Change of Coordinates in Two Dimensions

Transcription

1 CHAPTER 5 Change of Coordinates in Two Dimensions Suppose that E is an ellipse centered at the origin. If the major and minor aes are horiontal and vertical, as in figure 5., then the equation of the ellipse is (5.) a b where a and b are the lengths of the major and minor radii. Figure 5. Figure 5. u v However, if the aes of E are neither horiontal nor vertical, as in figure 5., then we do not have this simple form of the equation. What we do know, since the curves in figures 5. and 5. are the same but for a rotation, is this: for u and v as shown in figure 5., (5.) u a v b Now, the point here is that u and v can be epressed in terms of the cartesian coordinates, and in turn, and can be determined from u and v. This replacement of one pair of variables which determine a point b another is called a change of coordinates. We now show how to do this in the contet of the ellipse of figure. First, recall Proposition 3.3, and in particular eample 3.6. Introduce a unit vector L in the positive direction of the major ais, so that L is in the direction of the minor ais. Now, an point X can be represented b the vector X ul vl (see figure 5.3). The following proposition tells us how to find u and v. 9

2 Chapter 5 Change of Coordinates in Two Dimensions 0 Figure 5.3 Figure 5.4 u X v L u µ v L L L µ Proposition 5. Let L be a unit vector in the plane. Then an vector X can be written as (5.3) X ul vl where u X L v X L To see this, we calculate the dot products; (5.4) X L ul vl µ L ul L vl L u (5.5) X L ul vl µ L ul L vl L v since L L L L L L 0. In this wa, if we are given a geometric description of the ellipse, we can find its equation in the cartesian coordinates. Eample 5. Let E be the ellipse centered at the origin, with major radius of length 5, major ais the line 3 4 0, and minor radius of length. Find the equation of the ellipse. The point (4,3) is on the given line, so 4I 3J lies in the direction of the major ais. The length of this vector is 5, so we can take the unit vector in the direction of the major ais to be L 4I 3Jµ5. Thus, in the contet of the above discussion, (5.6) L 4 5 I 3 5 J L and a point X I J is on the ellipse if and onl if 3 5 I 4 5 J (5.7) u 5 v where u X L v X L Now, X L 45µ 35µ X L 35µ 45µ, so the equation of the ellipse is (5.8) 45µ 35µ 5 35µ 45µ 4

3 Ü5.0 This simplifies to , or In general, it happens that, in solving a particular problem, the situation can be easil realied in variables adapted to the problem, but the solution requires presentation in terms of an initial cartesian coordinate sstem. For eample, the above ellipse is easil described in terms of the variables u and v adapted to its aes, but to realie the ellipse b an equation, we had to represent u and v in terms of and. We now state the proposition which gives the procedure used in eample 5.. Proposition 5. Given a unit vector L cosθi sinθj we can write an vector X I J as X ul vl, where u X L and v X L ; that is (5.9) u cosθ sinθ v sinθ cosθ We refer to equations (5.9) as a change of coordinate b rotation through the angle θ. We can also reverse the roles of these variables and return to from u v just b rotating back through an angle θ. This gives us the equations (5.0) ucosθ vsinθ usinθ vcosθ Of course, this is just what we get b solving equations (5.9) for and in terms of u and v. Eample 5. The curve is smmetric about the aes. Write the curve in coordinates u v relative to these aes, as in figure 5.4. Since the line makes an angle of 45 Æ with the horiontal, the change of coordinates is accomplished b a rotation through 45 Æ. Since cos 45 Æ µ sin 45 Æ µ Ô, we have the relations (5.) u Ô v Ô u v Ô u v Ô Substituting for and in terms of u and v in, we get u v u v (5.) Ô Ô leading to u v the equation of a hperbola in the u v coordinates. As we have seen in the above eamples, a hperbola or ellipse leads to a quadratic equation, which will have a nonero term if the aes are not horiontal and vertical. This is alwas true; as well as the reverse: an quadratic equation is the equation of a conic curve. We now see how to find the standard description of the conic from the equation (first with an eample). Eample 5.3 Let C be the curve given b the equation (5.3) Find coordinates which put this in standard form. We want to make a substitution of the form (5.0) so that the coefficient of the uv term is 0. Making the substitution gives us (5.4) ucosθ vsinθ µ ucosθ vsinθ µ usinθ vcosθµ

