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The edge of the light cone is a circle if and only if the flashlight is pointed straight down, so the center of the beam is perpendicular to the surface of the ice (Fig 13-2A) The edge of the region
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of light is an ellipse if and only if the beam axis subtends an angle of more than p /10 with respect to the ice, but the entire light cone still lands on the surface (Fig 13-2B) The edge of the region of light is a parabola if and only if the beam axis subtends an angle of exactly p /10 with respect to the ice, so the top edge of the light cone is parallel to the surface (Fig 13-2C)
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Imagine the scenario described above with the flashlight How can you aim the flashlight so the edge of the light cone forms a half-hyperbola on the ice
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Edge of bright region is a circle
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Edge of bright region is an ellipse
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Flashlight
Edge of bright region is a parabola
Figure 13-2 At A, the edge of the light cone creates a circle on the
surface At B, the edge of the light cone creates an ellipse on the surface At C, the edge of the light cone creates a parabola on the surface The dashed lines show the edges of the light cones The dotted-anddashed lines show the central axes of the light cones
Conic Sections
Solution
The edge of the region of light is a half-hyperbola if and only if one of the following conditions is met: The beam s central axis intersects the lake at an angle of less than p /10 with respect to the surface of the ice (Fig 13-3A) The beam s central axis is aimed horizontally (Fig 13-3B) The beam s central axis is aimed into the sky at an angle of less than p /10 above the horizon (Fig 13-3C)
Flashlight
Edge of bright region is a half-hyperbola
Flashlight
Edge of bright region is a half-hyperbola
Flashlight
Edge of bright region is a half-hyperbola
Figure 13-3
At A, B, and C, the edge of the light cone creates a half-hyperbola on the surface The uppermost part of the light cone is above the horizon in all three cases The dashed lines show the edges of the light cones The dotted-and-dashed lines show the central axes of the light cones
Basic Parameters
Basic Parameters
Figure 13-4 illustrates generic examples of a circle (at A), an ellipse (at B), and a parabola (at C) in the Cartesian xy plane The circle and ellipse are closed curves, while the parabola is an open curve In the circle, r is represents the radius In the ellipse, a and b represent the semi-axes The longer of the two is called the major semi-axis The shorter of the two is called the minor semi-axis In these examples, the circle and the ellipse are centered at the origin, and the parabola s vertex (the extreme point where the curvature is sharpest) is at the origin
Specifications for a parabola Suppose that we re traveling in a geometric plane along a course that has the contour of a parabola At any given time, our location on the curve is defined by the ordered pair (x,y) To follow a parabolic path, we must always remain equidistant from a point called the focus and
r x a
Figure 13-4
Three basic conic sections in the Cartesian xy plane At A, a circle centered at the origin with radius r At B, an ellipse centered at the origin with semi-axes a and b At C, a parabola with the vertex at the origin
Conic Sections
Point (x, y)
Focus
f Vertex u = 2f + y u
Directrix
Figure 13-5
All the points on a parabola are at equal distances u from the focus and the directrix The focus and the directrix are at equal distances f from the vertex of the curve
a line called the directrix as shown in Fig 13-5, where the focus and the directrix both lie in the same plane as the parabola Let s call this distance u In this illustration, the focus of the parabola is at the coordinate origin (0,0) Now imagine a straight line passing through the focus and intersecting the directrix at a right angle This line forms the axis of the parabola In Fig 13-5, the parabola s axis happens to coincide with the coordinate system s y axis Along the axis line, the distance u is called the focal length, which mathematicians and scientists usually call f (Be careful here! Don t confuse this f with the name of a relation or a function) By drawing a line through the focus parallel to the directrix and perpendicular to the axis, we can divide u, measuring our distance from the directrix, into two line segments, one having length 2f and the other having length y Therefore u = 2f + y The focus is at the point (0,0) Therefore, the distance u is the length of the hypotenuse of a right triangle whose base length is x and whose height is y The Pythagorean theorem tells us that x2 + y2 = u2 If we divide the distance from the focus to point (x,y) on the curve by the distance from (x,y) to the directrix, we get a figure called the eccentricity of the curve The eccentricity is
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