Cylinders and quadratic surfaces (Sect. 12.6)

Transcription

1 Clindes nd qudtic sufces (Sect..6) Clindes. Qudtic sufces: Sphees, Ellipsoids, Cones, Hpeboloids, Pboloids, Sddles, + + =. + b + c =. + b c = 0. + b c + b c = 0. b c = 0. =, b + c =. Clindes Given cuve on plne, clled the geneting cuve, clinde is sufce in spce geneting b moving long the geneting cuve stight line pependicul to the plne contining the geneting cuve. A cicul clinde is the pticul cse when the geneting cuve is cicle. In the pictue, the geneting cuve lies on the -plne.

2 Clindes Find the eqution of the clinde given in the pictue. The intesection of the clinde with the = 0 plne is cicle with dius, hence points of the fom (,, 0) belong to the clinde iff + = nd = 0. Fo 0, the intesection of hoiontl plnes of constnt with the clinde gin e cicles of dius, hence points of the fom (,, ) belong to the clinde iff + = nd constnt. Summiing, the eqution of the clinde is + =. The coodinte does not ppe in the eqution. The eqution holds fo eve vlue of R. Clindes Find the eqution of the clinde given in the pictue. The geneting cuve is cicle, but this time on the plne = 0. Hence point of the fom (, 0, ) belong to the clinde iff + =. We conclude tht the eqution of the clinde bove is + =, R. The coodinte does not ppe in the eqution. The eqution holds fo eve vlue of R.

3 Clindes Find the eqution of the clinde given in the pictue. 4 pbol The geneting cuve is pbol on plnes with constnt. This pbol contins the points (0, 0, 0), (, 0, ), nd (, 0, 4). Since thee points detemine unique pbol nd = contins these points, then t = 0 the geneting cuve is =. The clinde eqution does not contin the coodinte. Hence, =, R. Clindes nd qudtic sufces (Sect..6) Clindes. Qudtic sufces: Sphees. Ellipsoids. Cones. Hpeboloids. Pboloids. Sddles.

4 Qudtic sufces Given constnts i, b i nd c, with i =,, 3, qudtic sufce in spce is the set of points (,, ) solutions of the eqution Remks: b + b + b 3 + c = 0. Thee e sevel tpes of qudtic sufces. We stud onl qudtic sufces given b b 3 = c. () The sufces below e ottions of the one in Eq. (), b 3 = c, b 3 = c. Clindes nd qudtic sufces (Sect..6) Clindes. Qudtic sufces: Sphees. Ellipsoids. Cones. Hpeboloids. Pboloids. Sddles. + + =.

5 Sphees Recll: We stud onl qudtic equtions of the fom: b 3 = c. A sphee is simple qudtic sufce, the one in the pictue hs the eqution + + =. ( = = 3 = /, b 3 = 0 nd c =.) Equivlentl, + + =. Sphees Remk: Line tems move the sphee ound in spce. Gph the sufce given b the eqution = 0. Complete the sque: + [ ( 4 ( 4 ] ( 4 ) ) ) = 0. Theefoe, + ( + 4 ) + = 4. This is the eqution of sphee centeed t P 0 = (0,, 0) nd with dius =.

6 Clindes nd qudtic sufces (Sect..6) Clindes. Qudtic sufces: Sphees, + Ellipsoids, Pboloids. Cones. Hpeboloids. Sddles. + =. + b + c =. Ellipsoids Given positive constnts, b, c, n ellipsoid centeed t the oigin is the set of point solution to the eqution + b + c =. Gph the ellipsoid, =. 3

7 Ellipsoids Gph the ellipsoid, =. On the plne = 0 we hve the ellipse + 3 =. 3 On the plne = 0, with ( < 0 < we hve the ellipse + = 3 0 ). Denoting c = (0 /4), then 0 < c <, nd c + 3 c =. 3 Clindes nd qudtic sufces (Sect..6) Clindes. Qudtic sufces: Sphees, + Ellipsoids, + b Cones, Hpeboloids. Pboloids. Sddles. + =. + c =. + b c = 0.

8 Cones Given positive constnts, b, cone centeed t the oigin is the set of point solution to the eqution = ± + b. Gph the cone, = Cones Gph the cone, = + +. On the plne = we hve the ellipse + =. On the plne = 0 > 0 we hve the ellipse + = 0, tht is, =.

9 Clindes nd qudtic sufces (Sect..6) Clindes. Qudtic sufces: Sphees, + Ellipsoids, + b Cones, + b Hpeboloids, Pboloids. Sddles. + =. + c =. c = 0. + b c =, b + c =. Hpeboloids Given positive constnts, b, c, one sheet hpeboloid centeed t the oigin is the set of point solution to the eqution + b c =. (One negtive sign, one sheet.) ellipse hpebol Gph the hpeboloid, + =. hpebol hpebols

10 Hpeboloids ellipse Gph the hpeboloid + hpebol =. hpebol hpebols Find the intesection of the sufce with hoiontl nd veticl plnes. Then combine them into qulittive gph. On hoiontl plnes, = 0, we obtin ellipses + = + 0. On veticl plnes, = 0, we obtin hpebols = 0. On veticl plnes, = 0, we obtin hpebols = 0. Hpeboloids Given positive constnts, b, c, two sheet hpeboloid centeed t the oigin is the set of point solution to the eqution b + c =. (Two negtive signs, two sheets.) ellipse Gph the hpeboloid, + =. hpebol hpebol

11 Hpeboloids ellipse Gph the hpeboloid + =. hpebol hpebol Find the intesection of the sufce with hoiontl nd veticl plnes. Then combine ll these esults into qulittive gph. On hoiontl plnes, = 0, with 0 >, we obtin ellipses + = + 0. On veticl plnes, = 0, we obtin hpebols + = + 0. On veticl plnes, = 0, we obtin hpebols + = + 0. Clindes nd qudtic sufces (Sect..6) Clindes. Qudtic sufces: Sphees, + Ellipsoids, + b Cones, + b Hpeboloids, + b Pboloids, Sddles. + =. + c =. c = 0. c =, b + b c = 0. + c =.

12 Pboloids Given positive constnts, b, pboloid centeed t the oigin is the set of point solution to the eqution = + b. ellipse Gph the pboloid, = +. pbol Pboloids. ellipse Gph the pboloid = +. Find the intesection of the sufce with hoiontl nd veticl plnes. Then combine ll these esults into qulittive gph. On hoiontl plnes, = 0, with 0 > 0, we obtin ellipses + = 0. pbol On veticl plnes, = 0, we obtin pbols = + 0. On veticl plnes, = 0, we obtin pbols = 0 +.

13 Clindes nd qudtic sufces (Sect..6) Clindes. Qudtic sufces: Sphees, + Ellipsoids, + b Cones, + b Hpeboloids, + b + =. + c =. c = 0. Pboloids, + b c = 0. Sddles, c =, b b c = 0. + c =. Sddles, o hpebolic pboloids Given positive constnts, b, c, sddle centeed t the oigin is the set of point solution to the eqution = b. Gph the pboloid, = +.

14 Sddles Gph the sddle = +. hpebol pbol pbol Find the intesection of the sufce with hoiontl nd veticl plnes. Then combine ll these esults into qulittive gph. On plnes, = 0, we obtin hpebols + = 0. On plnes, = 0, we obtin pbols = + 0. On plnes, = 0, we obtin pbols = 0 +.