4 Chapter 5 Change of Coordinates in Two Dimensions The coefficient of uv is (5.5) cosθ sinθ cos θ sin θ or sin θ µ cos θ µ This is ero when tan θ µ, or θ π8. For this value of θ we now compute the coefficients A and B of u and v : (5.6) A cos θ cosθ sinθ cos θ µ sin θ µµ Ô (5.7) B sin θ cosθ sinθ cos θ µ sin θ µµ Ô so that the equation for the curve in the u v coordinates is Ô (5.8) µu Ô µv 4 the equation of a hperbola. Following this eample, given an quadratic equation in and : (5.9) A B C D E F 0 we can find a rotation which eliminates the cross term. The resulting equation in the new variables u v is that of a conic section. Of course, there will be eceptional cases; for eample the equation 0 has no solutions, and the equation 0 is a pair of straight lines. But, if (5.9) defines a curve, it must be an ellipse, hperbola or parabola. If we introduce the new variables u and v b a rotation through an angle θ, the equation in the new coordinates is still quadratic in u and v; that is, the equation is of the form (5.0) A ¼ u B ¼ uv C ¼ v D ¼ u E ¼ v F ¼ 0 where the new coefficients are epressed in terms of θ and the old ones. B setting B ¼ 0, we see how to choose θ. So, let s make the substitution (5.0) in the equation (5.0). The part which is purel quadratic is (5.) A ucosθ vsinθµ B ucosθ vsinθ µ usinθ vcosθµ C usinθ vcosθµ The coefficient of uv in this epression is (5.) B ¼ Acosθ sinθ B cos θ sin θ µ C cosθ sinθ Set this to ero and solve for θ. Using double angle formulas, the equation is (5.3) A Cµsin θ µ Bcos θ µ 0 or tan θ B µ A C If A C, the denominator is ero, so we take θ π, or θ π4. Now, the equation (5.0) (with B ¼ 0) is of the form considered in chapter, and can be put in standard form with center at some other point.

5 Ü5.0 3 Eample 5.4 Let C be the curve given b the equation (5.4) Ô Ô Find coordinates which put C in standard form. First, we use (5.3) to find the angle of rotation: (5.5) tan θ µ Ô 3 3 Ô 3 so θ π 3 θ π 6 and (sinθ cosθ Ô 3), the substitution (5.0) is Ô 3u (5.6) v u Ô 3 We do this in two steps. First, the quadratic terms of (5.4) are, in the coordinates u v: (5.7) 4 3u Ô 3uv v µ Ô Ô Ô Ô 3 3u uv 3v µ 3 u 3uv 3v which reduces to v. Now incorporate the linear terms of (5.4) in terms of u v: (5.8) v 6 Ô Ô 3 3u which can be put in the standard form (5.9) v vµ 6 u Ô 3vµ 6 5 u 3 5 Thus the curve is a parabola, with ais at an angle of π3 with the -ais, which opens downward. We summarie this discussion as follows. Proposition 5.3 A curve given b the equation (5.30) A B C D E F 0 is a conic section. Rotate coordinates b the angle θ given b (5.3) tan θ µ B A C ; that is, make the substitution in (5.38): (5.3) ucosθ vsinθ usinθ vcosθ There is no uv term, so after completing the squares, the equation is in standard form. In particular, the aes of the conic are at an angle θ with the coordinate aes. Proposition 5.4 For a curve given b equation (5.38), If B 4AC 0, the curve is an ellipse. If B 4AC 0, the curve is a hperbola.

6 Chapter 5 Change of Coordinates in Two Dimensions 4 If B 4AC 0, the curve is a parabola. To indicate wh this is true, let us consider just the quadratic terms and start with an equation of the form (5.33) A B C Supposing A 0, we complete the square for the first two terms, rewriting (5.49) as (5.34) A B B B A A A C or (5.35) A B A 4AC 4A B If the coefficients of the squared terms are both positive, then there are no solutions for large and, so the curve is an ellipse. On the other hand, if the signs of the coefficients are different, there are alwas solutions for large and, so the curve must be a hperbola. Thus the shape is determined b the sign of 4AC B, and if we carefull follow through the argument, we arrive at proposition 5.4. Eample 5.5 Describe the curve. Since B 4AC 0, this is an ellipse. Since A C, we need to rotate coordinates b π4. We make the subsititution (5.36) u v Ô getting u v Ô (5.37) u uv v u v u uv v which reduces to 3u v. Ü5.. Special Coordinate Sstems Often a problem can be seen as that of understanding the motion of a particle relative to a fied point or a fied ais. In these cases it is useful to epress everthing in coordinates which emphasie positions relative to the fied point or ais. Ü5.. Polar coordinates First, we recall, from Chapter, polar coordinates in the plane. We consider the fied point as the origin of these coordinates, and take the positive -ais as the ero direction. Then an other direction is described b the angle between it and the positive ais, which we denote as θ. The distance of a

