Parabolic Mirrors and Telescopes
Date: 06/03/97 at 00:45:09 From: Derek Miyasaki Subject: Calculus, parabolas... Any suggestions (especially parts e and f)? Why Astronomers Use Telescopes with Parabolic Mirrors The star nearest our sun is Alpha Centauri, which is about 4 light years from earth. Alpha Centauri is so far away that when its light reaches earth, it is traveling in essentially parallel rays. To observe distant stars, astronomers use mirrors shaped like paraboloids, which are parabolas rotated about their axes. The reason they use a paraboloidal mirror is that it focuses all the light to a single point, the focus. This point is the image of the star in the paraboloidal mirror. In this project you will demonstrate the focusing property of parabolas. a) Suppose our mirror is shaped like the parabola y = kx^2, where k is any positive constant. Find the coordinates of its focus and the equation of its directrix in terms of k. *Definition* The line tangent to the curve with equation y = f(x) at a point (a,f(a)) on the curve is the line tangent through (a,f(a)) with slope f'(a). We call this line a tangent line. b) Find the equation of the line tangent to the parabola at a point (x1,y1) on the parabola. Then find the y-intercept of the tangent line. c) Consider the triangle formed by a point (x1,y1) on the parabola, the y-intercept of the tangent line at this point, and the focus. Draw a picture of this triangle. Prove this triangle is isosceles. d) Suppose an incoming light ray strikes a curve at a point (x1,y1). If the light ray makes an angle "z" with respect to the tangent line, then it is reflected at an equal angle to the tangent line. This result from physics is known by the phrase "the angle of incidence equals the angle of reflection." Using this fact, argue that incoming light rays parallel to the axis of the parabola are all reflected to the focus, independent of the point of incidence. Thus, a parabolic mirror focuses incoming light rays parallel to the axis to a point. e) The path followed by a ray of light from the star to the focus of the mirror has another special property. Draw a chord of the parabola that is above the focus and parallel to the directrix. Consider a ray of light parallel to the axis as it crosses the chord, hits the parabola and is reflected to the focus. Let d1 be the distance from the chord to the point of incidence (x1,y1) on the parabola and let d2 be the distance from (x,y) to the focus. Show that the sum of the distances d1+d2 is constant, independent of the particular point of incidence. f) Extend your argument from the parabolic cross section to the entire paraboloidal mirror, obtained by rotating this cross section about the y-axis. Thus, prove that all incident rays parallel to the axis of such a mirror focus to a point. Also prove that light traveling along different rays to the focus from a chord perpendicular to the axis will have traveled the same distance, even though these rays were reflected from different points of incidence on the mirror. Thus all of the light waves will arrive in phase, and so interfere constructively to produce a nice bright spot as the image of the star.
Date: 06/03/97 at 12:01:42 From: Doctor Anthony Subject: Re: Calculus, parabolas... Derek, a) The standard equation of a parabola is y^2 = 4ax. With this equation the focus is at (a,0) and the directrix is the line x = -a. We convert the equation y = kx^2 into this form if we write x^2 = 4ay. Then the focus would be at (0,a) and directrix would be y = -a. But x^2 = (1/k)y, giving 4a = 1/k, so a = 1/(4k). Thus the focus is (0,1/(4k)) and the directrix is the line y = -1/(4k). b) y = kx^2, so dy/dx = 2kx and the equation of the tangent at (x1,y1) is: y-y1 = 2kx1(x-x1) This cuts the y-axis where x = 0 y-y1 = -2k(x1)^2 = -2y1 y = -y1 So the tangent cuts the y-axis the same distance below the origin as the point (x1,y1) is above the x-axis. c) The three points are (0,1/(4k)), (x1,y1), and (0,-y1). The distance from the focus to the point (0,-y1) is y1+1/(4k). The distance from the focus to (x1,y1) is: sqrt(x1^2 + (y1-1/(4k))^2) = sqrt((y1/k) + y1^2 - y1/(2k) + 1/(4k)^2) = sqrt(y1^2 + y1/(2k) + 1/(4k)^2) = sqrt(y1+1/(4k))^2 = y1 + 1/4k So the distances from the focus to other points are equal, hence the triangle is isosceles. d) This follows from the fact that base angles of an isosceles triangle are equal. The angle between the incoming ray (parallel to the y-axis) and the tangent is angle of incidence, and this equals the angle between the tangent and the line to the focus from (x1,y1) (the angle of reflection). So every ray parallel to the y-axis will be reflected to pass through the focus. e) If the chord is the line y = c, then the distance from the point where this chord cuts the parabola to the focus is c + 1/(4k) (proved above). The distance d1 from y = c to y = y1 is c - y1, and so the distance covered by the ray which is incident on the parabola at (x1,y1) is c-y1 + y1 + 1/(4k) = c + 1/(4k), which is the same as for the other ray. This would apply to any other ray incident, say at (x2,y2), and shows that all rays crossing the chord have an equal path length to the focus. f) If we cut the parabolic dish by a plane through its axis, the arguments given above apply to the incoming rays lying in the plane. So all rays in all such planes will meet at the focus. -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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