Optical Properties and Applications of Conic Curves The first person to investigate conic curves was Menakumos (375-325 BC). About 100 years later, Apollonio studied conic curves in more detail and systematically. Their research on conic curve is very real: the curves obtained by plane truncated cone cutting with different inclination angles are investigated, that is to say, if the angle between the notch and the bottom is less than the angle between the bus and the bottom, the notch is elliptical; If the two angles are equal, the incision is parabolic; If the former is greater than the latter, the incision is hyperbolic. In addition, Apollo further studied the optical properties of elliptic conic curves. He found that if one side of the elliptical focus F is made into a mirror and a light source is placed at F, then all the light reflected by the elliptical mirror will pass through the other focus F. Heat is also reflected like light, so it will be burnt at this time. This is the origin of the name focus. It is said that this discovery was made when he studied the practice of ellipse (that is, the practice introduced at the beginning of the current textbook). Thanks to the German astronomer Kepler (A.D. 157 1- 1630), the conic curve really stepped onto the foreground from the background and entered the real world from the academic ivory tower. Kepler discovered the famous "Kepler's three laws" through long-term astronomical observation and analysis of recorded data, the first of which is: "The planet contains the sun. Draw an ellipse with the sun as the focus. " In this way, the curves studied by Menakumos and Apollo Neo out of their interest in mathematics appeared on the astronomical stage nearly 2000 years later. Later, Halley accurately predicted the closest point between Halley's comet and the earth by using the theory and calculation method of conic curve. 1758 After Halley's death, Halley's Comet met the Earth as scheduled, which caused a sensation in Europe and even the whole world, further promoting people's interest in the study of conic sections. There are three kinds of optical properties of conic curves, namely, those of ellipse, hyperbola and parabola. 1: Optical properties of an ellipse: Light or sound waves emitted from one focus of the ellipse are reflected at the periphery of the ellipse and then pass through the other focus of the ellipse. (as shown in figure 1) The fixed point in the definition of conic curve is called focus because of its optical focusing property. Assuming that the axial cross-sectional profile of a mirror is an ellipse, when you place a ray source at a fixed point F 1, all rays will be concentrated on another fixed point F2 after being reflected by the ellipse; And vice versa (see Figure 7-78). The phenomenon of light concentration is optically called focusing, so naturally these two fixed points F 1, F2 are called focus. This light characteristic of ellipse is often used to design some lighting equipment or heat collection devices. For example, if the heat source is placed at F 1, infrared rays can also be focused at F2. Heat the object at F2 temperature. Figure 1 2: Optical properties of hyperbola: If a light source or sound source is placed at one focus F2 of hyperbola, light or sound wave will be emitted from another focus F 1 after reflection. (as shown in Figure 2) The optical properties of hyperbola also have focusing properties, but it is inverse virtual focusing, that is, after the ray source placed at one focus of hyperbola is reflected by hyperbola, the inverse extension line of its reflection line must pass through another focus hyperbola. This inverse virtual focusing characteristic can also find practical application in the design of astronomical telescopes. Figure 2 3: Optical properties of parabola: After the light or sound wave from the focus of parabola is reflected on the circumference of parabola, the reflected light is parallel to. A parabola (as shown in Figure 3) is considered as an "ellipse" whose focus is at infinity. The optical characteristics of an ellipse in which light from one focus is focused on another focus are shown on a parabola, and the form is very different from that of an ellipse: imagine that the ray source is at that focus at infinity, and the light from infinity will reach another focus at a limited position after parabolic reflection, but the light from infinity can only be a ray beam parallel to the symmetry axis in your view of a limited position (for example, although the sun is far from the earth, it is not at infinity after all. Therefore, the light parallel to the axis of symmetry must be focused on the focus after parabolic reflection (see Figure 7-80). On the contrary, put the radiation source at the focus of the parabola (it is in a limited position). After parabolic reflection, all the light will converge to the focal point at infinity, so the reflected light can only be parallel to the axis of symmetry, that is, the light from the focal point will become a beam parallel to the axis of symmetry after parabolic reflection. The focusing characteristics of parabola make it the best choice for energy gathering device or directional emission device. For example, the longitudinal section lines of searchlights, car headlights and other reflectors are parabolas, and the light source is placed at its focus. After specular reflection, it can become a parallel beam, which increases the irradiation distance, and the irradiation direction can be controlled by rotating the parabolic symmetry axis. Generally, the bowl-shaped receiving or transmitting antenna for satellite communication is also obtained by rotating a parabola around the axis of symmetry. The receiver is placed at its focus, and the parabolic symmetry axis tracks and aims at the satellite, so that the weak electromagnetic wave signal rays emitted by the satellite are concentrated on the receiver to the maximum extent, and the receiving effect is guaranteed. On the contrary, installing a transmitter at the focus and tracking the symmetry axis at the satellite can make the emitted electromagnetic wave signal rays reach the receiver of the satellite in parallel, and also ensure the receiving effect. The most common solar water heaters also use parabolic mirrors to collect sunlight to heat the reservoir at the focus. Fig. 3 The optical properties of these three quadratic curves are widely used in life. The light emitted by a small light bulb (Figure 4) will scatter in all directions, but it can emit strong parallel light when it is installed in a flashlight (Figure 5). Why? The reason is that there is a reflector behind the small bulb in the flashlight, which is parabolic in shape, and its function is to take the light emitted from the focus as parallel light (parallel to the axis of the paraboloid). The searchlight (Figure 6) is also made using this principle. (Figure 4) (Figure 5) (Figure 6) According to the reversibility of light, a solar cooker for heating water and food can be designed (Figures 7 and 8). The solar cooker is equipped with a rotatable parabolic reflector. When its axis is parallel to the sunlight, the sunlight will be concentrated on the focal point after reflection, and the temperature at this point will be very high. Others, such as spotlights, radar antennas, satellite antennas and radio telescopes, are also made by using the optical properties of paraboloids. (Figure 7) (Figure 8) In addition, the spotlight of the film projector has a reflector in the shape of a rotating ellipse. In order to get the strongest light at the film door (where the film passes), the spotlight bulb and the film door should correspond to the two focuses of the ellipse respectively, as shown in the following figure: Because water wave, sound wave and light wave are all a form of wave, they have many similar properties. For example, analyze the reflection of water waves when they encounter elliptic surface, hyperbola surface and paraboloid: in order to let visitors walk in the exhibition hall hear clearly, according to the optical properties of conical curve and the related principles of sound waves, the exhibition hall is often designed as an ellipse. Quadratic curve is widely used in architecture because of its simple equation, changeable linear shape and some good mechanical properties. Especially popular in the current large-scale thin-shell suspended ceiling buildings, many of its longitudinal section lines are conical curves. The optical properties of conic curve are those of ellipse, hyperbola and parabola, which are widely used in life. We should constantly understand and explore its essence and use it to benefit mankind. There is no end to science!