CMSC 425: Lecture 5 More on Geometry and Geometric Programming

Transcription

1 CMSC 425: Lectue 5 Moe on Geomety and Geometic Pogamming Moe Geometic Pogamming: In this lectue we continue the discussion of basic geometic ogamming fom the eious lectue. We will discuss coodinate systems fo affine and Euclidean geomety, coss-oduct and oientation testing, and affine tansfomations. Local and Global Fames of Refeence: Last time we intoduced the basic elements of affine and Euclidean geomety: oints and (fee) ectos. Howee, as of yet we hae no mechanism fo eesenting these objects. Recall that oints ae to be thought of as locations in sace and (fee) ectos eesent diection and magnitude, but ae not tied down to a aticula location in sace. We seek a fame of efeence fom which to descibe ectos and oints. This is called a coodinate fame. Thee is a global coodinate fame (also called the wold fame) fom which all geometic objects ae descibed. It is conenient in geometic ogamming to define aious local fames as well. Fo examle, suose we hae a ehicle diing aound a city. We might attach a local fame to this ehicle in ode to descibe the elatie ositions of objects and chaactes within the ehicle. The osition of the ehicle itself is then descibed elatie to the global fame. This aises the uestion of how to conet between the local coodinates used to define objects within the ehicle to thei global coodinates. Bases, Vectos, and Coodinates: The fist uestion is how to eesent oints and ectos in affine sace. We will begin by ecalling how to do this in linea algeba, and genealize fom thee. We know fom linea algeba that if we hae 2-linealy indeendent ectos, u 0 and u 1 in 2-sace, then we can eesent any othe ecto in 2-sace uniuely as a linea combination of these two ectos (see Fig. 1(a)): fo some choice of scalas α 0, α 1. = α 0 u 0 + α 1 u 1, u 1 u 0 u 1 = 2u 0 + 3u 1 [F ] = (2, 3) u 0 e 1 y e 0 x w = 3e 0 + 2e 1 w [F ] = (3, 2) (a) (b) Fig. 1: Bases and linea combinations in linea algeba (a) and the standad basis (b). Thus, gien any such ectos, we can use them to eesent any ecto in tems of a ai of scalas (α 0, α 1 ). In geneal d linealy indeendent ectos in dimension d is called a basis. The most conenient basis to wok with consists of two ectos, each of unit length, that ae othogonal to each othe. Such a collection of ectos is said to be othonomal. The standad basis consisting of the x- and y-unit ectos is an examle of such a basis (see Fig. 1(b)). Lectue 5 1 Sing 2018

2 Note that we ae using the tem ecto in two diffeent senses hee, one as a geometic entity and the othe as a seuence of numbes, gien in the fom of a ow o column. The fist is the object of inteest (i.e., the abstact data tye, in comute science teminology), and the latte is a eesentation. As is common in object oiented ogamming, we should think in tems of the abstact object, een though in ou ogamming we will hae to get dity and wok with the eesentation itself. Coodinate Fames and Coodinates: Now let us tun fom linea algeba to affine geomety. Again, let us conside just 2-dimensional sace. To define a coodinate fame fo an affine sace we would like to find some way to eesent any object (oint o ecto) as a seuence of scalas. Thus, it seems natual to genealize the notion of a basis in linea algeba to define a basis in affine sace. Note that fee ectos alone ae not enough to define a oint (since we cannot define a oint by any combination of ecto oeations). To secify osition, we will designate an abitay oint, denoted O, to see as the oigin of ou coodinate fame. Let u 0 and u 1 be a ai of linealy indeendent ectos. We aleady know that we can eesent any ecto uniuely as a linea combination of these two basis ectos. We can eesent any oint by adding a ecto to O (in aticula, the ecto O). It follows that we can eesent any oint in the following fom: = α 0 u 0 + α 1 u 1 + O, fo some ai of scalas α 0 and α 1. This suggests the following definition. Definition: A coodinate fame fo a d-dimensional affine sace consists of a oint (which we will denote O), called the oigin of the fame, and a set of d linealy indeendent basis ectos. Gien the aboe definition, we now hae a conenient way to exess both oints and ectos. As with linea algeba, the most natual tye of basis is othonomal. Gien an othonomal basis consisting of oigin O and unit ectos e 0 and e 1, we can exess any oint and any ecto as: = α 0 e 0 + α 1 e 1 + O and = β 0 e 0 + β 1 e 1 fo scalas α 0, α 1, β 0, and β 1. In ode to conet this into a coodinate system, let us entetain the following notational conention. Define 1 O = O and 0 O = 0 (the zeo ecto). Note that these two exessions ae blatantly illegal by the ules of affine geomety, but this conention makes it ossible to exess the aboe euations in a common (homogeneous) fom (see Fig. 2): = α 0 e 0 + α 1 e O and = β 0 e 0 + β 1 e O. This suggests a nice method fo exessing both oints and ectos using a common notation. Fo the gien coodinate fame F = ( e 0, e 1, O) we can exess the oint and the ecto as (see Fig. 2). [F ] = (α 0, α 1, 1) and [F ] = (β 0, β 1, 0) Lectue 5 2 Sing 2018

