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12 quadric surfaces

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  • 1. Quadric SurfacesA trace of a surface is the intersection of the surfacewith a plane.
  • 2. Quadric SurfacesA trace of a surface is the intersection of the surfacewith a plane. The traces of a given surface S that weare interested in are traces parallel to the coordinateplanes,
  • 3. Quadric SurfacesA trace of a surface is the intersection of the surfacewith a plane. The traces of a given surface S that weare interested in are traces parallel to the coordinateplanes, i.e. the intersection of the surface with theplanes, x = c, y = c, or z = c as shown here.
  • 4. Quadric SurfacesA trace of a surface is the intersection of the surfacewith a plane. The traces of a given surface S that weare interested in are traces parallel to the coordinateplanes, i.e. the intersection of the surface with theplanes, x = c, y = c, or z = c as shown here. x=c S A trace with x = c
  • 5. Quadric SurfacesA trace of a surface is the intersection of the surfacewith a plane. The traces of a given surface S that weare interested in are traces parallel to the coordinateplanes, i.e. the intersection of the surface with theplanes, x = c, y = c, or z = c as shown here. x=c y=c S S A trace with x = c A trace with y = c
  • 6. Quadric SurfacesA trace of a surface is the intersection of the surfacewith a plane. The traces of a given surface S that weare interested in are traces parallel to the coordinateplanes, i.e. the intersection of the surface with theplanes, x = c, y = c, or z = c as shown here. z=c x=c y=c S S S A trace with x = c A trace with y = c A trace with z = c
  • 7. Quadric SurfacesA trace of a surface is the intersection of the surfacewith a plane. The traces of a given surface S that weare interested in are traces parallel to the coordinateplanes, i.e. the intersection of the surface with theplanes, x = c, y = c, or z = c as shown here. z=c x=c y=c S S S A trace with x = c A trace with y = c A trace with z = cBecause hand point–plotting is not useful in R3, onemethod to visualize a surface is the trace-method.
  • 8. Quadric SurfacesParallel traces are similar to thesuccessive ribs of a structure thatshow the contour of its surface.
  • 9. Quadric SurfacesParallel traces are similar to thesuccessive ribs of a structure thatshow the contour of its surface.With the trace–method, usingparallel planes, we construct aseries of traces on the surface Sto reveal the shape of the S.
  • 10. Quadric SurfacesParallel traces are similar to thesuccessive ribs of a structure thatshow the contour of its surface.With the trace–method, usingparallel planes, we construct aseries of traces on the surface Sto reveal the shape of the S.The planes x = c, y = c, or z = cproduce traces that areperpendicular to the x, y, and zaxes respectively.
  • 11. Quadric SurfacesParallel traces are similar to thesuccessive ribs of a structure thatshow the contour of its surface.With the trace–method, usingparallel planes, we construct aseries of traces on the surface Sto reveal the shape of the S. Traces with x = c.The planes x = c, y = c, or z = cproduce traces that areperpendicular to the x, y, and zaxes respectively. For example,the traces of S with the x = c,as c varies, are plane curves thatstacked perpendicularly to the x–axis.
  • 12. Quadric SurfacesParallel traces are similar to thesuccessive ribs of a structure thatshow the contour of its surface.With the trace–method, usingparallel planes, we construct aseries of traces on the surface Sto reveal the shape of the S. Traces with x = c.The planes x = c, y = c, or z = cproduce traces that areperpendicular to the x, y, and z xaxes respectively. For example,the traces of S with the x = c,as c varies, are plane curves thatstacked perpendicularly to the x–axis.
  • 13. Quadric SurfacesParallel traces are similar to thesuccessive ribs of a structure thatshow the contour of its surface.With the trace–method, usingparallel planes, we construct aseries of traces on the surface Sto reveal the shape of the S. Traces with x = c.The planes x = c, y = c, or z = cproduce traces that areperpendicular to the x, y, and z xaxes respectively. For example,the traces of S with the x = c,as c varies, are plane curves thatstacked perpendicularly to the x–axis.
