Computer Aided Geometric Design

Transcription

1 Copue Aided Geoei Design Geshon Ele, Tehnion sed on ook Cohen, Riesenfeld, & Ele Geshon Ele, Tehnion

2 Definiion 3. The Cile Given poin C in plne nd nue R 0, he ile ih ene C nd dius R is defined s he se of ll poins in he plne disne R fo he poin C. In se noion e ie, { P =, : P - C = R } C R Geshon Ele, Tehnion

3 Definiion 3. The Ellipse Given o poins, F nd F lled he foi nd nue P K F - F, n ellipse is defined s he se of ll poins F F he su of hose disnes fo he foi K. Th is, { P =, : P - F + P - F = K }. Geshon Ele, Tehnion 3

4 The Ellipse The is onining he foi is lled he jo is of he ellipse nd he is ohogonl o he jo is hough he ene, C = F + F /, is denoed he ino is. If ene C is soe loion C =, he ellipse equls, Geshon Ele, Tehnion 4.

5 Definiion 3.3 The Hpeol Given o poins, F nd F lled he foi nd nue P K 0, hpeol is defined s he se of ll poins F F he diffeene of hose disnes fo he foi K. Th is, { P =, : P - F - P - F = K }. Quesion: Wh if K = 0? Geshon Ele, Tehnion 5

6 עשת/ב רדא/'כ ו" Geshon Ele, Tehnion 6 The Hpeol The is onining he foi is lled he jo is of he hpeol nd he is ohogonl o he jo is hough he ene, C = F + F /, is denoed he ino is. If ene C is soe loion C =, he hpeol equls,. o

7 Definiion 3.4 The Pol Given fied poin F lled he fous nd fied line lled he diei, he se of poins equidisn iseo fo F nd he diei is lled pol. Diei F Geshon Ele, Tehnion 7

8 The Pol = -, 0 Le he diei e he line = - nd ssue F =, 0. Then,, nd squing he epession, 4. Geshon Ele, Tehnion 8

9 Coni Seions z Conside he one z = +, hee is el nue. Quesion: Wh is he ffe of? Quesion: Wh is he shpe of Plne-Cone ineseion? Geshon Ele, Tehnion 9

10 z = + Coni Seions z Conside he plne = k. Then, z = k +, o = z / - k. If k = 0, z =, o To Lines. z k 0,, o Hpeol. k k Quesion: Wh e he foi of he hpeol? =k =k Geshon Ele, Tehnion 0

11 Coni Seions z = + z Conside he plne z =. Then, / = +, o ile. Quesion: Wh if = 0? z= Geshon Ele, Tehnion

12 עשת/ב רדא/'כ ו" Geshon Ele, Tehnion Coni Seions Conside he plne z = +, 0. Then, o,,. z z = +

13 Coni Seions. z If =, hen he ineseion uve equls o, Pol., Quesion: Wh if = 0? Geshon Ele, Tehnion 3

14 עשת/ב רדא/'כ ו" Geshon Ele, Tehnion 4 Coni Seions If < hen le = - nd nd opleing he sque,, z.

15 עשת/ב רדא/'כ ו" Geshon Ele, Tehnion 5 Coni Seions Mulipling = -, nd dividing, e ge n Ellipse., z

16 עשת/ב רדא/'כ ו" Geshon Ele, Tehnion 6 Coni Seions If > hen le = - nd nd ulipling,. z.

17 עשת/ב רדא/'כ ו" Geshon Ele, Tehnion 7 Coni Seions If = 0, eihe = o e hve ossing lines. Oheise, 0, divide, Hpeol.. z

18 Iplii Qudi Funions s Conis We hve seen h ll oni seions e qudi iplii fos. Quesion: Ae ll qudi iplii fos oni seions? Geshon Ele, Tehnion 8

19 Iplii Qudi Funions s Conis Conside he genel qudi iplii fo of A + B + C + D + E + F = 0. Quesion: Is hee hnge of sis fo, o, suh h he se gph is dn he uve, A + C + D + E + F = 0? Geshon Ele, Tehnion 9

20 Definiion 3.7 Fo he qudi equion: A + B + C + D + E + F = 0, he quni B - 4AC, is lled he disiinn. Theoe 3.8 The disiinn is invin unde oions. Geshon Ele, Tehnion 0

21 Theoe 3.9 Eve iplii qudi is oni seion nd if B 4AC 0, 0, 0, he he he uve uve uve hpeol. Geshon Ele, Tehnion is is is n ellipse, pol, Poof Sine he disiinn is invin unde oions, oe hough he speil ngle so h B = 0 in he ne oed oodine sse. Then, B 4AC B' 4A' C' 4A' C'.

22 Theoe 3.0 An iplii funion f, = 0 is oni seion if nd onl if f is seond degee polnoil in nd. Quesion: Ho n e inuiivel onsu oni seions? Geshon Ele, Tehnion

23 5 Poins Consuion Quesion: Ho n degees of feedo does he qudi equion of A + B + C + D + E + F = 0 hve? These degees of feedo n e pesied using five poins i, i : A A A A A B B B B B Geshon Ele, Tehnion C C C C C D D D D D E E E E E F F F F F

24 עשת/ב רדא/'כ ו" Geshon Ele, Tehnion 4 5 Poins Consuion Con. O in i fo, Quesions: Wh is issing hee? F E D C B A 4, 4 3, 3 5, 5,,

