Reflection properties of conic sections pdf

The later group of conic sections is defined by their two specific conjugates, or geometric foci f 1, f 2, with the near focus for parabola coinciding with the gaussian infinity focus common to all conics with a given vertex radius r. It also offers a procedure for determining the caustic of the wave front. Consider a ray of light that is emanated from a point source at position f at time. Highintensity sound waves generated at the other focus are re. All rays coming from one specific focus pass through the other after reflection from a conic. Reflective properties the reflective property of a parabola states that when a wave of something light,radio,tv is shot at the parabola from its focus, the parabola will reflect that beam straight outwards in the direction of the axis. A conic section is the locus of points latexplatex whose distance to the focus is a constant multiple of the distance from latexplatex to the directrix of the conic.

Foci of a hyperbola from equation video khan academy. Applying problembased learning model and creative design. If the source of a signal is placed at one of the two focal points of an ellipse, the signal will be reflected to the other focal point. Reflective properties of conic sections wolfram demonstrations. A conic section approach to the relativistic reflection law.

Any ray sent from one of the focus of an ellipse gets reflected by the curve into a ray towards the other. Optical system design s15 reflector telescopes joseph a. Collectively they are referred to as conic sections. The resulting surface has the property that all incoming rays parallel to the axis are reflected through the focus of the parabola. Reflection we believed this conic sections unit was one of the most intriguing of the year, because of how abstract yet simple the idea was of a conic section. Conic sections each conic section or simply conic can be described as the intersection of a plane and a doublenapped cone. In the following activity, we are interested in their reflective properties, i. Applying problembased learning model and creative design to. The rotating line m is called a generator of the cone. In spite of a collection of over 600 lesson plans, i was surprised to find. It is evident that the properties, by which conic sections are defined, are really their own, and that they cannot be shared with any other curve. Lesson plan 3 conic sections university of georgia. Wavefront generated by reflection of a plane wave from a conic section conference paper pdf available in proceedings of spie the international society for optical engineering august 2008.

Dapoxetine viagra combo, generic viagra open discussion. Shaw montana state university conics a conic is a surface of revolution formed by spinning a conic section around the axis. Also, let and be the two points of intersection of the perpendiculars drawn from the point to two parallel lines called the directrices, with corresponding to the directrix closer to and corresponding to the directrix closer to. If a source of light or sound is placed at one focus of a surface. Do not use the intersection points of the circles to define the conic. These represent 2d curves formed by looking at the intersection of a transparent cone by a plane tilted at various angles with respect to the cone axis. Conic sections, that is, ellipses, hyperbolas, and parabolas, all have special reflective properties.

There are four types of curves that result from these intersections that are of particular interest. When the plane passes through the vertex, the resulting figure is a degenerate conic, as shown in. Fermat principle requires the trajectory to be such that it minimizes the travel time over all paths. Radio waves kroc radio station is 4 miles west and 6 miles north of the center of bigcity. The effective surface of reflection esr is defined as the loci of intersection of all beams, emanating from the source at a given time, with. A level cut gives a circle, and a moderate angle produces an ellipse. Develop and understanding of the reflective properties of the parabola, the ellipse, and the hyperbola 2.

Linkedin this website contains a link to the linkedin service from linkedin corporation, 2029 stierlin court, mountain view, ca 94043, usa. For hyperbolas and ellipses centered at the origin, using. The method uses the optical properties of conics to obtain both reflecting and focusing profiles as the envelope of a specific family of conic sections. Pdf wavefront generated by reflection of a plane wave from. The reflective property of a parabola states that when a wave of something light,radio,tv is shot at the parabola from its focus, the parabola will reflect. All mirrors in the shape of a nondegenerate conic section reflect light coming from or going toward one focus toward or away from the other focus. Figure out the use of the reflective properties in the context of lighting devices, receptors, and other technologies. A steep cut gives the two pieces of a hyperbola figure 3. Conicsections that ratio above is called the eccentricity, so we can say that any conic section is. The lack of proofs makes practical conic sections mostly a catalogue of interesting facts. Students find difficulty in working with the equations, the complexity of the numerous new terms, and realizing the graphs of equations and the reflection properties of standard conic sections. Here are those properties, as i understand them for the sake of a common description, ill suppose a beam is bouncing off each conic section. Select three of the points defining the conic near the yaxis on one locus, and the other two points near the yaxis of the other locus. Kahan page 34 only one of which can be satisfied in nondegenerate cases to get one equation that, after.

Te a c h e r s may wish to preselect the magazines or web sites that students will visit to find exa m p l e s of conic sections. The basic reflective principle angle of incidence angle of reflection for a flat surface and how it extends to a curved surface having a tangent line. The effective surface of reflection esr is defined as the loci of intersection of all beams, emanating from the source at a given time, with the moving mirror. To prove the reflection property of the parabola, we need to show that. One of the things i find most interesting about conic sections is their reflective properties. All points whose distance to the focus is equal to the eccentricity times the distance to the directrix for eccentricity 1 a hyperbola.