  • 14. Quadric SurfacesParallel traces are similar to thesuccessive ribs of a structure thatshow the contour of its surface.With the trace–method, usingparallel planes, we construct aseries of traces on the surface Sto reveal the shape of the S. Traces with x = c.The planes x = c, y = c, or z = cproduce traces that areperpendicular to the x, y, and z xaxes respectively. For example,the traces of S with the x = c,as c varies, are plane curves thatstacked perpendicularly to the x–axis.
  • 15. Quadric SurfacesParallel traces are similar to thesuccessive ribs of a structure thatshow the contour of its surface.With the trace–method, usingparallel planes, we construct aseries of traces on the surface Sto reveal the shape of the S. Traces with x = c.The planes x = c, y = c, or z = cproduce traces that areperpendicular to the x, y, and z xaxes respectively. For example,the traces of S with the x = c, Sas c varies, are plane curves thatstacked perpendicularly to the x–axis.
  • 16. Quadric SurfacesWe will examine quadratic surfaces, i.e. the graphs of2nd degree equations of the formAx2 + By2 + Cz2 + Dx + Ey + Fz + G = 0. We will studythe ones that can be transformed to one of the followingforms (with x2 as the leading term): x2 y2 z2 x2 y2 + + =1 z = 2 + 2 a2 b2 c 2 a2 b All positive square terms Constant term = 0 x2 y2 z2 + – =1 x2 y2 a 2 b 2 c 2 z = + Assuming no z2One negative square term a2 b2 x2 y2 z2 x2 – – =1 y2 Assuming no z2 a2 b2 c 2 z = – a2 b2Two negative square termsTo sketch then surfaces, if possible, find the square-sums who give traces that are ellipses or circles.
  • 17. Quadric SurfacesTo graph the trace of f(x, y, z) = 0 with y = c,for example, set y = c to obtain f(x, c, z) = 0,an xz-equation.
  • 18. Quadric SurfacesTo graph the trace of f(x, y, z) = 0 with y = c,for example, set y = c to obtain f(x, c, z) = 0,an xz-equation. The trace is the 2D graph off(x, c, z) = 0 in the plane y = c.
  • 19. Quadric SurfacesTo graph the trace of f(x, y, z) = 0 with y = c,for example, set y = c to obtain f(x, c, z) = 0,an xz-equation. The trace is the 2D graph off(x, c, z) = 0 in the plane y = c. The traces of quadricsurfaces, after setting a variable to constants, areconic (2nd degree) sections in the planes.
  • 20. Quadric SurfacesTo graph the trace of f(x, y, z) = 0 with y = c,for example, set y = c to obtain f(x, c, z) = 0,an xz-equation. The trace is the 2D graph off(x, c, z) = 0 in the plane y = c. The traces of quadricsurfaces, after setting a variable to constants, areconic (2nd degree) sections in the planes. Circles: Ellipses: b x2+y2=r2 r x2 + y2 a a2 =1 b2 Hyperbolas: b Parabolas: x2 – y 2 a a2 b2 = 1 y = ax2 Figures are shown centered at (0, 0), in gereral they centered at (h, k).
  • 21. Quadric Surfaces x2 + y2 + z2 = 1The graphs of a2 b2 c2
  • 22. Quadric Surfaces x2 + y2 + z2 = 1The graphs of a2 b2 c2Isolate the sum of two square terms, e.g. x2 and y2: x2 y2 = 1 – z2 a2 + b2 c2
  • 23. Quadric SurfacesThe graphs of a2 x2 + y2 + z2 = 1 b2 c2Isolate the sum of two square terms, e.g. x2 and y2: x2 y2 = 1 – z2 a2 + b2 c2If z > |c|, we getx2 + y2 = negative #,a2 b2so there is no graph in thez = c plane.
