Homogeneous Coordinates

Transcription

1 Homogeneous Coordinates Com S 477/77 Notes Yan-Bin Jia Aug, 07 Introduction Geometr lies at the core of computer graphics, computer-aided design, computer vision, robotics, geographic information sstems, etc This course begins with projective geometr b describing how points and lines can be represented b Cartesian and homogeneous coordinates We will introduce planar and spatial transformations to construct objects from geometric primitives, and to manipulate existing objects Then we will stud projections and look at how to render three-dimensional D objects on a computer screen This will be followed b an introduction to quaternions which constitute a ver powerful tool in dealing with rotations Later on in the course we will stud the geometr of curves and surfaces In graphics applications, geometric objects are defined in terms of a number of building blocks called graphical primitives These primitives ma correspond to points, lines, curves, and surfaces For example, a rectangle can be defined b its four sides or four vertices Each side is constructed from a line segment b appling a number of geometric operations, called transformations, which position, orientate, or scale the line primitives Five tpes of transformation are particularl relevant in applications, namel, translations, scalings, reflections, rotations, and shears Planar Transformations A linear transformation of the plane is a mapping L : R R from the plane to itself such that x x A +b, where a a A a a and b A linear transformation is also called an affine mapping or affine transformation b b Appendices are optional for reading unless specificall required

2 v v v 4 v v v v v 4 Figure : Affine transformation applied to a square of side Lemma The transformation given b maps the line cx+d +e 0, where c 0 or d 0, to the line a c a dx+a d a c + a b a b c a b a b d+a a a a e 0 If a d a c 0 and a c a d 0, then a a a a 0 and ever point on the original line is mapped to the point b d a e/d,b d a e/d T Proof Use the line s parametric form to find the image of an arbitrar point on the original line Then convert the obtained parametric coordinates of the image into an implicit equation Besides collinearit, affine transformation also preserves ratios of distances [, p 6], for instance, the midpoint of a line segment remains the midpoint after the transformation Example Consider the affine mapping with A 0 and b 4 A square with vertices v, v is mapped to a parallelogram with vertices v, v, v, v, and v 4, and v 4 As shown in Figure, the square becomes a parallelogram no longer centered at the origin The vertices are ordered clockwise b index in the image as opposed to counterclockwise in the square Translation A translation is an affine transformation with the matrix A I That is, a transformation maps ever point p to a new point p b adding a constant vector b b b It has the effect of

3 Figure : -gon before and after the translation T4, moving the point in the direction of the x-axis b b units, and in the direction of the -axis b b units We denote the translation b Transb,b The transformation that maps p back to p is the inverse translation T Trans b, b Scaling A scaling about the origin is an affine transformation where the matrix A diags x,s with s x 0 and s 0, and b 0 This transformation, denoted b Scales x,s, maps a point b multipling its x and coordinates b factors s x and s, respectivel Here s s x +s is the scaling factor The scaling is said to be an enlargement if s >, and a contraction if s < It is said to be uniform if s x s Scaling can be performed b a matrix multiplication We abuse the notation b letting x sx 0 0 s x sx 0 Scales x,s 0 s This matrix is called the scaling transformation matrix Figure : The same -gon in Figure after scaling S, Reflection Two common effects in CAD or computer drawing packages are the horizontal or vertical flip or mirror effects A flip of an object is obtained b appling a transformation known as reflection

4 l p q p Figure 4: Reflection p of a point p about a line l Figure 4 shows a fixed line l in the plane and a point p To determine the reflected image of p, move from p toward l in the direction normal to the line Let q be the intersection of the movement with l So q is the projection of p onto l and d p q gives the shortest distance from p to l A continuing movement from q for another distance of d will reach p, the reflection of p It is eas to verif that the reflection Ref x in the x-axis is the transformation x x, and the reflection Ref in the -axis is the transformation x x These two transformations can be denoted b matrices Ref x Ref Reflections in arbitrar lines can also be denoted b matrices We will derive such matrices after the introduction of homogeneous coordinates 4 Rotation about the Origin A rotation about the origin through an angle θ maps ever point p x to a point p x such that p and p are at the same distance from the origin and the angle from the vector p to the vector p is θ See Figure, p θ φ p x Figure : Rotation about the origin To determine the coordinates of the image point p, it is ver convenient for us to use polar coordinates Let x rcosφ,rsinφ T, where r is the distance from p to the origin and φ the 4

