Lesson OVERVIEW of lessons on parabolas

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OVERVIEW of lessons on parabolas

This file is your guide on my lessons on Parabolas under the topic Conic sections in this site. These lessons are:
    - Parabola definition, canonical equation, characteristic points and elements
    - Parabola focal property
    - Tangent lines and normal vectors to a parabola
    - Optical property of a parabola.

In the first lesson Parabola definition, canonical equation, characteristic points and elements the basic notions are introduced related to parabolas,
such as canonical equation of a parabola, a focus of a parabola, a focus distance of a parabola, and a parabola directrix.

In this lesson, parabola is defined as a curve in a plane
such that all its points {(

)} satisfy the equation

(1)

with real positive number

in some rectangular
coordinate system OXY in the plane (see Figure 1).

Figure 1. The parabola, its focus and its directix

In the second lesson Parabola focal property the focal property of a parabola is introduced and proved.
The focal property of a parabola states that

A curve on a plane is a parabola if and only if the distance from any point of the curve to the fixed point on the plane (focus)
is equal to the distance to the fixed straight line on the plane which does not pass through the focus.

The focal property of a parabola is the characteristic property.
Based on this property, one can define a parabola as a curve
in a plane such that for any point of the curve the distance
to the fixed point on the plane is equal to the distance to
the fixed straight line on the plane not passing through the
given fixed point.

This definition is equivalent to the algebraic definition
of parabolas of the lesson Parabola definition, canonical equation,
characteristic points and elements.

So, the two definitions describe actually the same class of curves
in a plane.

Figure 2. To the parabola focal property

In the third lesson Tangent lines and normal vectors to a parabola the formula is derived for a tangent line to a parabola in a plane.

Tangent line to a parabola at the point (,) has the equation . (2)

Usually, deriving this kind of formulas requires Calculus. In the lesson I am talking about I didn't use Calculus.
I simply checked that the straight line (2) passes through the given point ( x%5B0%5D

) and have only one common point with the circle.
This is enough for a straight line to be a tangent line to the smooth convex figure as a parabola is.

A corollary is derived from the formula (2). It relates to the normal vector of a tangent line to a parabola.

The "inward" normal vector to the tangent line of a parabola at the point (,) is (,).
The unit "inward" normal vector to the tangent line of a parabola at the point (,) is (,), where = .
.

In the lesson Optical property of a parabola the optical property of a parabola is considered and proved.

        Optical property of a parabola reads as follows (Figure 3):

        If to put the source of light into the focus of a parabola
        and if the internal surface of the parabola reflects the light rays
        as a mirror, then after reflection all the light rays will be directed parallel
        to the parabola axis.

Figure 3 displays the parabola with the focus point F. A source of light is placed
at the focus F. Light rays emitted by the source after reflecting all move forward
parallel to the parabola axis.

Figure 3. Optical property of a parabola

This optical property is equivalent to any of the following geometric facts (Figure 4):

1. For any parabola's point the angle between the tangent line to the parabola
   at this point and the focal radius to the point is congruent to the angle between
   the tangent line and the parabola axis.

2. For any parabola's point the angle between the normal to the parabola at this
   point and the focal radius to the point is congruent to the acute angle
   between the normal and the parabola axis.

3. For any parabola's point the normal to the parabola at this point bisects
   the angle between the focal radius to the point and the straight ray drawn
   from the point parallel to the parabola axis.

Figure 4. To the optical property of
a parabola ( alpha

)

4. For any parabola's point the tangent line to the parabola at this point bisects the angle between the continuation of the focal vector
to the point and the straight line passing through the point parallel to the parabola axis.

Figure 4 displays the parabola with the focus point F=(

), where

is the focal distance of the parabola. The focus is connected by the focal vector with
the point M at the parabola, which is chosen by an arbitrary way. The tangent line and the inward normal vector at the point M are displayed among with the straight
line drawn parallel to the parabola axis. The focal radius is continued. The optical property says that

- the angle alpha

between the focal vector and the tangent line is congruent to the angle beta

between the tangent line and the straight line passing
through the point parallel to the parabola axis: alpha

;

- the angle gamma

between the focal vector and the normal line is congruent to the angle delta

between the normal line and the straight line passing
through the point parallel to the parabola axis: gamma

.

