VECTORS AND THE GEOMETRY OF SPACE

1. 12 VECTORS AND THE GEOMETRY OF SPACE

2. VECTORS AND THE GEOMETRY OF SPACE • We have already looked at two special types of surfaces: • Planes (Section 12.5) • Spheres (Section 12.1)

3. VECTORS AND THE GEOMETRY OF SPACE • Here, we investigate two other types of surfaces: • Cylinders • Quadric surfaces

4. VECTORS AND THE GEOMETRY OF SPACE 12.6 Cylinders and Quadric Surfaces • In this section, we will learn about: • Cylinders and various types of quadric surfaces.

5. TRACES • To sketch the graph of a surface, it is useful to determine the curves of intersection of the surface with planes parallel to the coordinate planes. • These curves are called traces(or cross-sections) of the surface.

6. CYLINDER • A cylinderis a surface that consists of all lines (called rulings) that are parallel to a given line and pass through a given plane curve.

7. CYLINDERS Example 1 • Sketch the graph of the surface z = x2 • Notice that the equation of the graph, z = x2, doesn’t involve y. • This means that any vertical plane with equation y = k (parallel to the xz-plane) intersects the graph in a curve with equation z = x2. • So, these vertical traces are parabolas.

8. CYLINDERS Example 1 • The figure shows how the graph is formed by taking the parabola z = x2 in the xz-plane and moving it in the direction of the y-axis.

9. PARABOLIC CYLINDER Example 1 • The graph is a surface, called a parabolic cylinder, made up of infinitely many shifted copies of the same parabola. • Here, the rulings of the cylinder are parallel to the y-axis.

10. CYLINDERS • In Example 1, we noticed the variable y is missing from the equation of the cylinder. • This is typical of a surface whose rulings are parallel to one of the coordinate axes. • If one of the variables x, y, or z is missing from the equation of a surface, then the surface is a cylinder.

11. CYLINDERS Example 2 • Identify and sketch the surfaces. • x2 + y2 = 1 • y2 + z2 = 1

12. CYLINDERS Example 2 a • Here, z is missing and the equations x2 + y2 = 1, z = k represent a circle with radius 1 in the plane z = k.

13. CYLINDERS Example 2 a • Thus, the surface x2 + y2 = 1 is a circular cylinder whose axis is the z-axis. • Here, the rulings are vertical lines.

14. CYLINDERS Example 2 b • In this case, x is missing and the surface is a circular cylinder whose axis is the x-axis. • It is obtained by taking the circle y2 + z2 = 1, x = 0 in the yz-plane, and moving it parallel to the x-axis.

15. CYLINDERS Note • When you are dealing with surfaces, it is important to recognize that an equation like x2 +y2 = 1 represents a cylinder and not a circle. • The trace of the cylinder x2 + y2 = 1 in the xy-plane is the circle with equations x2 + y2 = 1, z = 0

16. QUADRIC SURFACE • A quadric surfaceis the graph of a second-degree equation in three variables x, y, and z.

17. QUADRIC SURFACES • The most general such equation is: • Ax2 + By2 + Cz2 + Dxy +Eyz +Fxz + Gx +Hy +Iz +J = 0 • A, B, C, …, J are constants.

18. QUADRIC SURFACES • However, by translation and rotation, it can be brought into one of the two standard forms: • Ax2 + By2 + Cz2 + J = 0 • Ax2 + By2 + Iz = 0

19. QUADRIC SURFACES • Quadric surfaces are the counterparts in three dimensions of the conic sections in the plane. • See Section 10.5 for a review of conic sections.

20. QUADRIC SURFACES Example 3 • Use traces to sketch the quadric surface with equation

21. QUADRIC SURFACES Example 3 • By substituting z = 0, we find that the trace in the xy-plane is: • x2 + y2/9 = 1 • We recognize this as an equation of an ellipse.

22. QUADRIC SURFACES Example 3 • In general, the horizontal trace in the plane z = k is: • This is an ellipse—provided that k2 < 4, that is, –2 < k < 2.

23. QUADRIC SURFACES Example 3 • Similarly, the vertical traces are also ellipses:

24. QUADRIC SURFACES Example 3 • The figure shows how drawing some traces indicates the shape of the surface.

25. ELLIPSOID Example 3 • It’s called an ellipsoid because all of its traces are ellipses.

26. QUADRIC SURFACES Example 3 • Notice that it is symmetric with respect to each coordinate plane. • This is a reflection of the fact that its equation involves only even powers of x, y, and z.

27. QUADRIC SURFACES Example 4 • Use traces to sketch the surface z = 4x2 + y2

28. QUADRIC SURFACES Example 4 • If we put x = 0, we get z = y2 • So, the yz-plane intersects the surface in a parabola.

29. QUADRIC SURFACES Example 4 • If we put x = k (a constant), we get z = y2 + 4k2 • This means that, if we slice the graph with any plane parallel to the yz-plane, we obtain a parabola that opens upward.

30. QUADRIC SURFACES Example 4 • Similarly, if y = k, the trace is z = 4x2 + k2 • This is again a parabola that opens upward.

31. QUADRIC SURFACES Example 4 • If we put z = k, we get the horizontal traces 4x2 + y2 = k • We recognize this as a family of ellipses.

32. QUADRIC SURFACES Example 4 • Knowing the shapes of the traces, we can sketch the graph as below.

33. ELLIPTIC PARABOLOID Example 4 • Due to the elliptical and parabolic traces, the quadric surface z = 4x2 +y2 is called an elliptic paraboloid. • Horizontal traces are ellipses. • Vertical traces are parabolas.

34. QUADRIC SURFACES Example 5 • Sketch the surface z = y2 – x2

35. QUADRIC SURFACES Example 5 • The traces in the vertical planes x = kare the parabolas z = y2 – k2, which open upward.

36. QUADRIC SURFACES Example 5 • The traces in y = k are the parabolas z = –x2 + k2, which open downward.

37. QUADRIC SURFACES Example 5 • The horizontal traces are y2 – x2 = k, • a family of hyperbolas.

38. QUADRIC SURFACES Example 5 • All traces are labeled with the value of k.

39. QUADRIC SURFACES Example 5 • Here, we show how the traces appear when placed in their correct planes.

40. HYPERBOLIC PARABOLOID Example 5 • Here, we fit together the traces from the previous figure to form the surface z = y2 – x2, a hyperbolic paraboloid.

41. HYPERBOLIC PARABOLOID Example 5 • Notice that the shape of the surface near the origin resembles that of a saddle. • This surface will be investigated further in Section 14.7 when we discuss saddle points.

42. QUADRIC SURFACES Example 6 • Sketch the surface

43. QUADRIC SURFACES Example 6 • The trace in any horizontal plane z = kis the ellipse

44. QUADRIC SURFACES Example 6 • The traces in the xz- and yz-planes are the hyperbolas

45. HYPERBOLOID OF ONE SHEET Example 6 • This surface is called a hyperboloid of onesheet.

46. GRAPHING SOFTWARE • The idea of using traces to draw a surface is employed in three-dimensional (3-D) graphing software for computers.

47. GRAPHING SOFTWARE • In most such software, • Traces in the vertical planes x = k and y = kare drawn for equally spaced values of k. • Parts of the graph are eliminated using hidden line removal.

48. GRAPHING SOFTWARE • Next, we show computer-drawn graphs of the six basic types of quadric surfaces in standard form. • All surfaces are symmetric with respect to the z-axis. • If a surface is symmetric about a different axis, its equation changes accordingly.

49. ELLIPSOID

50. CONE