7 Ü5. Special Coordinate Sstems 5 point on this line from the origin is denoted r. These equations relate the cartesian coordinates µ with the polar coordinates rθ: Ô (5.38) r cosθ r sinθ r θ arctan Polar coordinates have some ambiguities. Ever value of rθ µ determines a point in the plane. However, if r 0, the point is the origin, and θ doesn t make sense. Secondl, the values rθ µ and rθ πµ, and in fact, rθ nπµ for an n give the same point. The curve θ a is the ra of angle a emanating from the origin, and the curve r a is the circle of radius a centered at the origin (see figure 5.5). In three dimensions, we introduce two new coordinate sstems, the first oriented toward a fied ais, the -ais, and the second oriented toward the origin. Ü5.. Clindrical coordinates Here a point is described b its -coordinate and its polar coordinates in the plane (see figure 5.6). Figure 5.5 Figure 5.6 µ µ θ r The formulas for the change from cartesian coordinates are Ô (5.39) r cosθ r sinθ r θ arctan The equation r at is a circular clinder of radius a centered along 0; θ a describes the half plane with its edge along 0 making an angle a with the -plane, and a is a horiontal plane.

8 Chapter 5 Change of Coordinates in Two Dimensions 6 Figure 5.7 Figure 5.8 r 0 θ Ü5..3 Spherical coordinates These coordinates are oriented toward the origin, so that a point is described b its distance ρ from the origin and the ra from the origin on which it lies. We describe the ra b the angle φ it makes with the -ais and the angle θ it makes with the -plane (see figure 5.9). Figure 5.9 ρ cosφ φ ρ θ ρ sinθ We can read off from figure 5.9 the equations relating spherical coordinates with polar coordinates; (5.40) ρ sinφ cosθ ρ sinφ sinθ ρ cosφ Note that, although θ ranges through a whole circle, φ ranges from 0 to π. The curve ρ a is the sphere of radius a centered at the origin, θ a a half-plane, and φ a the half-cone with ais the -ais, making the angle a with its ais (see figures ). Eample 5.6 Describe the curves C : φ φ 0 ρ, C : θ θ 0 ρ R. Give their equations in clindrical and cartesian coordinates. The curve C is the intersection of the cone φ φ 0 with the sphere of radius R. If we think of this sphere as the globe, C is a circle of latitude. The radius of this circle is Rsinφ 0, and its center is on

9 Ü5. Surfaces; Graphs and Level curves 7 the -ais, at a distance Rcosφ 0 from the origin. In clindrical coordinates, this curve lies on the plane Rcosφ 0, and the clinder r Rsinφ 0 ; these are then the equations of C in clindrical coordinates. In rectangular coordinates the equations are Rsinφ 0 µ Rcosφ 0. The curve C is the intersection of the plane θ θ 0 with the sphere of radius R. If we think of this sphere as the globe, C is a circle of longitude. Its center is the origin and its radius is R. In clindrical coordinates, C is given b the equations r R θ θ 0. In cartesian coordinates, the equations are R tanθ 0. (When θ 0 π the second equation is 0.) Figure 5.0 Figure 5. Figure 5. ρ 0 ρ ρ 0 θ 0 φ 0 φ φ 0 Ü5.. Surfaces; Graphs and Level curves A relation among the variables defines a surface in three dimensions: the set of all which satisf the equation. For eample, we have seen that a linear relation a b c d 0 is the equation of a plane; that is, the set of all points µ which satisf that relation is a plane. Similarl, the sphere of radius R has the equation R. As we have observed alread, it is sometimes difficult to visualie a surface given b an equation in three variables. In this section we shall discuss various was of visualiing surfaces. In the case that the relation can be solved for in terms of and, then the surface is the graph of the function f µ. In this case it is a good idea to tr to sweep out the surfaces b the curves of intersection of the curve with the planes const. These are the level curves of the surface. Then we can imagine the surface as a stack of these level sets. In order to understand how the level sets stack, we ma want to look at representative profiles: these are the curves of intersection of the surface with planes perpendicular to the -plane. Eample 5.7 Draw the level curves of the surface 4, and sketch the surface. We see first of all, since is never negative, that 4. The level surface 4 is just the origin 00µ, but as decreases from 4 we get a famil of circles centered at the origin of ever increasing radius (the radius is Ô 4 ). Our surface then is a stack of circles. To see the shape of the stack, we look at a representative profile: the intersection of the surface with a plane through the -ais. For eample, for 0 we get the parabola 4, and now can safel sketch the graph.

