What is a Double Ordinate?
In the study of mathematics, a plane curve that is almost U-shaped and mirror-symmetrical is called a parabola. A parabola fits many superficially dissimilar mathematical expressions, and all of these descriptions can be proved to explain the same curve. The most popular definition of a parabola is that it includes a point of focus and a line called a directrix, however, the focus is not a point that lies on the directrix. Therefore, a parabola is defined as that locus of several points in the plane that is equidistant from the directrix and the focus. There is another way to define the parabola. This definition is derived from the concept of cones. A parabola belongs to the family of conic sections. A parabola is a section of a cone that is created from the intersection of a plane parallel to another plane, tangent to the cone, and a right circular conical section. It becomes clearer when one refers to the figure below.
The first figure describes how a parabola is a member of conic sections. From the second figure, we can learn important terms used for parabola. The axis of symmetry is defined as the perpendicular line, dropped onto the directrix, and is also passing through the focus of the parabola. The axis of symmetry divides the parabola into two identical halves, which are mirror-symmetrical to each other. Vertex is that point on the parabola where the axis of symmetry cuts the parabola. It is also that point at which the parabola is the most sharply curved. The distance measured between the focus and the vertex is termed as the focal length of the parabola. This measurement is along the axis of symmetry. Another important term to understand is the latus rectum. It is the chord line of the parabola, drawn parallel to the directrix and passing through the focus. The parabola can be in any direction, depending on how it has been cut from the cone. The most common directions are either upwards, downwards, left, right, or any arbitrary direction. One important property of a parabola is that it has the capacity to be rescaled and repositioned to match exactly on any other parabola. This simply implies that all parabolas are geometrically similar.
One important parameter of the parabola, which makes it a conic section of choice for many designers and architects is that, if a parabola is manufactured using a reflective material, then the incident light passing along parallel to the parabolic axis of symmetry will all reflect onto the focus of the parabola after reflection. This reflection is irrespective of where the light strikes the surface. Conversely, if a source of light is placed at the focus of the parabola, then all the light will be reflected off the concave surface and reflect, parallel to its axis of symmetry. This creates the concept of a collimated beam. This property of the parabola finds wide-scale adoption in the field of circular parabolic solar thermal energy design projects, where solar rays are reflected onto an absorber pipe for heating an industrial-grade oil for heat transfer. This concept also finds use in designing lighting fixtures to create uniform general lighting conditions. The parabola has numerous significant applications, from a parabolic antenna or parabolic microphone to vehicle headlight reflectors as well as the design of ballistic missiles. It is often used in physics, engineering, and other scientific and design applications.
The simplest expression for a parabola is y= x^2 and it describes the parabola with the vertex being on the origin.
However, a more generalized derivation for the mathematical expression of the parabola is as follows. Let us assume a parabola as represented below.
Let(x,y) be any point on the parabola having its vertex at (0,0), focus(0,p) and directrix y=−p.
The distance from the point (x,y) to point (x, -p) on the directrix can be calculated as d=y+p. The distance between (0, p) to (x,y) is calculated using the right triangle method rule.
On solving for d, by equating both the expressions for d, we can obtain the following equation.
x^2 = 4py
The above is the generalized expression for the definition of a parabola.
Another important term to know about the parabola is the concept of ordinate and double ordinate. An Ordinate of a parabola is defined as a line drawn perpendicular to the axis of symmetry of the parabola, beginning from the axis and terminating on a point on the parabola itself. Ordinates are constantly inside the parabola. The figure below represents many ordinates for parabola represented by y^2=x
On the other hand, a double ordinate is the double of an ordinate. This means that it is a line drawn perpendicular to the axis of symmetry of the parabola> the only difference is that instead of one, this meets the parabola at two points, as shown below. It must be noted that the latus rectum is actually a double ordinate, passing through the focus.
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