  • 24. Quadric SurfacesThe graphs of a2 x2 + y2 + z2 = 1 b2 c2Isolate the sum of two square terms, e.g. x2 and y2: x2 y2 = 1 – z2 a2 + b2 c2If z > |c|, we getx2 + y2 = negative #,a2 b2so there is no graph in thez = c plane.If z = |c|, we get 2x2 + y2 we get the two poles. 2 = a b 0,
  • 25. Quadric SurfacesThe graphs of a2 x2 + y2 + z2 = 1 b2 c2Isolate the sum of two square terms, e.g. x2 and y2: x2 y2 = 1 – z2 z a2 + b2 c2 cIf z > |c|, we get bx + y = negative #, 2 2 a ya2 b2 x z=0so there is no graph inthez = c plane. get x2If z = |c|, we y2 a 2 + b 2 = we get the two poles. y2 0,If z < |c|, we get x 2 + b2 = positive #, 2 awhich are ellipses, with the trace z = 0 being thelargest elliptic trace forming the equator.
  • 26. Quadric Surfaces x2 y2 z2 = 1The graphs of + – 2 a 2 b 2 c
  • 27. Quadric Surfaces x2 y2 z2 = 1The graphs of + – 2 a 2 b 2 cExample A. Graph x2 + y2 – z2 = 1
  • 28. Quadric Surfaces x2 y2 z2 = 1The graphs of + – 2 a 2 b 2 cExample A. Graph x2 + y2 – z2 = 1Solve for the sum–of–two–squares: x2 + y2 = 1 + z2
  • 29. Quadric Surfaces x2 y2 z2 = 1The graphs of + – 2 a 2 b 2 cExample A. Graph x2 + y2 – z2 = 1Solve for the sum–of–two–squares: x2 + y2 = 1 + z2Let z varies, we getcircles in the z-planes.
  • 30. Quadric Surfaces x2 y2 z2 = 1The graphs of + – 2 a 2 b 2 cExample A. Graph x2 + y2 – z2 = 1Solve for the sum–of–two–squares: x2 + y2 = 1 + z2Let z varies, we getcircles in the z-planes.When z = 0, we get thesmallest circle
  • 31. Quadric Surfaces x2 y2 z2 = 1The graphs of + – 2 a 2 b 2 cExample A. Graph x2 + y2 – z2 = 1Solve for the sum–of–two–squares: x2 + y2 = 1 + z2Let z varies, we getcircles in the z-planes.When z = 0, we get thesmallest circle and as|z| gets larger, the circlesget larger.
  • 32. Quadric Surfaces x2 y2 z2 = 1The graphs of + – 2 a 2 b 2 cExample A. Graph x2 + y2 – z2 = 1Solve for the sum–of–two–squares: x2 + y2 = 1 + z2Let z varies, we getcircles in the z-planes.When z = 0, we get thesmallest circle and as|z| gets larger, the circlesget larger. These circlesare stacked vertically.
  • 33. Quadric Surfaces x2 y2 z2 = 1The graphs of + – 2 a 2 b 2 cExample A. Graph x2 + y2 – z2 = 1Solve for the sum–of–two–squares: x2 + y2 = 1 + z2Let z varies, we getcircles in the z-planes.When z = 0, we get thesmallest circle and as|z| gets larger, the circles y –z 2 2 =1get larger. These circlesare stacked vertically.Set x = 0, we get thehyperbolic outline of the"stacked circles“.
  • 34. Quadric Surfaces x2 y2 z2 = 1The graphs of + – 2 a 2 b 2 cExample A. Graph x2 + y2 – z2 = 1Solve for the sum–of–two–squares: x2 + y2 = 1 + z2Let z varies, we getcircles in the z-planes.When z = 0, we get thesmallest circle and as|z| gets larger, the circles y –z =1 2 2get larger. These circlesare stacked vertically.Set x = 0, we get thehyperbolic outline of the"stacked circles“.
