1、Section 8.412Equations for SurfaceThe curve can be thought as a trace of a variable point.Similarly,a surface can also be regarded as the generating trace of a variablepoint(or a movable curve)that moves according to some given law.P3coordinate system Oxyz.Then these coordinates must satisfy somegiv
2、en conditions.If P moves in S,from those conditions,we can obtainand equation in the three variables x,y and z(,)x y zIt is clear that if the point P lies on S,then its coordinatesin the rectangularSuppose that the coordinates of P are(,)x y zmust satisfy this equation;conversely,if the coordinates
3、x,y,and z of the point P satisfy this equation,then the point P must lie on S.Thus,the equation(1)is called the equation of the surface S,and(,)0.F x y z (1)the surface S is called the Figure of the equation.Equations for Surface4When graphing a cylinder or other surface by hand or analyzing onegene
4、rated by a computer,it helps to look at the curves formed byintersecting the surface with planes parallelto the coordinate planes.zxyoThese curves are called cross sections or traces.Equations for Surface5Oxyz0PSRSolution The sphere S is the traceExample Find the equation of the sphere S with center
5、 and radius R.0000(,)P xy zP(,)P x y z of a variable pointsuch thatthe distance from P to P0 is R.or0|P PR uuu vThus,P lines on S iff2222000()()().xxyyzzRTherefore the equation of the sphereS in rectangular coordinate is222000()()().xxyyzzREquations for Surface6Cylinders(柱面柱面)Definition (Cylinder(柱面
6、柱面)A cylinder is the surface composed of all the lines that(2)pass through a given plane curve.(1)lie parallel to a given line in space;The curve is called as generating curve orand the linedirectrix(准线)for the cylinder,is called the generating line(母线)of the cylinder.In solid geometry,where the cyl
7、inder meanscircular cylinder,the generating curves arecircles,but now we allow generating curvesof any kind.7The Parabolic Cylinder y=x2200,xxyxSolution Suppose that the point2000(,)Qxxzlies on the parabola Example Find an equation for the cylinder make by the lines parallel to the z-axis that pass
8、through the parabola 2,0.yxzin the xy-plane.Then,for any value of z,will lie on the cylinder2000(,0)P xxthe pointparallel to the z-axis.2yx because it lies on the line0PthroughConversely,lies on the2000(,)Qxxzany pointcylinder.The equation of this cylinder is y=x2 .Finish.8CylindersIt is easy to see
9、 that a cylinder is determined uniquely by its directrix and a fixed line C,but the directrix of a cylinder is not unique.iff that the coordinates(,)M x y zS Obviously,(,)0f x y and z is arbitrary.of M satisfy(,)0.f x y Therefore,the equation of S isSuppose that the generating line of the cylinder S
10、 is parallelto the z-axis and its directrix is a curve in the xOy plane.(,)0,:0.f x yz Then the equation of the cylinder S is(,)0.f x y 9represents a cylinder(,)0g y z Similarly,the equation(,)0h z x or whose generating line is parallel to the x-axis or y-axis and itsdirectrix may be taken as(,)0,0.
