Basic tools: orientation tests

Transcription

1 Baic tool: oientation tet Rodigo Silveia Comutational Geomety Facultat d Infomàtica de Bacelona Univeitat Politècnica de Catalunya

2 Tuning ight o tuning left? Comutational Geomety, Facultat d Infomàtica de Bacelona, UPC Conide a euence of 3 oint:,,

3 Tuning ight o tuning left? Comutational Geomety, Facultat d Infomàtica de Bacelona, UPC Conide a euence of 3 oint:,, Baic uetion: do they make a ight tun? A left tun? O eha they ae colinea?

4 Tuning ight o tuning left? Comutational Geomety, Facultat d Infomàtica de Bacelona, UPC Conide a euence of 3 oint:,, Baic uetion: do they make a ight tun? A left tun? O eha they ae colinea?

5 Tuning ight o tuning left? Comutational Geomety, Facultat d Infomàtica de Bacelona, UPC Conide a euence of 3 oint:,, Baic uetion: do they make a ight tun? A left tun? O eha they ae colinea?

6 Tuning ight o tuning left? Comutational Geomety, Facultat d Infomàtica de Bacelona, UPC Conide a euence of 3 oint:,, Baic uetion: do they make a ight tun? A left tun? O eha they ae colinea? Thi i a fundamental baic oeation in geometic algoithm. Reuie a olution that i Efficient Numeically obut

7 Tuning ight o tuning left? Comutational Geomety, Facultat d Infomàtica de Bacelona, UPC Conide a euence of 3 oint:,, Baic uetion: do they make a ight tun? A left tun? O eha they ae colinea? Thi i a fundamental baic oeation in geometic algoithm. Reuie a olution that i Efficient Numeically obut Solution: igned aea of tiangle

8 Signed aea of tiangle Comutational Geomety, Facultat d Infomàtica de Bacelona, UPC Signed aea of The igned aea of tell u the aea of the tiangle, and the ign give the oientation (left-tuning o ight-tuning) Can the aea be zeo?

9 Signed aea of tiangle Comutational Geomety, Facultat d Infomàtica de Bacelona, UPC Signed aea of Comuting A( ) The igned aea of tell u the aea of the tiangle, and the ign give the oientation (left-tuning o ight-tuning) Can the aea be zeo? A( ) i twice the aea of the aallelogam defined by vecto and w w = Recall: in 2-dimenion, the aea of the aallelogam defined by two vecto i the deteminant of a 2 2 matix whoe column ae the two vecto =

10 Signed aea of tiangle Comutational Geomety, Facultat d Infomàtica de Bacelona, UPC Signed aea of A( ) = 1 x 2 y w x w y w = =

11 Signed aea of tiangle Comutational Geomety, Facultat d Infomàtica de Bacelona, UPC Signed aea of A( ) = 1 x w x 2 y w y w = = 1 ( ) x ( ) x 2 ( ) y ( ) y = 1 x x x x = 2 y y y y = 1 x x x 2 y y y 1 1 1

12 Signed aea of tiangle Comutational Geomety, Facultat d Infomàtica de Bacelona, UPC Signed aea of A( ) = 1 2 = 1 2 = 1 2 = 1 2 x w x y w y ( ) x ( ) x ( ) y ( ) y x x x x y y y y x x x y y y Doe it tun ight o left? w = = Fact: (check it youelf!) A( ) > 0 if and only if,, ae in counte-clockwie ode

13 Moe thing you can do with thi tet Comutational Geomety, Facultat d Infomàtica de Bacelona, UPC Point with eect to a line I oint to the ight o left of the oiented line l? l

14 Moe thing you can do with thi tet Comutational Geomety, Facultat d Infomàtica de Bacelona, UPC Point with eect to a line I oint to the ight o left of the oiented line l? l Euivalent uetion: do,, tun ight o left?

15 Moe thing you can do with thi tet Comutational Geomety, Facultat d Infomàtica de Bacelona, UPC Point with eect to a line I oint to the ight o left of the oiented line l? l Euivalent uetion: do,, tun ight o left? Line-egment inteection tet l Doe egment inteect line l?