  • 35. Quadric Surfaces x2 y2 z2Summary for the graphs of a2 + b2 – c2 = 1
  • 36. Quadric Surfaces x2 y2 z2Summary for the graphs of a2 + b2 – c2 = 1 x2 y2 z2i. Solve for the sum of squares: a2 + b2 = 1 + c2
  • 37. Quadric Surfaces x2 y2 z2Summary for the graphs of a2 + b2 – c2 = 1 x2 y2 z2i. Solve for the sum of squares: a2 + b2 = 1 + c2ii. Let z varies,for a fixed k, we getan ellipses in the z = k planes,
  • 38. Quadric Surfaces x2 y2 z2Summary for the graphs of a2 + b2 – c2 = 1 x2 y2 z2i. Solve for the sum of squares: a2 + b2 = 1 + c2ii. Let z varies,for a fixed k, we getan ellipses in the z = k planes,with z = 0 giving the smallestellipse x2/a2+y2/b2 =1 as the waistand progressively larger ellipsesstacked vertically.
  • 39. Quadric Surfaces x2 y2 z2Summary for the graphs of a2 + b2 – c2 = 1 x2 y2 z2i. Solve for the sum of squares: a2 + b2 = 1 + c2ii. Let z varies, z traces: x 2 2 y k2for a fixed k, we get a 2 + b = 1+ 2 c2an ellipses in the z = k planes, z = kwith z = 0 giving the smallestellipse x2/a2+y2/b2 =1 as the waistand progressively larger ellipsesstacked vertically. x2 y2 – z2 = 1 a2 + b2 c2
  • 40. Quadric Surfaces x2 y2 z2Summary for the graphs of a2 + b2 – c2 = 1 x2 y2 z2i. Solve for the sum of squares: a2 + b2 = 1 + c2ii. Let z varies, z traces: x 2 2 y k2for a fixed k, we get a 2 + b = 1+ 2 c2an ellipses in the z = k planes, z = kwith z = 0 giving the smallestellipse x2/a2+y2/b2 =1 as the waistand progressively larger ellipsesstacked vertically.Setting x = 0 or y =0, we have thehyperbolic outlines of the surface. x2 y2 – z2 = 1 a2 + b2 c2
  • 41. Quadric Surfaces x2 y2 z2Summary for the graphs of a2 + b2 – c2 = 1 x2 y2 z2i. Solve for the sum of squares: a2 + b2 = 1 + c2ii. Let z varies, z traces: x 2 2 y k 2for a fixed k, we get a 2 + b = 1+ c 2 2an ellipses in the z = k planes, z = kwith z = 0 giving the smallestellipse x2/a2+y2/b2 =1 as the waistand progressively larger ellipsesstacked vertically.Setting x = 0 or y =0, we have thehyperbolic outlines of the surface.Exercise: Draw a few traces along the x2 y2 – z2 a2 + b2 c2 = 1x and y axes and find their equations.
  • 42. Quadric Surfaces x2 + y2 = 1 + z2
  • 43. Quadric SurfacesIf we fix the a, b and c in x2 y2 z2 + 2 = 1+ 2 a2 b cbut we replace 1 by k so we have x2 + y2 z2 = k + c2 a2 b2 x2 + y2 = 1 + z2
  • 44. Quadric SurfacesIf we fix the a, b and c in x2 y2 z2 + 2 = 1+ 2 a2 b cbut we replace 1 by k so we have x2 + y2 z2 = k + c2 a2 b2then as k → 0 the waists of thecorresponding surfaces shrink, x2 + y2 = 1 + z2
  • 45. Quadric SurfacesIf we fix the a, b and c in x2 y2 z2 + 2 = 1+ 2 a2 b cbut we replace 1 by k so we have x2 + y2 z2 = k + c2 a2 b2then as k → 0 the waists of thecorresponding surfaces shrink, x 2 + y2 = 1 + z2and if k = 0 we have the equationz2 = x2 y2 + 2c 2 a2 bwhose waist is just one point as inthe case of z2 = x2 + y2 shown here. x2 + y2 = 0 + z2