11、h z xy or(,)0,0,g y zx In general,if the equation of a surface does not include one of the variables x,y or z,say z,then the surface is a cylinder and its generating line is parallel to the z-axis.Their figure is shown in the following.For example,222222221,1,xyxyabab22xpy are all cylinders.Cylinder
12、s10Cylinders11Cones(锥面锥面)Definition(Cone(锥面锥面)A cone S is the surface formed by a moving line L that passes througha fixed point M0 and moves along a fixed curve is called a cone.The line L is called the generating line of the cone,the curve is called the directrix of the cone,and thepoint M0 is cal
13、led the vertex of the cone.It is easy to see that a cylinder is determineduniquely by its directrix and its vertex M0,but the directrix of a cone is not unique.12and whose(0,0,0),O Suppose S is a cone whose vertex is the origin 0z0(,)0,:,f x yzz directrix iswhere is a constant.be an arbitrary point
14、on(,)P x y zLetthe cone S.Then P must lie on thegenerating line OP,and OP crosses0000(,).P xy zIt is not difficult to see thatthe coordinates of points P and P0 satisfy the following equations:000,xyzxyzCones(锥面锥面)at13and so00,0.z x z yfzz that is 00(,)0,f xy Since 0P we haveThis is the equation of
15、the cone whose vertex is0000,.z xz yxyzzthe origin and whose directrix is the curve0.zz in the plane:(,)0f x yCones(锥面锥面)14Elliptic Cone(0,0,0),OFor example,suppose S is a cone whose vertex is the origin and whosec22221,:,xyabzc directrix is where is a constant.That is,2222111,cxcyzzabThe equation o
16、f the cone S is00,0.z x z yfzz This is called an elliptic cone.222222.xyzabc15Graphing ConesThe elliptic cone222222xyzabcis symmetric with respect to the three coordinate planes.The sections cut by the coordinate planes are0:z the point(0,0,0).0:x the linesczyb 0:y the linesczxa Then the graphic of
17、this elliptic cone can be shown in the following.16Graphing Cones17Surfaces of Revolution（旋转曲面）（旋转曲面）Let C be a curve in a plane and let L be a fixed line in .A surface formed by the curve C revolving around the fixedstraight line L is called a surface of revolution and the curveand the fixed line L
18、 is called the axis of theC is called a generator of the surface of revolution,surface of revolution.18xOzy(,)0f y z 000(0,)My z Md Suppose that(,)0,:0f y zCx is a given curve in the yOz plane,and C revolves around the z-axis.be an arbitrary point(,)M x y zLet obtained from the point000(0,)My zC by
19、the revolution,so that0(,)0.f y z to z-axis0MThe distance0|yfrom22dxyfrom M to z-axis,and hence220.yxy in S,is equal to the distanceSurfaces of Revolution（旋转曲面）（旋转曲面）19220yxy then the coordinates of M must satisfy(,),M x y zS That is to say,if the last equation and this equation is the equation of t
20、he surface of revolution.Since the surface of revolution which is formed by the curve C revolving around the z-axis,we need only replace y in the equation22.xyby(,)0f y z Substituting the expression of 0(,)0f y z we get0yto the equation22(,)0.fxyz22(,)0f yxzis the equation of the surface Similarly,o
21、f revolution that is formed by C revolving around the y axis.Surfaces of Revolution（旋转曲面）（旋转曲面）20Surfaces of RevolutionThe equations of the surfaces formed by the parabola22,0ypzx revolving around the z-axis is222.xypzThis is called a paraboloid of revolution.21The equations of the surfaces formed b
22、y the hyperbola22221,0 xyabz revolving around the x-axis is222221.xyzab This is called a hyperboloid of two sheets of revolution.Surfaces of Revolution22The equations of the surfaces formed by the hyperbola22221,0 xyabz revolving around the y-axis is222221.xzyab This is called a hyperboloid of one s
23、heets of revolution.Surfaces of Revolution23Quadric Surfaces(二次曲面二次曲面)A surface is called a quadric surface,if it can be represented by asecond degree equation in the three variables x,y and z.The mostgeneral form is2220,AxByCzDxyEyzFxzGxHyJzKbut the equation can be where A,B,C,and so on are constan
24、ts,simplified by translation and rotation,as in the two-dimensional case.The basic quadric surfaces are ellipsoids,paraboloids,elliptic cones,and hyperboloids.(We can think of spheres as special ellipsoids.)24Quadric Surfaces（二次曲面）（二次曲面）The equation of an ellipsoid(椭球面椭球面)2222221(,0)xyza b cabcThe e