10 Chapter 5 Change of Coordinates in Two Dimensions 8 Figure 5.3 Eample 5.8 Do the same for the surface. Setting equal to the constant 0, we get the parabola 0, so the level curves are the famil of parabolas with ais the -ais, opening upward, with verte at 0 0 µ. We have shown tpical level curves in figure 5.4. Thus the surface is is a stack of parallel parabolas with verte moving linearl up the -ais; that is, the vertices lie on the line 0. To get a further idea of the shape of the surface we look at a profile constant, sa 0. There the surface is given b the parabola, a parabola opening downward. Putting this information together we get figure 5.5. Figure 5.4 Figure 5.5 As is clear from these eamples, sketching surfaces is an imprecise science, and the configuration of level sets gives an idea of the shape, but not ver precise. If we draw a large number of level sets on the plane, we can observe that at points where the level sets are close together, the surface is steep, and where the are far apart, the surface is quite flat. This is illustrated in figures 5.6 and 5.7: figure 5.6 is that of the surface, and figure 5.7 the configuration of its level sets in the -plane.

11 Ü5.3 Clinders and Surfaces of Revolution 9 Figure 5.6 Figure Ü5.3. Clinders and Surfaces of Revolution Starting with a curve C in a plane Π, the surface swept out b the translates of this curve in the direction perpendicular to Π is the clinder over the curve C. This is the case when the relation defining the surface S is independent of one of the variables. For eample, the surface S given b the relation is independent of, so if µ is a point satisfing this relation, then all points µ are on the surface, so S is the clinder over the hperbola C : in the -plane. Eample 5.9 Sketch the surface 9. Since the relation is independent of we just draw the parabola given b this equation on the -plane, and etend it b lines parallel to the -ais. Figure 5.8

12 Chapter 5 Change of Coordinates in Two Dimensions 30 If a surface has the propert that, for a particular line L, the intersection of the surface with a plane perpendicular to L is a circle centered on L, then the surface is a surface of revolution about the ais L. Suppose that S is a surface of revolution about the ais. Then the intersection of S with the half-plane 0 0 completel determines the surface. Let C be that curve of intersection, and suppose it is given b the equation f µ. If µ is a point on C, then ever point on the plane 0 whose distance from the ais is 0 is on S. That is, if Ô 0, then 0 f Ô µ, so 0 µ is on the surface S. Thus the equation of a surface Sof revolution is given b the equation f µ defining its profile just b replacing b Ô : S is the surface f Ô µ. Eample 5.0 Sketch the surface. This is the surface obtained b revolving the curve about the ais. Since that curve consists of the two lines, we get the cone in figure 5.4. Ü5.4. Quadric surfaces These are the surfaces which are given b a quadratic relation among the variables. B completing the square, and - if necessar - rotating the aes, we can reduce ever quadric surface to one of the surfaces in this section. It is a good eercise to trace out these surfaces using the technique of level sets and profiles from the preceding sections. The figures are collected together at the end of the tet. It is essential to become familiar with these surfaces, for the are the fundamental eamples for the rest of the course. Sphere of radius R R (Figure 5.9) The sphere is smmetric about all aes and planes through its center. The intersection of the sphere with an plane is a circle. The intersection of the sphere with a plane through its center is a great circle. a b c Ellipsoid (Figure 5.0) This equation is just that of the sphere, but with the coordinates replaced b a b c. The effect is that the sphere has been dilated in the directions b the factors a b c respectivel, producing the ellipsoid with vertices a00µ 0 b0µ, 00 cµ along the coordinate aes. Hperboloid of one sheet a b (Figure 5.) c First, let s consider the case a b c. Then the equation can be written as (5.4) Õ so the intersection of this surface with the plane 0 is a circle centered at the origin of radius 0. This is a stack of circles of ever increasing radius as we move awa from the -plane. If we set 0, we get the profile : a hperbola, and thus figure 5.. We could also have come to this figure b observing that this is the surface of revolution of this hperbola. Note that if we set in equation (5.4) we get an identit. Thus this line lies on the surface. More importantl, since this is a surface of revolution, if we revolve this line about the -ais, we generate the surface (see figure 5.). The line also lies on the surface and generates it b rotation. Now, for general abc, the level set of the surface at 0 is the ellipse

