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Conics around us Some properties of conics (curves of the second degree) Johannes Kepler, working with data painstakingly collected by Tycho Brahe without the aid of a telescope, developed three laws which described the motion of the planets across the sky. 1. The Law of Orbits: All planets move in elliptical orbits, with the sun at one focus. 2. The Law of Areas: A line that connects a planet to the sun sweeps out equal areas in equal times. 3. The Law of Periods: The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit. All planets move in elliptical orbits, with the sun at one focus. The eccentricity of an ellipse can be defined as the ratio of the distance between the foci to the major axis of the ellipse. The eccentricity is zero for a circle. Of the planetary orbits, only Pluto has a large eccentricity. Mercury .206 Venus .0068 Earth .0167 Mars .0934 Jupiter .0485 Saturn .0556 Uranus .0472 Neptune .0086 Pluto .25 – – – – v1 = 7.9 km/s = first cosmic velocity: circular orbit around the Earth v2 = 11.2 km/s = second cosmic velocity: leaves Earth (parabolic orbit) v3 = 16.7 km/s = third cosmic velocity: leaves Sun if launched from Earth (parabolic orbit) v4 = 31.8 km/s = falls on the Sun The largest "dish" antenna in the world, the radio telescope at Arecibo Observatory, Puerto Rico. It is 1000 feet (305 meters) in diameter; has a gain of about 10 million. Temperature up to 3,000 C (5,400 F) 1MegaWatt power delivered onto 0.5 m (1.5 feet) diameter spot Temperature up to 3,000 C (5,400 F) 1MegaWatt power delivered onto 0.5 m (1.5 feet) diameter spot Temperature up to 3,500 C (6,300 F) Temperature up to 3,500 C (6,300 F) Smaller is relative – I’ve seen it, it is not “small”! Solar furnace – Google Search 1MW & 3,500 C (6,300 F) Four Solaire a Odeillo, France The 2nd century AD author Lucian wrote that during the Siege of Syracuse (c. 214–212 BC), Archimedes destroyed enemy ships with fire. In 2005 a group of MIT students carried out an experiment with 127 one-foot (30 cm) square mirror tiles, focused on a mock-up wooden ship at a range of around 100 feet (30 m). Flames broke out on a patch of the ship, but only after the sky had been cloudless and the ship had remained stationary for around ten minutes. Definition. Ellipse is the locus of points P(x,y) such that the sum of the distances to two given points F1(f,0) and F2(-f,0) is the same for all points P(x,y). What does the sum of these two distances equal to? What dimension of the ellipse does it define? Y P (x,y) X F2(-f,0) F1 (f,0) The sum of the distances from any point P on the ellipse to its two foci is constant and equal to the major axis, |PF1 | + |PF2 | = 2a . In the method of coordinates, Upon squaring this equation twice we obtain the equation of an ellipse, Y P (x,y) X F2(-f,0) F1 (f,0) Excersize: The sum of the distances from any point inside the ellipse to the foci is less — and from any point outside the ellipse is greater — than the length of the major axis. Proof: Use the triangle inequality. B |AB|+|AC| ≥ |BC| |AB|-|AC| ≤ |BC| A C Fermat principle states that the light always travels along the shortest path. It is clear from the Figure that of all reflection points P on the line l (mirror) the shortest path between points F1 and F2 on the same side of it is such that points F1, P, and the reflection of F2 in l, F’2, lie on a straight line. If a ray of light is reflected in a mirror, then the reflection angle equals the incidence angle. Suppose a line l is tangent to an ellipse at a point P. Then l is the bisector of the exterior angle F1PF2 (and its perpendicular at point P is the bisector of F1PF2). A light ray passing through one focus of an elliptical mirror will pass through another focus upon reflection. Proof. Let X be an arbitrary point of l different from P. Since X is outside the ellipse, we have XF1 +XF2 > PF1 +PF2, i.e., of all the points of l the point P has the smallest sum of the distances to F1 and F2. This means that the angles formed by the lines PF1 and PF2 with l are equal. Definition. Parabola is the locus of points equidistant from a given point F (a,0) and a given line l, (x = -a) which does not contain this point. In method of coordinates, Horizontal parabola Parabolas can open up, down, left, right, or in some other arbitrary direction. Any parabola can be repositioned and rescaled to fit exactly on any other parabola — that is, all parabolas are similar. Not true for ellipses! Vertical parabola Parabola at an angle Y Y y = kx 2 ( x F ,y F ) (0 ,a ) y= ax +b (0,b) X O (0 ,-a ) y = -a O (-B /a,0) X Excersize: For the points inside a parabola the distance to the focus is less than the distance to the directrix, and