25、quation of an elliptic paraboloid(椭圆抛物面椭圆抛物面)2222xyzabcThe equation of a hyperbolic paraboloid(双曲抛物面双曲抛物面)2222,0yxzcbacThe equation of a hyperboloid of one sheet(单叶双曲面单叶双曲面)2222221xyzabcThe equation of a hyperboloid of two sheet(双叶双曲面双叶双曲面)2222221xyzabc 25centerprincipal axesprincipal planessemimajo
26、raxissemimeanaxissemiminoraxisGraphing Ellipsoids26Graphing Paraboloids27Graphing a Saddle Surface(马鞍面马鞍面)28Graphing Hyperboloids29Graphing Hyperboloids30Space CurvesGeneral form of equations of space curvesWe know that a straight line in space can be regarded as the line of intersection Likewise,a
27、space curve can be of two planes.regarded as the curve of intersection of two surfaces.Let 2(,)0F x y z and1(,)0F x y z be the equations of two surfaces.Then 12(,)0,(,)0F x y zF x y z represent a space curve C,and is called thegeneral equation of the space curve C.31Parametric equations of space cur
28、vesA space curve can be represented by parametric equations in thesame way as a straight line in space.In general,if the coordinates of a variable point on the curve C can bean interval I,that is(),(),(),xx tyy ttIzz t are called parametric equations of C.(,)x y zrepresented by functions of a variab
29、le t onEquations of Space Curves32Projections of Space Curves on Coordinate PlanesThe cylinder whose directrix is and whose generating line is parallelto the z axis is called a projecting cylinder of on the xOy plane.The curve of intersection of the projecting cylinder and the xOy plane iscalled the
30、 projecting curve(or projection)of on the xOy plane.33Let the general form equation of the space curve be12(,)0,(,)0.F x y zF x y z The key to find the equation of the projecting cylinder of on xOy iseliminating the variable z from the two equations to obtain an equation(,)0.x y Therefore,the equati
31、on of the projecting curve of on the xOy planeis(,)0,0.x yz Projections of Space Curves on Coordinate Planes34Example Find the equations of the projecting curves of the curve22224,:2zxyCxyy to the three coordinate planes,respectively.Solution C is a curve of intersection of the upper hemisphere224zx
32、y222xyydoes not contain theSince variable z,the equation of projection to xOy 222.xyyplane is just and the circular cylinder222.xyyProjections of Space Curves on Coordinate Planes35Solution(continued)Therefore,the equation of the projection curveof C on the xOy plane is42,(02).0,zyyx Eliminating the
33、 variable x from the equationequation of the projecting curve of C onthe yOz plane isof C,42zyso that thewe have222,0.xyyz 22224,:2zxyCxyy Projections of Space Curves on Coordinate Planes36Solution(continued)the xOy plane isSimilarly,eliminating the variable y from the equation C,we have24210,(|1,02
34、).40,xzzxzy 24210,4xzzso that the equation of the projecting curve of C on22224,:2zxyCxyy Finish.Projections of Space Curves on Coordinate PlanesCylindrical Coordinate System37Definition Cylindrical CoordinatesCylindrical coordinates represent a point P as the(,)z in space by ordered triplesright fi
35、gure.Equations Relating Rectangular(x,y,z)and Cylindrical(r,z)Coordinates222(,tan/).xyy xcos,sin,xyzzwhere0,02,.z Cylindrical Coordinate System38Example Find an equation for the saddle surface z=x2-y2 in cylindrical coordinates.222(,tan/).xyy xcos,sin,xyzzSolutioncos,sin,xyzz22222cossincos2.z Finish
36、.Spherical Coordinate System39Equations Relating Spherical Coordinates to Cartesian andCylindrical Coordinates222220,0,02,().rrxyzzcoszr sinr cossin cosxrsinsin sinyrDefinition Spherical CoordinatesSpherical coordinates represent a point P as the(,)r in space by ordered triplesright figure.cossinsin
37、sincosxryrzr备注备注40Equations Relating Spherical Coordinates to Cartesian andCylindrical Coordinates222220,0,02,().rrxyzzcoszr sinr cossin cosxrsinsin sinyrcossinsinsincosxryrzr 课本上的球面坐标三个量的定义与课件上完全一致，不过写成分量形式时，一个点的表示是：也就是说把第二个量和第三个量换了位置。在做作业表示一个点时，请按照书上的顺序写。(,)r Spherical Coordinate System41Example Find the equation in rectangular coordinates forthe surface .cossinsinsincosxryrzr 222220,0,02,().rrxyzz2sinr Solution2222sin4sin44cosrr422244cosrrr 222222()4().xyzxyTherefore,Finish.
