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Home , Ellipsoid, Hyperboloid, Paraboloid

Demo of some simple and quadratic surfaces

Yunkai Zhou

Department of Mathematics Southern Methodist University

(Prepared for -III, Math 2339)

Acknowledgement: The very nice free software K3dSurf was used for the plots.

Math 2339, SMU – p. 1/22 Left: x = cos(z); Right: Cylinder y = sin(z)

Math 2339, SMU – p. 2/22 Left: Cylinder x2 + y2 =1; Right: Three cylinders x = cos(z), y = sin(z), x2 + y2 =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 x2 y2 z2 + + =1 a2 b2 c2

Math 2339, SMU – p. 4/22 x2 y2 z2 Ellipsoid + + =1; y = b , (|b | < |b|) a2 b2 c2 1 1 (Notice the intersection is an . In fact the intersection of an ellipsoid with any that intersects with it is an ellipse.) Math 2339, SMU – p. 5/22 x2 y2 z2 Ellipsoid + + = 1; x = c , y = c , z = c a2 b2 c2 1 2 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 x2 y2 Elliptic z = + + c a2 b2

Math 2339, SMU – p. 7/22 x2 y2 Elliptic paraboloid z = + + c; x = c , y = c a2 b2 1 2 The intersection of an elliptic paraboloid (1) with any plane parallel to the z-axis is a ; (2) with any plane not parallel to the z-axis but intersects the paraboloid is an ellipse. Math 2339, SMU – p. 8/22 x2 y2 Hyperbolic paraboloid z = − a2 b2 (viewed from different angles) Notice the (paraboloid intersecting with any plane z = c =0), and the (paraboloid intersecting with planes x = c1, y = c2, or y = kx.)

Math 2339, SMU – p. 9/22 x2 y2 Hyperbolic paraboloid z = − a2 b2 (viewed from different angles) Notice the hyperbolas (paraboloid intersecting with any plane z = c =0), and the parabolas (paraboloid intersecting with planes x = c1, y = c2, or y = kx.)

Math 2339, SMU – p. 10/22 x2 y2 Hyperbolic paraboloid z = − a2 b2 (Restricted region plot, looks more like a saddle used in real life. The right figure also plots z = c, note the intersection is a .)

Math 2339, SMU – p. 11/22 x2 y2 Hyperbolic paraboloid z = − + a2 b2 (Changing the signs of the x2 and y2 terms changes the orientation of the saddle.)

Math 2339, SMU – p. 12/22 x2 y2 Hyperbolic paraboloid z = − + a2 b2 Notice the hyperbolas (paraboloid intersecting with any plane z = c =0), and the parabolas (paraboloid intersecting with planes x = c1, y = c2, or y = kx.) Math 2339, SMU – p. 13/22 x2 y2 of one sheet z2 + c = + , (c> 0) a2 b2 (The right figure plots the one-sheet hyperboloid intersecting with two planes x = c1 and y = c2)

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 x2 y2 Hyperboloid of one sheet z2 + c = + , (c> 0) a2 b2 Notice the hyperbolas (the intersecting with any plane x = c1,y = c2)

Math 2339, SMU – p. 15/22 x2 y2 Hyperboloid of one sheet z2 + c = + , (c> 0) a2 b2 (with c decreasing to 0, the hyperboloid gradually turns into .)

Math 2339, SMU – p. 16/22 x2 y2 Cone z2 = + a2 b2 The cone intersects with any plane passing the (0,0,0) point (e.g. c1x + c2y + c3z =0) in (1) two straight lines, (2) one straight , 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 x2 y2 (The cone z2 = + can intersect with any plane not passing the (0,0,0) point in a2 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 x2 y2 Hyperboloid of two sheets z2 − c = + , (c> 0) a2 b2 (with c increasing from 0, the hyperboloid turns from cone shape into more obvious two sheets.

Math 2339, SMU – p. 19/22 Hyperboloid of two sheets x2 y2 z2 − c = + , (c> 0); x = c , y = c a2 b2 1 2 Notice the hyperbolas of the hyperboloid intersecting with planes parallel to the z-axis.

Math 2339, SMU – p. 20/22 Comments: The previous plots seem to “favor” the z-axis. That is, the two standard quadratic forms are written as (I) Ax2 + By2 + Cz2 + 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 from A and B (in hyperboloid and cone), this “helps” to “favor” the z-axis. And (II) Ax2 + By2 + 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” to z can be bestowed to either x or y. This will lead to different orientation of the quadratic surfaces, but the essential of the surfaces do not change. (See the next slide for two examples.)

Math 2339, SMU – p. 21/22 y2 z2 Left: Elliptic paraboloid x = + − d. (”favor” x) b2 c2 2 2 2 x z Right: One-sheet hyperboloid y + b = + , (b> 0). (”favor” y) a2 c2 Notice the orientation of each surface.

Math 2339, SMU – p. 22/22