  • 46. Quadric SurfacesIf we fix the a, b and c in x2 y2 z2 + 2 = 1+ 2 a2 b cbut we replace 1 by k so we have x2 + y2 z2 = k + c2 a2 b2then as k → 0 the waists of thecorresponding surfaces shrink, x 2 + y2 = 1 + z2and if k = 0 we have the equationz2 = x2 y2 + 2c 2 a2 b z = ±ywhose waist is just one point as inthe case of z2 = x2 + y2 shown here.The x = 0 traces give the outlinez = ±y in the yz–plane. x +y 2 2 = 0 + z2
  • 47. Quadric Surfaces z2 x2 y2The graphs of – – 2 = 1 a2 b2 c
  • 48. Quadric Surfaces z2 x2 y2The graphs of a2 – b2 – c2 = 1 x2 y2 = z2 – 1Solve for the sum of squares: 2 + 2 a b c2
  • 49. Quadric Surfaces z2 x2 y2The graphs of a2 – b2 – c2 = 1 x2 y2 = z2 – 1Solve for the sum of squares: 2 + 2 a b c2 x2 + y2 z2it’s equation if we set k = –1 in 2 = k + 2 a2 b c
  • 50. Quadric Surfaces z2 x2 y2The graphs of a2 – b2 – c2 = 1 x2 y2 = z2 – 1Solve for the sum of squares: 2 + 2 a b c2 x2 + y2 z2it’s equation if we set k = –1 in 2 = k + 2 a2 b cIn this case, we havenon–empty level curves if z2 – 1 ≥ 0, i.e. |z| ≥ c. c2
  • 51. Quadric Surfaces z2 x2 y2The graphs of a2 – b2 – c2 = 1 x2 y2 = z2 – 1Solve for the sum of squares: 2 + 2 a b c2 x2 + y2 z2it’s equation if we set k = –1 in 2 = k + 2 a2 b cIn this case, we havenon–empty level curves if z2 – 1 ≥ 0, i.e. |z| ≥ c. c2For |z| > c, we get large andlarger ellipses .
  • 52. Quadric Surfaces z2 x2 y2The graphs of a2 – b2 – c2 = 1 x2 y2 = z2 – 1Solve for the sum of squares: 2 + 2 a b c2 x2 + y2 z2it’s equation if we set k = –1 in 2 = k + 2 a2 b cIn this case, we havenon–empty level curves if z2 – 1 ≥ 0, i.e. |z| ≥ c. c2For |z| > c, we get large andlarger ellipses .The surfacex2 + y2 = z2 – 1is shown here with x +y 2 2 = z2 – 1some x and z traces.
  • 53. Quadric SurfacesSummary of The graphs of z2 = x2 + y2 + k: z2 = x2 + y2 + 1 z2 = x2 + y2 + 0 z2 = x2 + y2 – 1 x2 y2 z2 x2 y2 + + =1 z = 2 + 2 a 2 b2 c 2 a2 b All positive square terms Constant term = 0 x2 y2 z2 Hence we have addressed + – =1 a2 b2 c2 One negative square term cover these cases. x2 y2 z2 The following is a summary – – =1 a2 b2 c 2 and the remaining cases. Two negative square terms
  • 54. Quadric Surfaces x2 y2 z2 x2 y2 z2 + + =1 + – =0 a2 b2 c 2 a2 b 2 c 2All positive square terms Constant term = 0x2 y2 z2 z2 x2 y2 + – =1 – – =1a2 b2 c 2 a2 b 2 c 2One negative square term Two negative square termsThe remaining cases have a missing square term.We will assume that there is no z2. We may in thesecases, express z as a function of x and y.Again, there are two cases; the sum or the differenceof squares. x2 – y2 z = x2 + y2 2 2 z= 2 a b a b2 Assuming no z2 Assuming no z2
  • 55. Quadric SurfacesTraces with z = c are called level curves since zcoordinates give the height of the poitnt.
  • 56. Quadric SurfacesTraces with z = c are called level curves since zcoordinates give the height of the poitnt.Example A. a. Sketch the traces of z – x2 – y2 = 0with z = –1, 0, 1, 4.