5 Figure 6: Rotation of a -gon about the origin b 0 degrees polar angle Then we have x rcosθ +φ rcosθcosφ rsinθsinφ xcosθ sinθ, rsinθ +φ rsinθcosφ+rcosθsinφ xsinθ +cosθ More succinctl, the coordinates of p can be obtained from those of p through a matrix multiplication: x p cosθ sinθ x sinθ cosθ The orthogonal matrix cosθ sinθ Rotθ sinθ cosθ is called the rotation matrix The inverse transform is the transpose Rotθ T which rotates vectors back through θ See Figure for an example Example Consider a planar R robot manipulator see Figure 7 consisting of two rigid links The first link is attached to the base b a revolute joint J which permits the link to rotate about the joint The second link is attached to the first link b another revolute joint J The robot s end effector is attached to the second link The pose ie, position and orientation of the end effector is controlled b exerting internal torques to turn the links about the two joints We set up a world x,-coordinate sstem with the origin at J The second link has its own u,v- coordinate sstem with J as the origin Let d be the distance between J and J, θ be the rotation angle from the x-axis to the axis of the first link, θ be the rotation angle from the axis of the first link to that of the second link this angle as shown in the figure is negative The pose of the second link is obtained b appling a rotation Rotθ +θ followedb a translationtransdcosθ,dsinθ Given the u,v coordinate of a point p with respect to the second link, the x, coordinates of p in the world coordinate sstem is obtained b the transformation x cosθ +θ sinθ +θ sinθ +θ cosθ +θ A square matrix Q is orthogonal if QQ T Q T Q I u + v dcosθ dsinθ

6 v p θ u end effector J J base θ x Figure 7: R robot manipulator ucosθ +θ vsinθ +θ +dcosθ usinθ +θ +vcosθ +θ +dsinθ When the joint angles are known, the world coordinates of an object can be determined from its local coordinates with respect to the robot s end effector The calculation is referred to as the forward kinematics of the robot manipulator We can generalize the above result to a robot manipulator with n revolute joints The aim is to express the concatenations of all rotations and translations associated with the joints with one matrix multiplication This will be possible with the assistance of homogeneous coordinates Shear Let a fixed direction be represented b the unit vector v v x v A shear about the origin of factor r in the direction v maps a point p to the point p p+drv, where d is the signed distance from the origin to the line through p in the direction v Suppose p x The unit vector normal to the line is n v v x Note that we choose the normal vector such that v n Therefore the distance d is given as d p n v x xv The shear transformation then maps p to p x+rvx v xv x p+drv +rv x v xv Thus the shear transformation matrix is rvx v Shearv,r rvx rv +rv x v In particular, a shear along the x-axis has v 0 and thus Shear 0,r 6 r 0

7 v Figure 8: Shearing in v, b a factor r The portion of the -gon to the left of v is extended along the direction v while the portion to the right of the vector is pulled back in the direction v Example 4 The shear in the direction v, with a factor r Shear, T, Appling the shear to the -gon in Example, we have has transformation matrix Consecutive Transformations In man applications it is desirable to appl more than one transformations to an object In robotics, for instance, an object has often undergone both translation and rotation after being manipulated In vision, the image of an object results from a projection of its model after some translation and rotation It would be nice to concatenate all transformations into one equivalent transformation for the convenience of computation Example A point p x, undergoes a rotation about the origin through an angle π The resulting point p has coordinates x cos π x sin π sin π cos π x A projection is more general than affine transformation 7

8 Next, appl to p a shear about the origin of factor in the direction of the unit vector, The new point p has coordinates x x + + x + x + x The concatenated transformation from p to p is thus represented b the matrix + A problem is encountered when translations are involved We would need to combine a matrix addition for the translation with a matrix multiplication for the other transformations This is ver awkward There is a remed, though, with the introduction of homogeneous coordinates Then all transformations will be represented b matrices, and performed b matrix multiplications And concatenation of transformations will be represented b the matrix product of the transformation matrices Furthermore, an inverse transformation, which maps ever image point back to its original position, will be obtained b taking a matrix inverse Recall that an affine transformation maps the point p x to p x Ap+b To represent the mapping as a matrix multiplication, we introduce the homogeneous coordinates x,, such that x Ap+b A b 0 We can easil verif that the approach of homogeneous coordinates also works for the other linear transformations we have learned so far: scaling, rotation, reflection, and shear All we need is to let b 0 in the new transformation matrix The problem is solved! But there is more about homogeneous coordinates we ought to know 4 Definition of Homogeneous Coordinates To formall introduce homogeneous coordinates, let us first recall that a relation on a set S is a subset of S S such that u is related to v whenever u,v is in the subset We also write x 8

