Demo of some simple cylinders and quadratic surfaces

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1 Demo of some simple cylinders and quadratic surfaces Yunkai Zhou Department of Mathematics Southern Methodist University (Prepared for Calculus-III, Math 2339) Acknowledgement: The very nice free software K3dSurf was used for the plots. Math 2339, SMU p. 1/22

2 Left: Cylinder x = cos(z); Right: Cylinder y = sin(z) Math 2339, SMU p. 2/22

3 Left: Cylinder x 2 +y 2 = 1; Right: Three cylinders x = cos(z), y = sin(z), x 2 +y 2 = 1 intersecting each other. Notice the intersection of the three cylinders is the well-known space curve helix r (t) = cos(t), sin(t), t Math 2339, SMU p. 3/22

4 Ellipsoid x2 a 2 + y2 b 2 + z2 c 2 = 1 Math 2339, SMU p. 4/22

5 Ellipsoid x2 a + y2 2 b + z2 2 c = 1 ; y = b 1, ( b 2 1 < b ) (Notice the intersection is an ellipse. In fact the intersection of an ellipsoid with any plane that intersects with it is an ellipse.) Math 2339, SMU p. 5/22

6 Ellipsoid x2 a + y2 2 b + z2 2 c = 1; x = c 1, y = c 2 2, z = c 3 (The intersection of an ellipsoid with any plane (not necessarily parallel to the coordinate planes) is an ellipse.) Math 2339, SMU p. 6/22

7 Elliptic paraboloid z = x2 a 2 + y2 b 2 +c Math 2339, SMU p. 7/22

to the z-axis is a parabola; (2) with any plane not parallel to the

8 Elliptic paraboloid z = x2 a 2 + y2 b 2 +c; x = c 1, y = c 2 The intersection of an elliptic paraboloid (1) with any plane parallel to the z-axis is a parabola; (2) with any plane not parallel to the z-axis but intersects the paraboloid is an ellipse. Math 2339, SMU p. 8/22

any plane z = c 0), and the parabolas (paraboloid intersecting

9 Hyperbolic paraboloid z = x2 a y2 2 b 2 (viewed from different angles) Notice the hyperbolas (paraboloid intersecting with any plane z = c 0), and the parabolas (paraboloid intersecting with planes x = c 1, y = c 2, or y = kx.) Math 2339, SMU p. 9/22

10 Hyperbolic paraboloid z = x2 a y2 2 b 2 (viewed from different angles) Notice the hyperbolas (paraboloid intersecting with any plane z = c 0), and the parabolas (paraboloid intersecting with planes x = c 1, y = c 2, or y = kx.) Math 2339, SMU p. 10/22

11 Hyperbolic paraboloid z = x2 a y2 2 b 2 (Restricted region plot, looks more like a saddle used in real life. The right figure also plotsz = c, note the intersection is a hyperbola.) Math 2339, SMU p. 11/22

12 Hyperbolic paraboloid z = x2 a 2+y2 b 2 (Changing the signs of thex 2 and y 2 terms changes the orientation of the saddle.) Math 2339, SMU p. 12/22

13 Hyperbolic paraboloid z = x2 a 2+y2 b 2 Notice the hyperbolas (paraboloid intersecting with any plane z = c 0), and the parabolas (paraboloid intersecting with planes x = c 1, y = c 2, or y = kx.) Math 2339, SMU p. 13/22

Hyperboloid of one sheet z 2 +c = x2 a + y2 2 b2, (c > 0) (The right figure plots the one-sheet hyperboloid intersecting with two planes x = c 1 and y = c 2 ) Notice the hyperbolas

14 Hyperboloid of one sheet z 2 +c = x2 a + y2 2 b2, (c > 0) (The right figure plots the one-sheet hyperboloid intersecting with two planes x = c 1 and y = c 2 ) Notice the hyperbolas (hyperboloid intersecting with any plane parallel to the z-axis) (hyperboloid intersecting with any plane not parallel to the z-axis may be a hyperbola or an ellipse.) Math 2339, SMU p. 14/22

15 Hyperboloid of one sheet z 2 +c = x2 a + y2 2 b2, (c > 0) Notice the hyperbolas (the surface intersecting with any plane x = c 1,y = c 2 ) Math 2339, SMU p. 15/22

16 Hyperboloid of one sheet z 2 +c = x2 a + y2 2 b2, (c > 0) (withcdecreasing to 0, the hyperboloid gradually turns into cone shape.) Math 2339, SMU p. 16/22

17 Cone z 2 = x2 a + y2 2 b 2 The cone intersects with any plane passing the (0,0,0) point (e.g. c 1 x+c 2 y +c 3 z = 0) in (1) two straight lines, (2) one straight line, or (3) a single point. The cone can intersect with any plane not passing the (0,0,0) point in (1) parabola, (2) ellipse, or (3) hyperbola. Math 2339, SMU p. 17/22

18 (The cone z 2 = x2 a 2 + y2 can intersect with any plane not passing the (0,0,0) point in b2 (1) parabola, (2) ellipse, or (3) hyperbola. That is why these curves are called conic sections.) (Acknowledgement: The above figure is from Wikipedia.com on conic sections.) Math 2339, SMU p. 18/22

19 Hyperboloid of two sheets z 2 c = x2 a + y2 2 b2, (c > 0) (withcincreasing from 0, the hyperboloid turns from cone shape into more obvious two sheets. Math 2339, SMU p. 19/22

20 Hyperboloid of two sheets z 2 c = x2 a + y2 2 b 2, (c > 0); x = c 1, y = c 2 Notice the hyperbolas of the hyperboloid intersecting with planes parallel to the z-axis. Math 2339, SMU p. 20/22

21 Comments: The previous plots seem to favor the z-axis. That is, the two standard quadratic forms are written as (I) Ax 2 +By 2 +Cz 2 +J = 0, ABC 0, which includes the ellipsoid, the hyperboloid (one-sheet & two-sheets), and the cone. Except for the ellipsoid, C has a different sign fromaand B (in hyperboloid and cone), this helps to favor the z-axis. And (II) Ax 2 +By 2 +Cz = 0, ABC 0, which includes the elliptic paraboloid, and the hyperbolic paraboloid. The only linear term is assigned to the z variable, that is why the z-axis is favored again. The above special treatment toz can be bestowed to eitherxory. This will lead to different orientation of the quadratic surfaces, but the essential shapes of the surfaces do not change. (See the next slide for two examples.) Math 2339, SMU p. 21/22

22 Left: Elliptic paraboloid x = y2 b + z2 d. 2 ( favor x) c2 Right: One-sheet hyperboloid y 2 +b = x2 a + z2 2 c2, (b > 0). ( favor y) Notice the orientation of each surface. Math 2339, SMU p. 22/22

4 = 1 which is an ellipse of major axis 2 and minor axis 2. Try the plane z = y2

4 = 1 which is an ellipse of major axis 2 and minor axis 2. Try the plane z = y2 12.6 Quadrics and Cylinder Surfaces: Example: What is y = x? More correctly what is {(x,y,z) R 3 : y = x}? It s a plane. What about y =? Its a cylinder surface. What about y z = Again a cylinder surface