  • 57. Quadric SurfacesTraces with z = c are called level curves since zcoordinates give the height of the poitnt.Example A. a. Sketch the traces of z – x2 – y2 = 0with z = –1, 0, 1, 4.Set z = x2 + y2, let z = –1, 0, 1, 4 and we get thefollowing xy-equations:z = –1: –1 = x2 + y2z = 0: 0 = x2 + y2z = 1: 1 = x2 + y2z = 4: 4 = x2 + y2
  • 58. Quadric SurfacesTraces with z = c are called level curves since zcoordinates give the height of the poitnt.Example A. a. Sketch the traces of z – x2 – y2 = 0with z = –1, 0, 1, 4.Set z = x2 + y2, let z = –1, 0, 1, 4 and we get thefollowing xy-equations:z = –1: –1 = x2 + y2 (no solution)z = 0: 0 = x2 + y2z = 1: 1 = x2 + y2z = 4: 4 = x2 + y2
  • 59. Quadric SurfacesTraces with z = c are called level curves since zcoordinates give the height of the poitnt.Example A. a. Sketch the traces of z – x2 – y2 = 0with z = –1, 0, 1, 4.Set z = x2 + y2, let z = –1, 0, 1, 4 and we get thefollowing xy-equations:z = –1: –1 = x2 + y2 (no solution)z = 0: 0 = x2 + y2z = 1: 1 = x2 + y2z = 4: 4 = x2 + y2 z=0 y x
  • 60. Quadric SurfacesTraces with z = c are called level curves since zcoordinates give the height of the poitnt.Example A. a. Sketch the traces of z – x2 – y2 = 0with z = –1, 0, 1, 4.Set z = x2 + y2, let z = –1, 0, 1, 4 and we get thefollowing xy-equations:z = –1: –1 = x2 + y2 (no solution) z=4z = 0: 0 = x2 + y2 z=1z = 1: 1 = x2 + y2z = 4: 4 = x2 + y2 z=0 y x
  • 61. Quadric SurfacesTraces with z = c are called level curves since zcoordinates give the height of the poitnt.Example A. a. Sketch the traces of z – x2 – y2 = 0with z = –1, 0, 1, 4.Set z = x2 + y2, let z = –1, 0, 1, 4 and we get thefollowing xy-equations:z = –1: –1 = x2 + y2 (no solution) z=4z = 0: 0 = x2 + y2 z=1z = 1: 1 = x2 + y2z = 4: 4 = x2 + y2 z=0We note that for z < 0, there is yno trace and as z →∞, we havesuccessive higher and larger xcircles.
  • 62. Quadric SurfacesNext we reconstruct the traces along the x–axis forthe outline of the vertically stacked circles.
  • 63. Quadric SurfacesNext we reconstruct the traces along the x–axis forthe outline of the vertically stacked circles.b. Sketch the traces of z – x2 – y2 = 0 withx = –2, –1, 0, 1, 2.
  • 64. Quadric Surfaces Next we reconstruct the traces along the x–axis for the outline of the vertically stacked circles. b. Sketch the traces of z – x2 – y2 = 0 with x = –2, –1, 0, 1, 2.Set z = x2 + y2, let x = –2, –1, 0, 1, 2 and get thefollowing yz-equations:
  • 65. Quadric Surfaces Next we reconstruct the traces along the x–axis for the outline of the vertically stacked circles. b. Sketch the traces of z – x2 – y2 = 0 with x = –2, –1, 0, 1, 2.Set z = x2 + y2, let x = –2, –1, 0, 1, 2 and get thefollowing yz-equations:x = –2: z = 4 + y2x = –1: z = 1 + y2x = 0: z = y2x = 1: z = 1 + y2x = 2: z = 4 + y2
  • 66. Quadric Surfaces Next we reconstruct the traces along the x–axis for the outline of the vertically stacked circles. b. Sketch the traces of z – x2 – y2 = 0 with x = –2, –1, 0, 1, 2.Set z = x2 + y2, let x = –2, –1, 0, 1, 2 and get thefollowing yz-equations:x = –2: z = 4 + y2x = –1: z = 1 + y2x = 0: z = y2x = 1: z = 1 + y2x = 2: z = 4 + y2 x=0 x