- the angle beta

between the tangent line and the straight line passing through the point parallel to the parabola axis is congruent to the angle phi

between the tangent line and the continuation of the focal vector: beta

. Angles

and

of the Figure 4 are congruent too: alpha

.
Actually, all four angles alpha

and

of the Figure 4 are congruent: alpha

.

Recall the physical law of reflection: the angle of incidence is equal to the angle of reflection measured form the normal. It is consistent with the geometric optical
properties above.

The proof of the optical properties is fully elementary. It uses the explicit formulas for the focal vector r1 (Figure 4), including formulas for
its length from the lesson Parabola focal property, and explicit formula for the parabola's normal vector of the lesson Tangent lines to a parabola.

Note that
      If to reverse the direction of the light beams and to direct parallel rays of light onto a parabolic mirror along its axis,
      then all the light rays will be collected after reflection at the parabola focus.

This statement does not require a separate proof. It is fully supported by the geometric properties above and, therefore, is just proved.

Two bright spots are central in this series of lessons. The first is the elementary proof of equivalency the algebraic and the geometric definitions of a parabola
(the lesson Parabola focal property). The second is the elementary proof of the optical property of a parabola (the lesson Optical property of a parabola).

By combining the parabola focal property and the parabola optical property we can formulate even stronger statement:

    If to put the source of light into the parabola's focus point, and
    if the internal surface of the parabola reflects the light rays as a mirror, and
    if the speed of light is constant for the media filling the parabola interior, then
        - after reflection, all the light rays emitted by the source will be directed parallel to the parabola's axis, and
        - after reflection, the wave front for all the light rays emitted by the source at the same time moment will be a straight line
           parallel to the parabola directrix.

In other words, after reflection the wave front of the reflected light rays is flat.

In order to this statement would be true, I should probably add an assumption that the time delay produced by the reflection act is the same (or negligible)
for all the points at the parabola internal surface.

The reflective property of the parabola has numerous practical applications.
If a light source is placed at the focus of a parabola the result will be a parallel beam of light directed outward along the parabola axis.
This is how projectors, car lights, flashlights, headlights, and searchlights work.

In the present days parabolic mirrors are used at the solar energy plants to concentrate sunlight on thermal collectors containing water like vessels
or tubes to absorb the heat energy, to heat the water and to get the water steam.

Also, see the lessons

Practical problems from the archive related to ellipses and parabolas

        Problem 1. A whispering gallery has an elliptical ceiling. A person standing at one focus of the ellipse can whisper and be heard
                           by another person standing at the other focus, because all the sound waves that reach the ceiling from one focus
                           are reflected to the other focus. A hall 100 feet in length is to be designed as a whispering gallery.
                           If the foci are located 25 feet from the center, how high will the ceiling be at the center?

        Problem 2. The reflector of a flashlight is in the shape of a paraboloid of revolution.
                           Its diameter is 4 inches and its depth is 1 inch.
                           How far from the vertex should the light bulb be placed so that the rays will be reflected parallel to the axis?

        Problem 3. The Lovell Telescope is a radio-telescope located at the Jodrell Bank Observatory in Cheshire, England.
                           The dish of the telescope has the shape of paraboloid with diameter of 250 feet.
                           The distance from the vertex of the dish to its focus is 75 feet.
                               A.   Find an equation of a cross section of the paraboloid.
                                     Assume that the dish has its vertex at (0,0) and vertical axis of symmetry.
                               B.   Find the depth of the dish.

        Problem 4. The antenna of a radio-telescope is a paraboloid measuring 81 feet across with depth of 16 feet.
                           Determine the distance from the vertex to the focus of this antenna.

A circle of maximum radius tangent to and inscribed in parabola

        Problem 1. Find a circle of maximum radius tangent to and inscribed in parabola   y = 4x%5E2

.

For similar lessons on ellipses see OVERVIEW of lessons on ellipses in this site.
For similar lessons on hyperbolas see OVERVIEW of lessons on hyperbolas in this site.

Use this file/link ALGEBRA-II - YOUR ONLINE TEXTBOOK to navigate over all topics and lessons of the online textbook ALGEBRA-II.

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