curved surface of the right circular cylinder, which is generated by rotating the line around it, called generating line.

If the directive curve is not a circle, but another curve, other types of cylinder are generated, such as the elliptical cylinder, whose directrix is an ellipse.

A circle can also act as a guideline to generate another curve, such is the case of the epitrochoid, a curve in the plane generated by a point, which in turn is on a smaller circumference that rolls without sliding, around the directrix.

It is easier to visually appreciate it through the following animation:

The drive curve on cylindrical surfaces

Cylindrical surfaces are classified according to their guide curve in cylinders:

-Circulars

-Ellipticals

-Parabic

-Hyperbolic

When a cylindrical surface has a directrix that lies in a plane perpendicular to that of the generatrix line, the equation for that surface is the same as the equation for the directrix.

Cylinders belong to the group of quadric surfaces, whose equation is of the second degree with three variables. The general form is:

Ax² + By² + Cz² + Dxy + Exz + Fyz + Gx + Hy + Iz + K = 0

Where the coefficients A, B, C ... are real numbers.

Cylinders are the most common and useful three-dimensional geometric bodies to be found, especially straight circular cylinders, but the other types of cylinders described below also have applications in engineering and design.

Straight circular cylinder

Its directrix is a circumference C that lies in a plane perpendicular to the cylinder, as shown in figure 1, since the generatrix line, which runs through C to form the lateral surface, is perpendicular to C.

The equation for the circumference C in the xy plane, centered at the origin (0,0) is:

x² + and² = R²

Where R, the radius of the circumference will obviously be the radius of the cylinder. The height h of the cylinder extends along the z axis, perpendicular to the xy plane.

Elliptical cylinder

The directrix is an ellipse in the xy plane centered at the origin (0,0), whose equation is:

The generatrix is a line perpendicular to the xy plane, which moves around the ellipse to give rise to the lateral surface. The ellipse can be at any height z on the xy plane.

For example, the ellipse equation:

4x² + 9y² = 36

It is the directive curve that gives rise to the elliptical cylinder whose equation is 4x² + 9y² = 36, plus z = 0. Adding this last expression, it is clear that it is the surface.

Parabolic cylinder

In this case the drive curve is a parabola, which can be of the form y = x². Thus the cylinder is directed along the z axis and is formed by stacking parabolas with vertex at (0,0) along said axis.

The parabolic cylinder has applications in solar energy, since some collectors have mirrors in this way, by means of which the sunlight is concentrated in the focus. Through this point a straight pipe is passed through which an oil reaches temperatures of up to 400ºC.

Hyperbolic cylinder

In the hyperbolic cylinder, the equation of the directrix is the hyperbola centered at the origin:

The cylinder is formed by stacking hyperbolas along the z axis.

Surface of revolution

The directive curve of a surface of revolution is the same axis of revolution, the line around which the curve that generates the surface rotates.

The rotating curve can have an arbitrary shape, in this way a surface is generated like the one seen in this animation:

If you rotate another line around the guideline, you get the familiar right circular cylinder. In the same way, other surfaces of revolution can be obtained, such as conical, spherical and toroidal surfaces of revolution.

Conical surface

A conical surface is generated by the movement of a generating line that always passes through the fixed plane curve or guideline curve and through the fixed point called vertex, which does not belong to the directive plane.

The vertex or point divides the cone into two parts, called leaves or branches.

Solved exercises

- Exercise 1

Find the lateral area of the right circular cylinder of height 25 cm, whose directrix curve is the circumference of radius 6 cm, centered at the origin.

Solution

The lateral area of the cylinder is the product of the length of the directive circumference and the height. If R is the radius of the circumference and h is the height of the cylinder, the area is given by:

A = 2πR x h = 2πx 6 cm x 25 cm = 942.5 cm²

- Exercise 2

We have the following equation that corresponds to a quadric surface:

x² + and² + 2z² + 2xz - 2yz = 1

Indicate what surface it is and what is the equation of the directive curve.

Solution

Making z = k, where k is constant, we obtain:

x² + and² + 2k² + 2kx - 2ky = 1

We rearrange the terms as follows:

(x² + 2kx) + (and²- 2ky) = 1- 2k²

You have to complete squares in each of the left parentheses, for this, k is added and subtracted², so as not to alter any of the parentheses:

(x² + 2kx + k² - k² ) + (and²- 2ky + k² - k²) = 1- 2k²

(x² + 2kx + k²) - k² + (and²- 2ky + k²) - k² = 1- 2k²

In this way it remains:

(x + k)² + (and - k)² = 1

As it is the equation of a circle with center (-k, k) and radius 1, the surface is a right circular cylinder, also with radius 1, as long as the generating line is perpendicular to said circle.

For example, setting k = 0, the equation reduces to the circle centered at the origin (0,0) whose radius is 1:

x² + and² = 1

References

Gaussians. Represent surfaces in three dimensions. Recovered from: gaussianos.com.
Kindle, J. Theory and problems of analytical geometry. McGraw Hill. Schaum series.
Surfaces as geometric places. Recovered from: algebra.frlp.utn.edu.ar.
Suárez, M. Surfaces. Recovered from: materia.unq.edu.ar.
Quadric surfaces. Recovered from: Sistemas.fciencias.unam.mx.