  • 67. Quadric Surfaces Next we reconstruct the traces along the x–axis for the outline of the vertically stacked circles. b. Sketch the traces of z – x2 – y2 = 0 with x = –2, –1, 0, 1, 2.Set z = x2 + y2, let x = –2, –1, 0, 1, 2 and get thefollowing yz-equations:x = –2: z = 4 + y2x = –1: z = 1 + y2x = 0: z = y2x = 1: z = 1 + y2x = 2: z = 4 + y2 x=0 x=1 x
  • 68. Quadric Surfaces Next we reconstruct the traces along the x–axis for the outline of the vertically stacked circles. b. Sketch the traces of z – x2 – y2 = 0 with x = –2, –1, 0, 1, 2.Set z = x2 + y2, let x = –2, –1, 0, 1, 2 and get thefollowing yz-equations:x = –2: z = 4 + y2x = –1: z = 1 + y2x = 0: z = y2x = 1: z = 1 + y2x = 2: z = 4 + y2 x=0 x=1 x x=2
  • 69. Quadric Surfaces Next we reconstruct the traces along the x–axis for the outline of the vertically stacked circles. b. Sketch the traces of z – x2 – y2 = 0 with x = –2, –1, 0, 1, 2.Set z = x2 + y2, let x = –2, –1, 0, 1, 2 and get thefollowing yz-equations:x = –2: z = 4 + y2x = –1: z = 1 + y2x = 0: z = y2x = 1: z = 1 + y2 x = –2x = 2: z = 4 + y2 x = –1 x=0 x=1 x x=2
  • 70. Quadric Surfaces Next we reconstruct the traces along the x–axis for the outline of the vertically stacked circles. b. Sketch the traces of z – x2 – y2 = 0 with x = –2, –1, 0, 1, 2.Set z = x2 + y2, let x = –2, –1, 0, 1, 2 and get thefollowing yz-equations:x = –2: z = 4 + y2x = –1: z = 1 + y2x = 0: z = y2x = 1: z = 1 + y2 x = –2x = 2: z = 4 + y2 x = –1We put parts a and b together x=0to visualize the surface. x=1 x x=2
  • 71. Quadric SurfacesThe graph of of z – x2 – y2 = 0 or z = x2 + y2
  • 72. Quadric SurfacesThe graph of of z – x2 – y2 = 0 or z = x2 + y2
  • 73. Quadric SurfacesThe graph of of z – x2 – y2 = 0 or z = x2 + y2
  • 74. Quadric SurfacesThe graph of of z – x2 – y2 = 0 or z = x2 + y2
  • 75. Quadric SurfacesThe graph of of z – x2 – y2 = 0 or z = x2 + y2
  • 76. Quadric Surfaces x2 y2 z2 x2 y2 z2 + + =1 + – =0 a2 b2 c 2 a2 b c 2 2All positive square terms Constant term = 0 x2 y2 z2 z2 x2 y2 + – =1 – – =1 a2 b2 c 2 a2 b 2 c 2One negative square term Two negative square terms x2 y2 x2 y2z = + z = – a 2 b 2 a 2 b2Assuming no z2 Assuming no z2
  • 77. Quadric Surfaces x2 y2 z2 x2 y2 z2 + + =1 + – =0 a2 b2 c 2 a2 b c 2 2All positive square terms Constant term = 0 x2 y2 z2 z2 x2 y2 + – =1 – – =1 a2 b2 c 2 a2 b 2 c 2One negative square term Two negative square terms x2 y2 x2 y2z = + z = – a 2 b 2 a 2 b2Assuming no z2 Assuming no z2The graph of z = x2 – y2
  • 78. Quadric Surfaces x2 y2 z2 x2 y2 z2 + + =1 + – =0 a2 b2 c 2 a2 b c 2 2 All positive square terms Constant term = 0 x2 y2 z2 z2 x2 y2 + – =1 – – =1 a2 b2 c 2 a2 b 2 c 2 One negative square term Two negative square terms x2 y2 x2 y2z = + z = – a 2 b 2 a 2 b2Assuming no z2 Assuming no z2The graph of z = x2 – y2There is no ellipse to track.