3 = 3 e e O [F ] = (3, 2, 1) O e 1 e 0 = 2 e e O [F ] = (2, 1, 0) Fig. 2: Coodinate fames and (affine) homogeneous coodinates. These ae called (affine) homogeneous coodinates. In summay, to eesent oints and ectos in d-sace, we will use coodinate ectos of length d+1. Points hae a last coodinate 1 of 1, and ectos hae a last coodinate of 0. Poeties of homogeneous coodinates: The choice of aending a 1 fo oints and a 0 fo ectos may seem to be a athe abitay choice. Why not just eese them o use some othe scala alues? The eason is that this aticula choice has a numbe of nice oeties with esect to geometic oeations. Fo examle, conside two oints and whose coodinate eesentations elatie to some fame F ae [F ] = (1, 4, 1) and [F ] = (4, 3, 1), esectiely (see Fig. 3). Conside the ecto =. If we aly the diffeence ule that we defined last time fo oints, and then conet this ecto into it coodinates elatie to fame F, we find that [F ] = ( 3, 1, 0). Thus, to comute the coodinates of we simly take the comonent-wise diffeence of the coodinate ectos fo and. The 1-comonents nicely cancel out, to gie a ecto esult. [F ] = (1, 4, 1) [F ] = (4, 3, 1) O e 1 e 0 [F ] = (1 4, 4 3, 1 1) Fig. 3: Coodinate aithmetic. = ( 3, 1, 0) In geneal, a nice featue of this eesentation is the last coodinate behaes exactly as it should. Let u and be eithe oints o ectos. Afte a numbe of oeations of the foms u + o u o αu (when alied to the coodinates) we hae: If the last coodinate is 1, then the esult is a oint. 1 Some conentions lace the homogenizing coodinate fist athe than last. Thee ae actually good easons fo doing this, as we will see below in ou discussion of oientation testing. But we will stick with standad engineeing conentions and lace it last. Lectue 5 3 Sing 2018

4 If the last coodinate is 0, then the esult is a ecto. Othewise, this is not a legal affine oeation. This fact can be oed igoously, but we won t woy about doing so. Coss Poduct: The coss oduct is an imotant ecto oeation in 3-sace. You ae gien two ectos and you want to find a thid ecto that is othogonal to these two. This is handy in constucting coodinate fames with othogonal bases. Thee is a nice oeato in 3-sace, which does this fo us, called the coss oduct. The coss oduct is usually defined in standad linea 3-sace (since it alies to ectos, not oints). So we will ignoe the homogeneous coodinate hee. Gien two ectos in 3-sace, u and, thei coss oduct is defined as follows (see Fig. 4(a)): u = u y z u z y u z x u x z u x y u y x. u u u u = (u ) (a) (b) Fig. 4: Coss oduct. A nice mnemonic deice fo emembeing this fomula, is to exess it in tems of the following symbolic deteminant: e x e y e z u = u x u y u z x y z. Hee e x, e y, and e z ae the thee coodinate unit ectos fo the standad basis. Note that the coss oduct is only defined fo a ai of fee ectos and only in 3-sace. Futhemoe, we ignoe the homogeneous coodinate hee. The coss oduct has the following imotant oeties: Skew symmetic: u = ( u) (see Fig. 5(b)). It follows immediately that u u = 0 (since it is eual to its own negation). Nonassociatie: Unlike most othe oducts that aise in algeba, the coss oduct is not associatie. That is ( u ) w u ( w). Lectue 5 4 Sing 2018