This property is such that the angle between a line drawn from one focus to a point on the. Thus, conic sections are the curves obtained by intersecting a right circular cone by a plane. The basic defining distance relationships for each of the three conics. These curves were known to the ancient greeks, who. The special reflective properties of each of the three conic sections. If the source is placed at one of the two focal points of a hyperbola, the signal will be reflected directly away from the other focal point. The teacher should actually use the playdoh f i r st before trying the lesson with the students. A source of light positioned in the yellow point emanates radial rays. This is the factor that determines what shape a conic section. It begins with their reflection properties and considers a few ways these properties are used today. The locus of the point is the conic section required, where satisifes. Jul 12, 2017 in this video i go over an example on ellipses that looks at the amazing reflective property of ellipses. The back side reflection property for a convex hyperbolic reflector says that light directed at the near focus is reflected through the far focus.

This principle is used in lithotripsy,a treatment for kidney stones. We found it interesting how all these different shapes formed from a intersecting plane had mathematical formulas. However beyond these we encounter other properties, some of which are not easily seen to be. If we take the intersection of a plane with a cone, the section so obtained is called a. For an introduction to the huge variety of applications of conic sections and their reflection properties, see the series of articles by whitt 17, 18, 19.

Algebra introduction to conic sections the intersection of a cone and a plane is called a conic section. Mar 04, 20 subject conic sections applications conics conic sections are the curves which result from the intersection of a plane with a cone. Class xi chapter 11 conic sections maths page 1 of 49 website. We obtain dif ferent kinds of conic sections depending on the position of the intersecting plane with respect to the cone and the angle made by it with the vertical axis of the cone. Pdf wavefront generated by reflection of a plane wave. Students should notice that the differing conics all have differing angles to the slant of the cones sides. We also find the equation of a tangent to a parabola using techniques from calculus, and we use this to prove the reflective property of the parabola. Conics are all around us focal properties of conics. We consider the reflection of light, from a stationary source, off of a uniformly moving flat mirror, and derive the relativistic reflection law using wellknown properties of conic sections. But, ancient as they are, these curves have exceedingly modern applications for example, the television dish aerial that brings sports and. These properties that the conic sections share are often presented as the following definition, which will be developed further in the following section. Conic sections, reflection and applications ltcc online. Let be a point on the conic section, and be two points called foci. Because conic sections arise naturally, have many useful properties, and are symmetric, they are important curves in mathematics, science, engineering, architecture, and design.

As is known, if a ray of light is reflected in a mirror, then the reflection angle. Subject conic sections applications conics conic sections are the curves which result from the intersection of a plane with a cone. Wavefront generated by reflection of a plane wave from a conic section conference paper pdf available in proceedings of spie the international society for optical engineering. Kahan page 34 only one of which can be satisfied in.

Find the equation of the circle with centre 0, 2 and radius 2. They remain important today, partly for their many and diverse applications. A lot of properties of conic sections especially, parabolas have been proved to be characteristic ones 67 8 9101112. Conic sections and certain hypersurfaces of revolution derived from them are characterized as the only nondegenerate smooth euclidean hypersurfaces having reflection properties. Conic sections in the complex zplane september 1, 2006 3. This applet demonstrates the special reflective properties of the conic sections. Conic sections conic sections are the curves that result from the intersection of a plane and a double cone. There are a few sections that address technological applications of conic sections, but the practical in the title seems mainly meant to distinguish the books approach from tedious proofs that abound in most books on the subject.

The vertex separates the cone into two parts called nappes. Like parabolas, ellipses have an interesting reflection property that has practical conse quences. Information recall access the knowledge youve gained regarding different types of conic sections additional learning be sure to check out the. Students will be able to state the reflection property for a flat.

Euclidean hypersurfaces with reflection properties. To prove the reflection property of the parabola, we need to. These curves were studied and revered by the ancient greeks, and were written about extensively by both euclid and appolonius. It begins with their reflection properties and considers a few ways these. In the case of the parabola, the second focus needs to be thought of as infinitely far away, so that the light rays going. This information is used to provide the services and any support thereto as set out and referenced in the documentation. The effective surface of reflection esr is defined as the loci of intersection of all beams, emanating from the source at a given time, with the moving. This assignment will assess your understanding of the reflective properties of the conic sections and how the conic sections can be used. Conic sections the parabola and ellipse and hyperbola have absolutely remarkable properties. See more ideas about conic section, precalculus and teaching math. Use the conic tool to draw a conic on top of the two loci that were just created. Now for a hyperbola, you kind of see that theres a very close relation between the ellipse and the hyperbola, but it is kind of a fun thing to ponder about. The conic sections, or conics, are curves obtained by making sections, or cuts, at particular angles through a cone. Conic sections 239 in the following sections, we shall obtain the equations of each of these conic sections in standard form by defining them based on geometric properties.