ch 12.6: cylinders and quadric surfaces - faculty...
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![Page 1: Ch 12.6: Cylinders and Quadric Surfaces - Faculty Webfacultyweb.kennesaw.edu/ykang4/file_1/math2203/Math2203_Lectures_Ch_12... · If a = b = c, the ellipsoid is a sphere. Hyperboloid](https://reader030.vdocuments.us/reader030/viewer/2022040713/5e183e5d3993df3bdc37661f/html5/thumbnails/1.jpg)
Ch 12.6: Cylinders and Quadric Surfaces
DefinitionA cylinder is a surface that consists of all lines (called rulings)that are parallel to a given line and pass through a given planecurve.
Example) Sketch the graph of surface z = x2.
![Page 2: Ch 12.6: Cylinders and Quadric Surfaces - Faculty Webfacultyweb.kennesaw.edu/ykang4/file_1/math2203/Math2203_Lectures_Ch_12... · If a = b = c, the ellipsoid is a sphere. Hyperboloid](https://reader030.vdocuments.us/reader030/viewer/2022040713/5e183e5d3993df3bdc37661f/html5/thumbnails/2.jpg)
Parabolic cylinderGiven z = x2,
![Page 3: Ch 12.6: Cylinders and Quadric Surfaces - Faculty Webfacultyweb.kennesaw.edu/ykang4/file_1/math2203/Math2203_Lectures_Ch_12... · If a = b = c, the ellipsoid is a sphere. Hyperboloid](https://reader030.vdocuments.us/reader030/viewer/2022040713/5e183e5d3993df3bdc37661f/html5/thumbnails/3.jpg)
ExamplesSketch the surfaces.
1. x2 + y2 = 12. y2 + z2 = 1
![Page 4: Ch 12.6: Cylinders and Quadric Surfaces - Faculty Webfacultyweb.kennesaw.edu/ykang4/file_1/math2203/Math2203_Lectures_Ch_12... · If a = b = c, the ellipsoid is a sphere. Hyperboloid](https://reader030.vdocuments.us/reader030/viewer/2022040713/5e183e5d3993df3bdc37661f/html5/thumbnails/4.jpg)
Quadric SurfacesA quadric surfaces is the graph of a second-degree equation inthree (3) variables x , y , and z . The standard form of the equationis
Ax2 + By2 + Cz2 + Dxy + Eyz + Fxz + Gx + Hy + Iz + J = 0
where A,B,C , · · · , J are constants.
Examples:
I x2 + y2
9 + z2
4 = 1I z = 4x2 + y2
I z = y2 − x2
![Page 5: Ch 12.6: Cylinders and Quadric Surfaces - Faculty Webfacultyweb.kennesaw.edu/ykang4/file_1/math2203/Math2203_Lectures_Ch_12... · If a = b = c, the ellipsoid is a sphere. Hyperboloid](https://reader030.vdocuments.us/reader030/viewer/2022040713/5e183e5d3993df3bdc37661f/html5/thumbnails/5.jpg)
ExampleUse traces to sketch the quadric equations x2 + y2
9 + z2
4 = 1.
![Page 6: Ch 12.6: Cylinders and Quadric Surfaces - Faculty Webfacultyweb.kennesaw.edu/ykang4/file_1/math2203/Math2203_Lectures_Ch_12... · If a = b = c, the ellipsoid is a sphere. Hyperboloid](https://reader030.vdocuments.us/reader030/viewer/2022040713/5e183e5d3993df3bdc37661f/html5/thumbnails/6.jpg)
Ellipsoid x2 + y2
9 + z2
4 = 1
![Page 7: Ch 12.6: Cylinders and Quadric Surfaces - Faculty Webfacultyweb.kennesaw.edu/ykang4/file_1/math2203/Math2203_Lectures_Ch_12... · If a = b = c, the ellipsoid is a sphere. Hyperboloid](https://reader030.vdocuments.us/reader030/viewer/2022040713/5e183e5d3993df3bdc37661f/html5/thumbnails/7.jpg)
ExampleUse traces to sketch the quadric equations z = 4x2 + y2.
![Page 8: Ch 12.6: Cylinders and Quadric Surfaces - Faculty Webfacultyweb.kennesaw.edu/ykang4/file_1/math2203/Math2203_Lectures_Ch_12... · If a = b = c, the ellipsoid is a sphere. Hyperboloid](https://reader030.vdocuments.us/reader030/viewer/2022040713/5e183e5d3993df3bdc37661f/html5/thumbnails/8.jpg)
Elliptic paraboloid: z = 4x2 + y 2
![Page 9: Ch 12.6: Cylinders and Quadric Surfaces - Faculty Webfacultyweb.kennesaw.edu/ykang4/file_1/math2203/Math2203_Lectures_Ch_12... · If a = b = c, the ellipsoid is a sphere. Hyperboloid](https://reader030.vdocuments.us/reader030/viewer/2022040713/5e183e5d3993df3bdc37661f/html5/thumbnails/9.jpg)
ExampleSketch the quadric equations z = y2 − x2.
![Page 10: Ch 12.6: Cylinders and Quadric Surfaces - Faculty Webfacultyweb.kennesaw.edu/ykang4/file_1/math2203/Math2203_Lectures_Ch_12... · If a = b = c, the ellipsoid is a sphere. Hyperboloid](https://reader030.vdocuments.us/reader030/viewer/2022040713/5e183e5d3993df3bdc37661f/html5/thumbnails/10.jpg)
Hyperbolic Paraboloid: z = y 2 − x2
![Page 11: Ch 12.6: Cylinders and Quadric Surfaces - Faculty Webfacultyweb.kennesaw.edu/ykang4/file_1/math2203/Math2203_Lectures_Ch_12... · If a = b = c, the ellipsoid is a sphere. Hyperboloid](https://reader030.vdocuments.us/reader030/viewer/2022040713/5e183e5d3993df3bdc37661f/html5/thumbnails/11.jpg)
Graphs of Quadric surfaces
![Page 12: Ch 12.6: Cylinders and Quadric Surfaces - Faculty Webfacultyweb.kennesaw.edu/ykang4/file_1/math2203/Math2203_Lectures_Ch_12... · If a = b = c, the ellipsoid is a sphere. Hyperboloid](https://reader030.vdocuments.us/reader030/viewer/2022040713/5e183e5d3993df3bdc37661f/html5/thumbnails/12.jpg)
Class ExerciseSketch the following surfaces:
1. 2x2 + y2 = 22. 2x2 − y2 + z2 = 0