5 Bilinea: The coss oduct is linea in both aguments. Fo examle: u (α ) = α( u ), u ( + w) = ( u ) + ( u w). Peendicula: If u and ae not linealy deendent, then u is eendicula to u and, and is diected accoding the ight-hand ule. Angle and Aea: The length of the coss oduct ecto is elated to the lengths of and angle between the ectos. In aticula: u = u sin θ, whee θ is the angle between u and. The coss oduct is usually not used fo comuting angles because the dot oduct can be used to comute the cosine of the angle (in any dimension) and it can be comuted moe efficiently. This length is also eual to the aea of the aallelogam whose sides ae gien by u and. This is often useful. The coss oduct is commonly used in comute gahics fo geneating coodinate fames. Gien two basis ectos fo a fame, it is useful to geneate a thid ecto that is othogonal to the fist two. The coss oduct does exactly this. It is also useful fo geneating suface nomals. Gien two tangent ectos fo a suface, the coss oduct geneate a ecto that is nomal to the suface. Oientation: Gien two eal numbes and, thee ae thee ossible ways they may be odeed: <, =, o >. We may define an oientation function, which takes on the alues +1, 0, o 1 in each of these cases. That is, O 1 (, ) = sign( ), whee sign(x) is eithe 1, 0, o +1 deending on whethe x is negatie, zeo, o ositie, esectiely. An inteesting uestion is whethe it is ossible to extend the notion of ode to highe dimensions. The answe is yes, but athe than comaing two oints, in geneal we can define the oientation of d + 1 oints in d-sace. We define the oientation to be the sign of the deteminant consisting of thei homogeneous coodinates (with the homogenizing coodinate gien fist). Fo examle, in the lane and 3-sace the oientation of thee oints,, is defined to be O 2 (,, ) = sign det x x x y y y, O 3 (,,, s) = sign det x x x s x y y y s y z z z s z What does oientation mean intuitiely? The oientation of thee oints in the lane is +1 if the tiangle P QR is oiented counte-clockwise, 1 if clockwise, and 0 if all thee oints ae collinea (see Fig. 5). In 3-sace, a ositie oientation means that the oints follow a ight-handed scew, if you isit the oints in the ode P QRS. A negatie oientation means a left-handed scew and zeo oientation means that the oints ae colana. Note that the ode of the aguments is significant. The oientation of (,, ) is the negation of the oientation of (,, ). As with deteminants, the swa of any two elements eeses the sign of the oientation. Lectue 5 5 Sing 2018.

6 O(,, ) = +1 O(,, ) = 0 O(,, ) = 1 s s O(,,, s) = 1 O(,,, s) = +1 Fig. 5: Oientations in 2 and 3 dimensions. You might ask why ut the homogeneous coodinate fist? The answe a mathematician would gie you is that is eally whee it should be in the fist lace. If you ut it last, then ositie oiented things ae ight-handed in een dimensions and left-handed in odd dimensions. By utting it fist, ositiely oiented things ae always ight-handed in oientation, which is moe elegant. Putting the homogeneous coodinate last seems to be a conention that aose in engineeing, and was adoted late by gahics eole. The alue of the deteminant itself is the aea of the aallelogam defined by the ectos and, and thus this deteminant is also handy fo comuting aeas and olumes. Late we will discuss othe methods. Oientation testing is a ey useful tool, but it is (suisingly) not ey widely known in the aeas of comute game ogamming and comute gahics. Fo examle, suose that we hae a bullet ath, eesented by a line segment. We want to know whethe the linea extension of this segment intesects a taget tiangle, abc. We can detemine this using thee oientation tests. To see the connection, conside the thee diected edges of the tiangle ab bc and ca. Suose that we lace an obsee along each of these edges, facing the diection of the edge. If the line asses though the tiangle, then all thee obsees will see the diected line assing in the same diection elatie to thei edge (see Fig. 6). (This might take a bit of time to conince youself of this. To make it easie, imagine that the tiangle is on the floo with a, b, and c gien in counteclockwise ode, and the line is etical with below the floo and aboe. The line hits the tiangle if and only if all thee obsees, when facing the diection of thei esectie edges, see the line on thei left. If we eese the oles of and, they will all see the line as being on thei ight. In any case, they all agee.) b a c Fig. 6: Using oientation testing to detemine line-tiangle intesection. It follows that the line asses though the tiangle if and only if O 3 (,, a, b) = O 3 (,, b, c) = O 3 (,, c, d). Lectue 5 6 Sing 2018

7 (By the way, this tests only whethe the infinite line intesects the tiangle. To detemine whethe the segment intesects the tiangle, we should also check that and lie on oosite sides of the tiangle. Can you see how to do this with two additional oientation tests?) Lectue 5 7 Sing 2018