  • 79. Quadric Surfaces x2 y2 z2 x2 y2 z2 + + =1 + – =0 a2 b2 c 2 a2 b c 2 2All positive square terms Constant term = 0 x2 y2 z2 z2 x2 y2 + – =1 – – =1 a2 b2 c 2 a2 b 2 c 2One negative square term Two negative square terms x2 y2 x2 y2z = + z = – a 2 b 2 a 2 b2Assuming no z2 Assuming no z2The graph of z = x2 – y2There is no ellipse to track.We may track parabolas byletting x = c, we get z = c2 – y2which are parabolas withmaximum at (0, c) in the x planes.
  • 80. Quadric Surfaces x2 y2 z2 x2 y2 z2 + + =1 + – =0 a2 b2 c 2 a2 b c 2 2All positive square terms Constant term = 0 x2 y2 z2 z2 x2 y2 + – =1 – – =1 a2 b2 c 2 a2 b 2 c 2One negative square term Two negative square terms x2 y2 x2 y2z = + z = – a 2 b 2 a 2 b2Assuming no z2 Assuming no z2The graph of z = x2 – y2 Traces with x = c.There is no ellipse to track.We may track parabolas by xletting x = c, we get z = c – y 2 2which are parabolas withmaximum at (0, c) in the x planes.
  • 81. Quadric Surfaces x2 y2 z2 x2 y2 z2 + + =1 + – =0 a2 b2 c 2 a2 b c 2 2All positive square terms Constant term = 0 x2 y2 z2 z2 x2 y2 + – =1 – – =1 a2 b2 c 2 a2 b 2 c 2One negative square term Two negative square terms x2 y2 x2 y2z = + z = – a 2 b 2 a 2 b2Assuming no z2 Assuming no z2The graph of z = x2 – y2 Traces with x = c.There is no ellipse to track.We may track parabolas by xletting x = c, we get z = c – y 2 2which are parabolas withmaximum at (0, c) in the x planes.
  • 82. Quadric Surfaces x2 y2 z2 x2 y2 z2 + + =1 + – =0 a2 b2 c 2 a2 b c 2 2All positive square terms Constant term = 0 x2 y2 z2 z2 x2 y2 + – =1 – – =1 a2 b2 c 2 a2 b 2 c 2One negative square term Two negative square terms x2 y2 x2 y2z = + z = – a 2 b 2 a 2 b2Assuming no z2 Assuming no z2The graph of z = x2 – y2 Traces with x = c.There is no ellipse to track.We may track parabolas by xletting x = c, we get z = c – y 2 2which are parabolas withmaximum at (0, c) in the x planes.
  • 83. Quadric SurfacesIf c = 0, the vertex is at z = 0. xTraces with x = c.
  • 84. Quadric SurfacesIf c = 0, the vertex is at z = 0. The larger the |x| is,the higher the vertex z = c thus the parabola is.These parabolas are stacked in the x-direction. xTraces with x = c.
  • 85. Quadric SurfacesIf c = 0, the vertex is at z = 0. The larger the |x| is,the higher the vertex z = c thus the parabola is.These parabolas are stacked in the x-direction. xTraces with x = c.
  • 86. Quadric SurfacesIf c = 0, the vertex is at z = 0. The larger the |x| is,the higher the vertex z = c thus the parabola is.These parabolas are stacked in the x-direction.We obtain the saddle with the origin as a saddle point. xTraces with x = c.
  • 87. Quadric SurfacesIf c = 0, the vertex is at z = 0. The larger the |x| is,the higher the vertex z = c thus the parabola is.These parabolas are stacked in the x-direction.We obtain the saddle with the origin as a saddle point.The saddle point is the pass between two hills. xTraces with x = c. The surface z = x2 – y2 with (0, 0, 0) as the saddle point