9 u v when u and v are related A relation is reflexive if u u for all u S; it is smmetric if u v whenever v u; it is transitive if u w whenever u v and v w The relation is an equivalence if it is reflexive, smmetric, and transitive The relation > on R is transitive, but not reflexive or smmetric The relations and are both reflexive and transitive, but not smmetric The most familiar equivalence relation on R is An equivalence relation on the set Z of integers is congruence modulo integer m > 0 Let be an equivalence relation on S The subset of S consisting of all elements related to an element s is the equivalent class of s and denoted as [s] For example, the congruence mod4 induces four equivalence classes [0],[],[], and [], where [i] {i+4k k integer} for i 0,,, Let us now focus on the relation on the set S R \{0,0,0} defined below: x,,z u,v,w iff x,,z ru,v,w for some r 0 It is eas to show that the relation is reflexive, smmetric, and transitive Hence it is an equivalence relation The equivalence class of x,,z is the set [x,,z] {rx,,z r R and r 0} Homogeneous coordinates are equivalence classes of the relation defined b The coordinates x,,z is identified with rx,,z with r 0 The projective plane P is defined to be the set of all equivalence classes, that is, {[x,,z] x 0, 0, orz 0} An equivalence class is referred to as a point in the projective plane Operations of the projective plane are carried out b taking a representative from each equivalence class For a class [u,v,w] with w 0, we use u/w,v/w, as the representative So there is a one-to-one correspondence between points x, of the Cartesian plane and points [u, v, w], w 0 in the projective plane with w 0 Points at Infinit Homogeneous coordinates of the form x,,0 do not correspond to a point in the Cartesian plane Instead, the correspond to the unique point at infinit in the direction x, To see this, consider the line through a point, sa, a,b, and with direction x, It has the parametric form a+tx,b+ t Ever point on the line thus has homogeneous coordinates a+tx,b+t,, and equivalentl, a t +x, b t +, t As t tends to infinit, that is, as a point moves along the line to infinit, the latter homogeneous coordinates become x,,0 Moving in the directions of x, and x, will end up at the same infinit point represented b x,,0 as well as x,,0 Hence the projective plane P can be seen as the plane R plus all the points at infinit, each of which along a different direction The plane P also makes sense of the notion that two parallel lines intersect at infinit, as we will see in the example below Example Consider the parallel lines x + and x + Let u,v,w be the homogeneous coordinates of a point x, on the first line Then x u w and v w We have u w + v w and thus the homogeneous equation of the line is u+v w Similarl, the second line has the homogeneous equation u+v w 4 9

10 Equations and 4 have solutions of the form r,r,0 That is, the solutions are all homogeneous coordinates of the point,,0, which is the unique intersection of the two parallel lines in the direction, Similarl, we conclude that all lines parallel to x+ intersect in a unique point at infinit in the direction, 6 Point and Line Geometr in Homogeneous Coordinates We have seen that a point x, in the Cartesian plane has homogeneous coordinates tx,,, t 0 These coordinates would correspond to a line through the origin excluded if the were Cartesian coordinates in the -dimensional space When homogeneous coordinates are viewed as Cartesian coordinates, the dimensions of the geometric object the describe increase b Thegeometr of alineinthecartesian planeis reviewed inappendixa Ithas general equation ax+b +c 0 Suppose u,v,w are the homogeneous coordinates of a point x, on the line; hence x u/w and v/w Substituting for x and in the line equation and multipling through b w, ields the conditions for u,v,w to be the homogeneous coordinates of a point on the line: au+bv +cw 0 Equation is known as the homogeneous line equation The line is uniquel specified b the coefficients a,b, and c, or an multiple ra,rb, and rc with r 0 Therefore it is natural to specif the line b the homogeneous coordinates l a,b,c Since an non-zero multiple of l defines the same line, it is useful to consider l as a vector of which onl the direction matters Let p u, v, w be a point in homogeneous coordinates Then in order for p to lie on the line, the dot product of p and l must vanish, that is, p l 0 6 The identit 6 allows us to easil determine the line through two distinct points as well as the point of intersection of two lines Suppose l is the vector that represents a line through two distinct points p and p, all in homogeneous coordinates Then from 6 we have p l 0 and p l 0 Thus l is perpendicular or orthogonal to both p and p To determine l it suffices to find a vector perpendicular to p and p since onl the direction matters We choose the cross product b letting l p p or an multiple of p p The equation of the line through two points can be determined b taking the cross product of their homogeneous coordinates Example 4 The line l passing through, and 4, satisfies equations l,, 0, l 4,, 0 In Cartesian coordinates the same equation would describe a plane through the origin and with normal a,b,c 0