for the points outside the parabola the opposite is true. Suppose a line l is tangent to a parabola at point P. Let P’ be the projection of P to the directrix. Then l is the bisector of angle FPP’. If a point light source, such as a small light bulb, is placed in the focus of a parabolic mirror, the reflected light forms plane-parallel beam. Light rays incident along the axis of a parabolic mirror upon reflection pass through its focus (the beam is focused to F). Proof. Let point P belong to a parabola and l’ be a bisector of the angle FPP’, where |PP’| is the distance to the directrix l. Then, for any point Q on l’, |FQ| = |QP’| ≥ |QQ’|. Hence, all points Q on l, except for Q = P, are outside the parabola, so l’ is tangent to the parabola at point P. Definition. Hyperbola is the locus of points where the absolute value of the difference of the distances to the two foci is a constant equal to 2a, the distance between its two vertices. In method of coordinates, or, where This definition accounts for many of the hyperbola's applications, such as trilateration; this is the problem of determining position from the difference in arrival times of synchronized signals, as in GPS. Excersize: Let d be the difference of the distances from any point on the hyperbola to the foci F1 and F2 and let Γ be the branch of the hyperbola inside which F1 lies. Then for the points X outside (inside) Γ the quantity |XF2| − |XF1| is less (greater) than d. Suppose a line l is tangent to a hyperbola at point P. Then l is the bisector of the angle F1PF2, where F1 and F2 are the foci of the hyperbola. If a point light source, such as a small light bulb, is placed in the focus F1 of a hyperbolic mirror, the reflected beam looks like it emerges from its image at the other focus F2. The Hubble Telescope has used two of these giant hyperbolic Mirrors Ellipse is the locus of centers of all circles tangent to two given nested circles (F1,R) and (F2,r). Its foci are the centers of these given circles, F1 and F2, and the major axis equals the sum of the radii of the two circles, 2a = R+r (if circles are externally tangential to both given circles, as shown in the figure), or the difference of their radii (if circles contain smaller circle (F2,r)). Hyperbola is the locus of the centers of circles tangent to two given nonnested circles. Its foci are the centers of these given circles, and the vertex distance 2a equals the difference in radii of the two circles. Indeed, consider circles (F1,R) and (F2,r) that are not nested. Then the loci of the centers O of circles externally tangent to these two satisfy |OF1| - |O F2| = R - r. As a special case, one given circle may be a point located at one focus; since a point may be considered as a circle of zero radius, the other given circle—which is centered on the other focus—must have radius 2a. This provides a simple technique for constructing a hyperbola. Parabola is the locus of the centers of circles passing through a given point and tangent to a given line. The point is the focus of the parabola, and the line is the directrix. Indeed, If the radius of one of the given circles is zero, then it shrinks to a point, and if the radius of the other given circle becomes infinitely large, then the “circle” becomes just a straight line. Parabola is the locus of points such that the ratio of the distance to a given point (focus) and a given line (directrix) equals 1 (by its definition). Ellipse can be defined as the locus of points P for which the distance to a given point (focus F2) is a constant fraction of the perpendicular distance to a given line, called the directrix, |PF2|/|PD| = e < 1. Hyperbola can also be defined as the locus of points for which the ratio of the distances to one focus and to a line (called the directrix) is a constant e. However, for a hyperbola it is larger than 1, |PF2|/|PD| = e > 1. This constant is the eccentricity of the hyperbola. By symmetry a hyperbola has two directrices, which are parallel to the conjugate axis and are between it and the tangent to the hyperbola at a vertex. Using the method of coordinates, we can write the equation which describes ellipse, hyperbola and parabola alike, or, This can be transformed to a canonical form A curve of second degree is a set of points whose coordinates in some (and therefore in any) Cartesian coordinate system satisfy a second order equation: In stereometry, an ellipse is defined as a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Hyperbola is the curve of intersection between a right circular conical surface and a plane that cuts through both halves of the cone. Parabola is a unique conic section, created from the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface.