Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Quadric surfaces, which are the graphs of second-degree equations in xyz-coordinate systems. Six common types of quadric surfaces are introduced: ellipsoids, hyperboloids of one sheet, hyperboloids of two sheets, elliptic cones, elliptic paraboloids, and hyperbolic paraboloids. The document also explains how to sketch these surfaces using rough techniques.
Typology: Lecture notes
1 / 13
This page cannot be seen from the preview
Don't miss anything!
Traceofsurtace "
" line that^ results^ when^ a (^) surface (^) is (^) cut by a (^) plane (^).
t : "
7=0×242-- 2- (^) =L ; (^) z
ta =L -^ x'- y ' " hemisphere " x'+42=1 - I
dogoinonwr.MU/hD8w-:X--0Z--Vt-yT;Z7ox---1zZ-2--1-y y '
Fi 2-^ ' - ftp.y#X--tzZ--J1-fz-y2z2=- y ' 72=
Ext fcx,yl= - fifty , 2-^ n^ i^. ' ' n ' (^) ' ongoing 7=0^ ;^
'-
atodshmaoiaiurnrsoosriawsunnghsosrmrhi n r. Nssnriuonorrnllnls Z
5. Quadric surfaces The equation in an xyz - coordinate system is Ax^2^ By^2^ Cz^2 Dxy Exz Fyz Gx Hy Iz J 0 which is called a second-degree equation in (^) x y , and z. The graphs of such equations are called quadric surfaces or sometimes quadrics. Six common types of quadric surfaces are shown in Table 1 ½ellipsoids, hyperboloids of one sheet, hyperboloids of two sheets, elliptic cones, elliptic paraboloids, and hyperbolic paraboloids. (The constants a, b, and c that appear in the equations in the table are assumed to be positive.) Observe that none of the quadric surfaces in the table have cross-product terms in their equations. This is because of their orientations relative to the coordinate axes. Later in this section we will discuss other possible orientations that produce equations of the quadric surfaces with no cross-product terms. In the special case where the elliptic cross sections of an elliptic cone or an elliptic paraboloid are circles, the terms circular cone and circular paraboloid are used. (a) (b) (c) (^206207) Solid (^) Analytic Geometry b.
Table 1 ( From: Calculus Early Transcendentals, 10th edition, Howard Anton, Irl C. Bevens, Stephen Davis, page 823 ) surface equation 2 2 2 2 2 2 1 x y z a b c
The traces in the coordinate planes are ellipses, as are the traces in those planes that are parallel to the coordinate planes and intersect the surface in more than one point. 2 2 2 2 2 2 1 x y z a b c
The trace in the (^) xy - plane is an ellipse, as are the traces in planes parallel to the xy - plane. The traces in the yz - plane and xz - plane are hyperbolas, as are the traces in those planes that are parallel to these and do not pass through the x - or (^) y - intercepts. At these intercepts the traces are pairs of intersecting lines. 2 2 2 2 2 2 1 z x y c a b
There is no trace in the xy - plane. In planes parallel to the xy - plane that intersect the surface in more than one point the traces are ellipses. In the yz - and xz - planes, the traces are hyperbolas, as are the traces in those planes that are parallel to these. ELLIPSOID HYPERBOLOID OF ONE SHEET HYPERBOLOID OF TWO SHEET
ภาคตัดกรวยแบบเลื่อนขนาน ทบทวนภาคตัดกรวย (^) 206112 QUICK REFERENCE ( =/#^ Isiah^ :^ c=VaIb (a^ >^ b) = I
206112 QUICK REFERENCE (^) Quadric Surfaces ที่มา : หนังสือ Thomas Calculus
A rough sketch of the hyperboloid of two sheet can be obtained by first plotting the intersections with the z - axis, then sketching the elliptical traces in the planes z r 2 c , and then sketching the hyperbolic traces that connect the z - axis intersections and the endpoints of the axes of the ellipses. (It is not essential to use the planes z r 2 c , but these are good choices since they simplify the calculations slightly and have the right spacing for a good sketch.) The next example illustrates this technique. Example 9 Sketch the graph of the hyperboloid of two sheet 2 2 2 1 4
y z x A rough sketch of the elliptic cone can be obtained by first sketching the elliptical traces in the planes z r 1 and then sketching the linear traces that connect the endpoints of the axes of the ellipses. The next example illustrates this technique. Example 10 Sketch the graph of the elliptic cone 2 2 2 4
y z x 2 2 2 2 ^2 ^2 1 (^!^ 0,^!^ 0,^ !0) z x y a b c c a b 2 2 2 2 ^2 (^!^ 0,^ !0) x y z a b a b ZZEOO ildiiawhooa^ - ,^ o^ - XIII. - I een (^) z
27,4-5-12=1 IIIa ! 3 tho.^ Co^ ,rE,z,
XI (^) -1452=
A rough sketch of the elliptic paraboloid can be obtained by first sketching the elliptical trace in the plane z 1 and then sketching the parabolic traces in the vertical coordinate planes to connect the origin to the ends of the axes of the ellipse. The next example illustrates this technique. Example 11 Sketch the graph of the elliptic paraboloid 2 2 4 9
x y z A rough sketch of the hyperbolic paraboloid can be obtained by first sketching the two parabolic traces that pass through the origin (one in the plane x 0 and the other in the plane y 0 ). After the parabolic traces are drawn, sketch the hyperbolic traces in the planes z r 1 and then fill in any missing edges. The next example illustrates this technique. Example 12 Sketch the graph of the hyperbolic paraboloid 2 2 4 9
y x z 2 2 2 ^2 (^!^ 0,^ !0) x y z a b a b 2 2 2 2 ^2 (^!^ 0,^ !0) y x z a b b a
2 2 2 2 ^2 2 1 x y z a b c 2 2 2 2 ^2 2 1 x y z a b c 2 2 2 2 ^2 2 1 z x y c a b 2 2 2 2 2 0 x y z a b 2 2 2 2 0 x y z a b 2 2 2 2 0 y x z b a 5.2 Techniques for identifying quadric surfaces The equations of the quadric surfaces in Table 1 have certain characteristics that make it possible to identify quadric surfaces that are derived from these equations by reflections. These identifying characteristics, which are shown in Table 2, are based on writing the equation of the quadric surface so that all of the variable terms are on the left side of the equation and there is a 1 or a 0 on the right side. These characteristics do not change when the surface is reflected about a coordinate plane or planes of the form x y x , z , or^ y z , thereby making it possible to identify the reflected quadric surface from the form^ of its equation. Table 2 identifying a quadric surface from the form of its equation equation characteristic No minus signs One minus sign Two minus signs No linear terms One linear term; two quadratic terms with the same sign One linear term; two quadratic terms with opposite signs classification Ellipsoid^ Hyperboloid of one sheet Hyperboloid of two sheets Elliptic cone Elliptic paraboloid Hyperbolic paraboloid Example 15 Identify the surfaces (a) 2 2 2 3 x 4 y 12 z 12 0 (b) 2 2 4 x 4 y z 0