11 Hence we have the line in homogeneous coordinates: l,, 4,, 4, 7,9, which give the line 4x We can verif this equation using the original points, and 4, Next, suppose p is the intersection of two lines l and l all in homogeneous coordinates Then from 6 we have l p 0 and l p 0 Inotherwords, pisorthogonal tobothl andl whenall areseenasvectors Henceit issufficientto take p l l or an multiple of it as the homogeneous coordinates of the point of intersection Example The intersection point p of the lines x and x satisfies Hence And the two lines intersects at the point, 7,8 p 0 and, 4, p 0 p, 7,8, 4,,,7 7, 7 in the Cartesian plane Example 6 The two parallel lines x 0 and x do not intersect in the Cartesian plane In homogeneous coordinates, their intersection point p is,,0,,, 6,0, which is at infinit 7 Projective Space Homogeneous coordinates of the three-dimensional D space R are derived in a similar manner as those of the plane A point x,,z in R is represented b the vector x,,z,, or b an multiple rx,r,rz,r with r 0 Conversel, the homogeneous coordinates s,u,v,w with w 0 represent the point x,,z s/w,u/w,v/w in the D space A point of the form s,u,v,0 corresponds to the point at infinit in the D space in the direction of the vector s,u,v The set of all homogeneous coordinates s, u, v, w is called the three-dimensional projective space and denoted P Example The homogeneous coordinates,,,,,,,,,,, represent the same point,, in R In Cartesian ie, Euclidean coordinates a plane is described b an equation of the form ax + b + cz + d 0 To obtain the corresponding equation in homogeneous coordinates, we substitute x s/w, u/w, and z v/w in the equation and multipl both sides b w, ielding as+bu+cv +dw 0 7 Recall that a line in the plane is represented b a line vector in homogeneous coordinates Similarl, a plane in the space is specified b a plane vector n a,b,c,d Thus from 7 a point p in homogeneous coordinates s,u,v,w lies on a plane with plane vector n if and onl if p n 0

12 7 Plane through Three Distinct Points We have learned that the line through two points in the plane can be obtained b carring out a cross product The analogous problem in space is to determine the unique plane, represented b n, that passes through three distinct points p i s i,u i,v i,w i, i,, The plane vector n satisfies n p 0, n p 0, n p 0 Namel, n is perpendicular to three vectors p, p, and p The condition for this to occur is that e e e e 4 n k s u v w s u v w, s u v w where k 0, e,0,0,0, e 0,,0,0, e 0,0,,0, and e 4 0,0,0, For convenience, we choose k Thedeterminant above is a vector determinant in the sensethat all e i s are treated as scalars in the expansion into a summation Accordingl, we can verif that n p i 0 e p i e p i e p i e 4 p i s u v w s u v w s u v w s i u i v i w i s u v w s u v w s u v w Example The plane through the points,4,,,7,, and,,9 is described b the line vector e e e e e e 9 +e 4 7 e e +9e +4e 8e Intersection of Three Planes Analogousl, the point of intersection p of three planes respectivel determined b vectors n, n, and n satisfies n p 0, n p 0, n p 0