16 Moe thing you can do with thi tet Comutational Geomety, Facultat d Infomàtica de Bacelona, UPC Point with eect to a line I oint to the ight o left of the oiented line l? l Euivalent uetion: do,, tun ight o left? Line-egment inteection tet l Doe egment inteect line l? inteect line l and ae on ooite ide of l ( i to the left and i to the ight of l) OR ( i to the ight and i to the left of l)

17 Moe thing you can do with thi tet Comutational Geomety, Facultat d Infomàtica de Bacelona, UPC Halfline-egment inteection h Doe egment inteect halfline h?

18 Moe thing you can do with thi tet Comutational Geomety, Facultat d Infomàtica de Bacelona, UPC Halfline-egment inteection h Doe egment inteect halfline h? Euivalent uetion(): and ae on ooite ide of h (o we can aume i to the left of h)

19 Moe thing you can do with thi tet Comutational Geomety, Facultat d Infomàtica de Bacelona, UPC Halfline-egment inteection h Doe egment inteect halfline h? Euivalent uetion(): and ae on ooite ide of h (o we can aume i to the left of h) h

20 Moe thing you can do with thi tet Comutational Geomety, Facultat d Infomàtica de Bacelona, UPC Halfline-egment inteection h Doe egment inteect halfline h? Euivalent uetion(): and ae on ooite ide of h (o we can aume i to the left of h) h h h

21 Moe thing you can do with thi tet Comutational Geomety, Facultat d Infomàtica de Bacelona, UPC Halfline-egment inteection h Doe egment inteect halfline h? Euivalent uetion(): and ae on ooite ide of h (o we can aume i to the left of h),, make a ight tun h h h

22 Moe thing you can do with thi tet Comutational Geomety, Facultat d Infomàtica de Bacelona, UPC Segment-egment inteection Doe egment inteect egment?

23 Moe thing you can do with thi tet Comutational Geomety, Facultat d Infomàtica de Bacelona, UPC Segment-egment inteection Doe egment inteect egment?

24 Moe thing you can do with thi tet Comutational Geomety, Facultat d Infomàtica de Bacelona, UPC Segment-egment inteection Doe egment inteect egment? Euivalent uetion(): and ae on ooite ide of

25 Moe thing you can do with thi tet Comutational Geomety, Facultat d Infomàtica de Bacelona, UPC Segment-egment inteection Doe egment inteect egment? Euivalent uetion(): and ae on ooite ide of

26 Moe thing you can do with thi tet Comutational Geomety, Facultat d Infomàtica de Bacelona, UPC Segment-egment inteection Doe egment inteect egment? Euivalent uetion(): and ae on ooite ide of and ae on ooite ide of

27 Summay Comutational Geomety, Facultat d Infomàtica de Bacelona, UPC Thing we can do with the oientation tet Tet if,, make a ight tun Tet if a oint i to the left of a line Tet if a egment and a line inteect Tet if a egment and a halfline inteect Tet if two egment inteect Tet if a oint i inide a tiangle All by evaluating the ign of a coule of deteminant!

28 Summay Comutational Geomety, Facultat d Infomàtica de Bacelona, UPC Thing we can do with the oientation tet Tet if,, make a ight tun Tet if a oint i to the left of a line Tet if a egment and a line inteect Tet if a egment and a halfline inteect Tet if two egment inteect Tet if a oint i inide a tiangle All by evaluating the ign of a coule of deteminant! Anothe ueful tet: oint in cicle See note on coue webage

29 Summay Thing we can do with the oientation tet Tet if,, make a ight tun Tet if a oint i to the left of a line Tet if a egment and a line inteect Tet if a egment and a halfline inteect Tet if two egment inteect Tet if a oint i inide a tiangle All by evaluating the ign of a coule of deteminant! Anothe ueful tet: oint in cicle See note on coue webage Quetion? Comutational Geomety, Facultat d Infomàtica de Bacelona, UPC