13 Ü5.4 Quadric surfaces 3 (5.4) a b 0 c Thus, the surface is a stack of similar ellipses of sie increasing as we move awa from the -plane, with, again, a hperbolic profile along each plane through the -ais. We observe that this surface has the same shape as that in the case a b c, ecept for dilations along the coordinate aes. Again, this surface is generated b lines as in the first case, for figure 5. still describes the surface, but for a change in scale in the coordinate directions. In particular, the pair of lines ling on the surface that go through the point a00µ are the lines (5.43) a b c Hperboloid of one sheet a b (Figure 5.3) c The level sets 0 of this surface are the hperbolas (5.44) a b 0 c whose vertices lie on the -ais, but move further and further from the origin as 0 moves awa from the origin. To get a better view of this surface, we look at the level curves 0 : (5.45) b c 0 a There is no curve for a, and for larger values of, we get a famil of ever increasing ellipses. This leads easil to figure 5.8. Elliptical Cone a b 0 (Figure 5.4) c Here the level curves are the ever-widening ellipses (5.46) and the profile in the plane 0 is given b (5.47) b c a b c or b c a pair of lines. The profile in the plane 0 is the pair of lines (5.48) a c This gives us figure 5.4. If a b, the level sets are circles, and the profiles are the same, for the surface is the surface of revolution of the curves given b (5.48). Now, the above list ehausts all possibilities for surfaces given b quadratic equations of the form (5.49) A B C D E F G 0

14 Chapter 5 Change of Coordinates in Two Dimensions 3 with none of A B C equal to ero. Each of the surfaces illustrated in the figures has the origin as center, and has a particular ais (the -ais, ecept in the case of the hperboloid of two sheets, in which case it is the -ais). For the general equation (5.49), the center might be an point, and the ais could be one of the other coordinate aes. For, if we complete the square in each of the variables in equation (5.49) we end up with an equation of the form (5.50) A 0 µ B 0 µ C 0 µ H If H 0 we have two cases: A B C all of the same sign, in which case there is no surface. Otherwise we get a cone. The ais is identified b the coefficient which is of a sign different from the other two. If H 0, we can divide b H, leading to one of the previous cases. The number of negative signs determines the surface; no negatives: ellipsoid; one negative: hperboloid of one sheet; two negative: hperboloid of two sheets. Finall, there are two more surfaces; corresonding to the cases where one or more of the coefficients of the quadratic term is ero Elliptical Paraboloid a b (Figure 5.5) We look at the level curves 0. If 0 0 we get no curve. For 0 0 we get a famil of ellipses of ever increasing sie. The profile on the plane 0 is the parabola b, and on the 0 the parabola a. This gives us enough information for figure 5.5. Hperbolic Paraboloid a (Figure 5.6) b We look at the level curves 0. We get the hperbolas (5.5) a b 0 If 0 0, the ais of the hperbola is the ais, but if 0 0, the ais is the -ais. As 0 increases, the vertices move awa from the -ais. The level curve for 0 0 consists of the two lines a b. To get a better grip on the surface, we look at its profiles. The profile for 0 is the parabola a, which has the -ais as ais, and opens upwards. The profile for 0 is the parabola b, which opens downward. putting this information together gives us figure 5.5. We can get a different (and perhaps more readable) view of this surface b interchanging the and coordinates. Figure 5.6 is that of the hperbolic paraboloid (5.5) a b

15 Ü5.4 Quadric surfaces 33 Figure 5.9: Sphere. Figure 5.0: Ellipsoid Figure 5.: Hperboloid of one sheet Figure 5.: Hperboloid of one sheet. Figure 5.3: Hperboloid of two sheets Figure 5.4: Elliptic cone

16 Chapter 5 Change of Coordinates in Two Dimensions 34 Figure 5.5: Elliptic paraboloid Figure 5.6: Hperboloid paraboloid