13 Denote n i s i,u i,v i,w i Hence the homogeneous coordinates of p are given as e e e e 4 s u v w s u v w s u v w or an multiple of the above vector determinant Example The point of intersection of the three planes x+ +z, 7x 4z, +z +8 0 is obtained b computing the determinant e e e e e +7e 4e +7e Namel, the intersection has homogeneous coordinates 99, 7, 4, 7 and Cartesian coordinates 99 7, 7 7, 4 7 A Line in the Plane Before getting into transformations right awa, let us first get warmed up with some basic line geometr Recall the general equation of a line: ax+b +c 0, where a 0 or b 0 8 The equation above is the implicit form of the line We normalize the coefficients and obtain a a +b x+ b a +b + c 0 9 a +b a For convenience, let us introduce the unit vector n a +b, b a +b T Now, move the term involving c to the right hand side of equation 9 and rewrite the remaining terms on the left hand side as a dot product: x c n 0 a +b For an two points x and x on the line, we have that n x x 0 Thus, the unit vector n is perpendicular to the line Equation 0 states that the distance to the c line from the origin is a +b The vector n points toward the line when c < 0 and awa from the line when c > 0 We easil see that the vector a b, just like n, is perpendicular to the line while its orthogonal vector b a is parallel to the line

14 ax + b + c 0 b a n p Figure 9: A line The line through a point p p x p in the direction of the vector v vx v can also be defined parametricall as xt p+tv t px +tv x p +tv From this parametric or explicit form, we immediatel derive the implicit form of the line b eliminating t from x p x +v x t and p +v t: v x v x +p v x p x v 0 Conversel, given the implicit form 8, we ma set v b a parallel to the line in deriving the parametric form A point on the line can be chosen b setting x 0 in case b 0 or 0 otherwise and determining or x, respectivel Example Consider two lines a x+b +c 0 and a x+b +c 0 Their directions are v b,a and w b,a, respectivel Let θ be the angle between the two lines, more specificall, from v to w Then, the identities v w v w cosθ and v w v w sinθ give rise to cosθ sinθ a a +b b a +b a, +b a b a b a +b a +b Hence tanθ a b a b a a +b b The two lines are parallel if and onl if θ 0, that is, if and onl if a b a b B Visualization of the Projective Plane Two models exist for us to visualize the projective plane and understand homogeneous coordinates geometricall The are the line model and the spherical model 4

15 The line model represents the point with homogeneous coordinates λu,v,w, λ 0 b the line through the origin with direction u, v, w A one-to-one correspondence exists between the points x, in the plane R and the lines parametrized as tx,,, t R Another one-to-one correspondence exists between the point x, and the point x,, in the plane w w x,, w u v Figure 0: The line model of the projective plane But points at infinit, which have homogeneous coordinates of the form x,,0, are not on the w plane Instead, the correspond to lines in the w 0 plane Lines in the Cartesian plane R correspond to planes in the u-v-w space This is the difficult with the line model Example The two parallel lines x+ and x+ in the previous example correspond to the planes in the u-v-w space given b equations u+v w, u+v w The planes intersect in a line t,t,0 through the origin in the w 0 plane This line corresponds to the point at infinit,,0, which is intersection of the two parallel lines w u + v w v u + v w u t, t, 0 Figure : Intersection of plane images of parallel lines in the Cartesian plane In the spherical model of the projective plane, a point with homogeneous coordinates x,, z maps to the point of the intersection of the correspondingline tx,,z and the unit spherecentered

16 at the origin x + +z In other words, x,,z x,,z ± x + +z The two image points are antipodal Since the two antipodal points correspond to the same point [x,,z] in the projective plane, it suffices to consider the upper half sphere together with half of the equator w tx,, z u v Figure : Spherical model of the projective plane Antipodal points represent the same homogeneous point Ever point at infinit in the Cartesian plane R has homogeneous coordinates x,,0 B it corresponds to two antipodal points on the equator Thus the sphere or half sphere provides a wa of visualizing all homogeneous coordinates A line in the plane R corresponds to a great circle which is the intersection of the sphere with the plane containing the origin and the line elevated to the z plane The intersection of two parallel lines correspond to the intersection points of the two great circles on the sphere, namel, two antipodal points on the equator which represent two points at infinit in the direction of these lines References [] D Marsh Applied Geometr for Computer Graphics and CAD Springer-Verlag, 999 [] E W Weisstein CRC Concise Encclopedia of Mathematics, nd ed Chapman & Hall/CRC, 00 6