25 5 Poins Consuion Con. Seeking oe inuiive ppoh, onside he fou lines hough he fou given poins, L 3 P L L P 3 L hough P nd P L hough P 3 nd P 4 L 4 P 4 L 3 hough P nd P 3 P L 4 hough P 4 nd P Geshon Ele, Tehnion 5

26 5 Poins Consuion Con. Le L i, = i + i + i. Then, L L L L Pj 0 P 0 j fo fo j j, 3,4 L 3 P P 3 L L 3 4 P j P j 0 0 fo fo j j,3 4, L 4 P P 4 Geshon Ele, Tehnion 6

27 5 Poins Consuion Con. Le L i, L j, = i + i + i j + j + j. Osevion: L i, L j, is qudi equion in nd. Osevion: L L P j, j =,,3,4 equl zeo! No onside he sufe onsn, z f, L L, L3 L4,. Quesion: To h is f P j, j =,,3,4 equl? Geshon Ele, Tehnion 7

28 z f, L L, L3 L4,. 5 Poins Consuion Con. Quesion: Ho n e pesie? Define fifh poin, P 5, nd ensue h f o, P 5 0 L L P 5 L 3 L 4 P L P5 L P5. P 5 L P L P L 4 L 3 P P L L P 3 P 4 Geshon Ele, Tehnion 8

29 Using Tngens Le L 3 L 4 nd hene L L f, = L L + L 3 T Quesion: Wh does he shpe of f, look like? P 5 P 5 is denoed he shoulde poin: M = P +P 4 /, nd he -oni equls = P 5 - M / T - M. L 3 =L 4 P =P M P 3 =P 4 Geshon Ele, Tehnion 9

30 Using Tngens P 5 = - M + T. Then L L he oni is Pol, Hpeol, Ellipse, if if if,,. L 3 =L 4 P =P P 5 M T P 3 =P 4 Geshon Ele, Tehnion 30

31 Coni As s Rionl Funions Assue T, P, nd P e no on he se line. Then, {U = P - T, V = P - T } spns he XY plne. Eve poin in he XY plne n e ien s u, v = T+uP - T+vP - T. Quesion: Is his oodine sse igid-oion invin? T U=0,V= P V 0 = T +u P, -T +v P, -T, 0 = T +u P, -T +vp, -T. P U=,V=0 U Geshon Ele, Tehnion 3

32 Coni As s Rionl Funions Quesion: Wh e he u, v oodines of he T P line? The T P line? The P P line? L : v = 0, L : u = 0, L 3 : u + v - = 0. T L 3 U=0,V= P P U=,V=0 V U L : T + u P - T = -u T + u P, L : T + vp - T = -v T + vp, L 3 : T + u P - T + -u P - T = u P + -u P. Geshon Ele, Tehnion 3

33 Coni As s Rionl Funions In uv oodines e hve 0 = L L + L 3 = uv + u + v -. Seing = -/ -, one ges Cu, v =-uv - u+v- = 0. T v u,v u U=0,V= P P U=,V=0 V U Fo 0, nd 0 u, v suh h u + v, Cu, v = 0 is inside he ingle P T P. Conside poin P on Cu, v = 0, P = u, v. Geshon Ele, Tehnion 33

34 u v, 0 v u v Coni As s Rionl Funions A Conside he io And he io TA u P A u u v v TA P A v v Geshon Ele, Tehnion 34 T A =0, v A =u, 0 u v u u,v U=0,V= P P U=,V=0 V U

35 Coni As s Rionl Funions u v v, u v u And onside he podu of hese o ios 4 u v u v T A =0, v A =u, 0 u,v U=0,V= P P U=,V=0 V U On poin u, v, in uve Cu, v, -uv = u+v- o u v u v nd 4 4. Geshon Ele, Tehnion 35

36 Theoe 3.3 U=0,V= P V A =0, v If P, T, P, A nd A e T u,v s ove, hen he podu A =u, 0 of he ios P U=,V=0 U TA A P TA A P uv u v is onsn fo he hole oni seion. Geshon Ele, Tehnion 36

37 Coni As s Rionl Funions u Beuse 4, e hve, u nd siill v. Geshon Ele, Tehnion 37

38 עשת/ב רדא/'כ ו" Geshon Ele, Tehnion 38 Coni As s Rionl Funions Going k o he oni uve, e hve. P T P T P T P T T P v T P u T T U V P P u,v

39 Coni As s Rionl Funions P T P. In ode o peeize s ionl fo,,, nd us sisf he folloing onsins,. = k, k = 4/- onsn.., p, o 0, nd, 0. 3., us e onoone fo,. Quesion: Wh, p o 0,? Wh is hee onoonii onsin? Geshon Ele, Tehnion 39

40 עשת/ב רדא/'כ ו" Geshon Ele, Tehnion 40 Coni As s Rionl Funions One possile soluion fo, is nd : o is veified. nd 3 e ivil o veif s ell.. 4 K

41 עשת/ב רדא/'כ ו" Geshon Ele, Tehnion 4 Coni As s Rionl Funions Then, Hene, eve oni seion n e ien s ionl qudi pei funion.. P T P P T P P T P

42 Eple 3.9 A of ile Assue ile of dius spnning degs. Fo i =,, i TA i i A P i TA i i A P TA i TA i os / os / T Thus, K os / P A A Quesion: Wh ill e he effe, if n, of, /? P P Geshon Ele, Tehnion 4

43 עשת/ב רדא/'כ ו" Geshon Ele, Tehnion 43 Hoogeneous Coodines The ionl fo of qudis equls, Le = -, = - -, 3 = -. Then,. P T P.,,, P T P P T P