lecture6-april11.pdf - Lecture 6 Friday April 11 CYLINDERS AND QUADRIC SURFACES(§12.6 We have seen that a linear equation like ax by cz = d gives a

lecture6-april11.pdf - Lecture 6 Friday April 11 CYLINDERS...

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Unformatted text preview: Lecture 6 - Friday, April 11 CYLINDERS AND QUADRIC SURFACES (§12.6) We have seen that a linear equation like ax + by + cz = d gives a plane in R3. What about surfaces defined by quadratic equations (quadratic in x, y, z)? The situation in R2 In R2 a quadratic equation gives one of the following curves: • y = ax 2 → parabola a>0 y a<0 y x x 41 • ( ax )2 + ( yb )2 = 1 → ellipse y x • hyperbola y y x x ( yb )2 − ( ax )2 = 1 ( ax )2 − ( yb )2 = 1 Remark: All the curves can be shifted 42 Scaling and translating • Scaling: y 2 2 1 −2 −1 −1 y 1 x 1 2 −2 x 2 + y2 = 1 1 x −2 −1 −1 2 1 y −2 −1 −1 2 −2 ( x2 )2 + y2 = 1 x 1 −2 (2x)2 + y2 = 1 Dividing x by a means stretching the object by a factor a along the x axis • Translating: 2 y 2 1 −2 −1 −1 1 2 −2 x 2 + y2 = 1 y 1 x −2 −1 −1 x 1 2 −2 (x − 1)2 + y2 = 1 Subtracting a from x means moving the object by a in direction of the x-axis. 43 2 Def: Given a plane curve in R3, a cylinder is a set of all parallel lines through this curve. Example: ( x4 )2 + z 2 = 1 describes an elliptic cylinder z 1 x 4 z y x 44 Example: z = y2 describes an parabolic cylinder z y z y x Observe: Intersecting any parallel translation of the yzplane with the cylinder gives the same plane curve. 45 Recall: x 2 + y2 + z 2 = r 2 defines a sphere of radius r. z y x 46 An ellipsoid is of the form ( ax )2 + ( yb )2 + ( zc )2 = 1. z y x Intersection with any coordinate planes (= trace) gives an ellipse 47 An cone is of the form ( zc )2 = ( ax )2 + ( yb )2. z y x Intersections with shifted xy-plane are ellipses. For a = b = c, the intersection with z = z0 is a sphere of radius |z0|. z0 radius 48 An elliptic paraboloid is of the form z c = ( ax )2 + ( yb )2. z y x Intersections with shifted xy-plane are ellipses. For a = b = c = 1, the intersection with z = z0 is a sphere √ of radius z0. z0 radius 49 An hyperboloid of one sheet is of the form y z x ( )2 + ( )2 − ( )2 = 1. a b c z y x Intersections with shifted xy-plane are ellipses For a = qb = c = 1, intersection with z = z0 is sphere of radius 1 + z02 z0 radius 50 An hyperboloid of two sheets is of the form ( ax )2 + ( yb )2 = ( zc )2 − 1. z y x Intersections with xy-plane are ellipses For a = b = c = 1, intersection with z = z0 is sphere of √ radius z 2 − 1 z0 radius 51 Example 1: What type of surface is z = x 2 + y2? Solution: Intersections with shifted xy-plane is a sphere of √ radius z ⇒ elliptic paraboloid. Example 2: What type of surface is 36x 2 − 9y2 + 4z 2 = 36? Solution: After scaling/shrinking along x, y and z-axis, the surface is of the same type as x 2 − y2 + z 2 = 1. Hence this is a hyperboloid of one sheet. 52 Example: Classify the surface given by x 2 + 2z 2 − 6x − y + 4z + 11 = 0. Idea: Bring eq. into form (x−?)2 − y + (z−?)2 =?. We rewrite the equation (x 2 − 6x) + 2(z 2 + 2z) = y − 11 ⇔ (x 2 − 6x + 32) + 2(z 2 + 2z + 1) = y − 11 + 2 + 9 ⇔ (x − 3)2 + 2(z + 1)2 = y This is an elliptic paraboloid, shifted by (3, 0, −1). The central axis is (3, 0, −1) + t(0, 1, 0) for t ≥ 0 z y x (3, 0, −1) 53 ...
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