be ye transformed by the renewing of your mind Romans 12:2

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1 Lecture 12: Coordinate Free Formulas for Affine and rojectie Transformations be ye transformed by the reing of your mind Romans 12:2 1. Transformations for 3-Dimensional Computer Graphics Computer Graphics in 2-dimensions is grounded on simple transformations of the plane. Turtle Graphics and LOGO are based on conformal transformations; Affine Graphics and CODO on affine transformations. The turtle uses translation, rotation, and uniform scaling to naigate on the plane, and traces of the turtle s path generate a wide ariety of shapes, ranging from simple polygons to complex fractals. Affine Graphics uses affine transformations to position, orient, and scale objects as well as to model shape. The ertices of simple figures like polygons and stars are typically positioned using translation and rotation; ellipses are generated by scaling circles nonuniformly along preferred directions; fractals are built using iterated function systems. Computer Graphics in 3-dimensions is grounded in transformations of 3-space. But in 3- dimensions Computer Graphics employs not only nonsingular conformal and affine transformations, but also affine projections and projectie transformations. rojections play a key role in 3-dimensional Computer Graphics because we need to display 3-dimensional geometry on a 2-dimensional screen. Orthogonal projections of 3-dimensional models from seeral different directions are traditional in many engineering applications. erspectie projection plays a central role in Computer Graphics because, as in classical art, we often want to display 3-dimensional shapes in perspectie just as they appear to the human eye. Visual realism requires projectie transformations. Here we are going to study 3-dimensional affine and projectie transformation, using the coordinate free techniques of ector geometry. 2. Affine and rojectie Transformations An affine transformation is a transformation that maps points to points, maps ectors to ectors, and preseres linear and affine combinations. In particular, A is an affine transformation if A(u + ) = A(u) + A() A(c) = c A() A( + ) = A() + A() As a consequence of Equation (1), non-degenerate affine transformations in 3-dimensions map lines to lines, triangles to triangles, and parallelograms to parallelograms just as in 2-dimensions. The most important affine transformations in 3-dimensional Computer Graphics are the rigid (1)

2 motions (translation, rotation, and mirror image), scaling (both uniform and non-uniform), and orthogonal projection. erspectie projection is not an affine transformation. Indeed, as we shall see shortly, perspectie projection is not defined at eery point nor is perspectie well-defined for any ector. Neertheless, in Section 5.2 we will derie a simple, straightforward formula for computing perspectie projection on any point where perspectie is well-defined. In this lecture we are going to derie explicit formulas for each of the most important affine and projectie transformations in 3-dimensional Computer Graphics -- translation, rotation, mirror image, uniform and non-uniform scaling, and orthogonal and perspectie projection -- using coordinate free ector algebra. In the next lecture, we will apply these formulas to construct matrices to represent each of these transformations. Before we proceed, we need to recall the following results. In most of our deriations we shall be required to decompose an arbitrary ector into components and parallel and perpendicular to a fixed unit ector u (see Figure 1). Recall from Lecture 9 that = + = ( u)u = ( u)u Be sure that you are comfortable with these formulas before you continue because we shall use these equations again and again throughout this lecture. (2) Figure 1: Decomposing a ector into components parallel and perpendicular to a fixed ector u. u 3. Rigid Motions Rigid motions are transformations that presere lengths and angles. The rigid motions in 3- dimensions are composites of three basic transformations: translation, rotation, and mirror image. 3.1 Translation. Translation is defined by specifying a distance and a direction. Since a distance and direction designate a ector, a translation is defined by specifying a translation ector. The 2

3 formulas for translation in 3-dimensions are similar to the formulas for translation in 2-dimensions: points are affected by translation, ector are unaffected by translation (see Figure 2). Let w be a translation ector. If is a point and is a ector, then = = + w w Figure 2: Translation. oints are affected by translation; ectors are not affected by translation. w (3) 3.2 Rotation. Rotation in 3-dimensions is defined by specifying an angle and an axis of rotation. The axis line L is typically described by specifying a point Q on L and a unit direction ector u parallel to L. We shall denote the angle of rotation by θ (see Figure 3). u θ Q L Figure 3: Rotation around the axis line L through the angle θ. The Formula of Rodrigues computes rotation around the axis L through the angle θ by setting = (cosθ) + (1 cosθ)( u)u + (sinθ)u (4) = Q+ (cosθ)( Q)+ (1 cosθ)(( Q) u)u + (sinθ)u ( Q) Notice that since = Q + ( Q) it follows that = Q + ( Q) = Q + ( Q). so the rotation formula for points follows immediately from the rotation formula for ectors. Therefore it is enough to proe Equation (4) for ectors. 3

4 To derie the Formula of Rodrigues for ectors, split into two components: the component parallel to u and the component perpendicular to u (see Figure 4, left). Then = + so by linearity = +. Since is parallel to the axis of rotation, =. Hence we need only compute the effect of rotation on. Now lies in the plane perpendicular to u; moreoer, u is perpendicular to and lies in the plane perpendicular u (see Figure 4, right). Therefore using our standard approach to rotation in the plane, we find that = (cosθ) + (sinθ)u. u u θ (sinθ)u (sinθ )u Q θ (cosθ ) L = + = (cosθ) + (sinθ)u Figure 4: Decomposing into components parallel and perpendicular to the axis of rotation (left), and rotation in the plane perpendicular to the axis of rotation (right). Now we hae = + = = (cosθ) + (sinθ)u. Therefore, = + (cosθ) + (sinθ)u. (*) 4

5 But since u is a unit ector, = ( u)u = = ( u)u. Substituting these two identities into (*) yields = ( u)u + (cosθ)( ( u)u) + (sinθ)u ( ( u)u) Thus, since cross product distributes through addition and u u = 0, = (cosθ) + (1 cosθ)( u)u + (sinθ)u, which is the Formula of Rodrigues for ectors. 3.3 Mirror Image. A mirror in 3-dimensions is a plane S, typically specified by a point Q on S and a unit ector n normal to S. (see Figure 5, left). n S Q S Q Figure 5: A mirror plane S specified by a point Q and a normal ector n (left), and the decomposition of a ector into components parallel and perpendicular to the normal ector (right). If is a point and is a ector, then the formulas for mirror image are = 2( n)n = + 2 (( Q) n)n Notice that since = Q + ( Q) it follows that = Q + ( Q) = Q + ( Q). so the formula for mirror image for points follows immediately from the formula for mirror image for ectors. Therefore once again it is enough to proe the mirror image formula for ectors. (5) 5

6 To derie the mirror image formula for ectors, we again split into two components: the component parallel to n and the component perpendicular to n (see Figure 5, right). Then = + so by linearity = + Since lies in the plane S, =. Moreoer, since is parallel to the normal ector n, Therefore, But =. = + =. (**) = = ( w)w. Substituting these identities into (**) yields = 2( n)n, which is the mirror image formula for ectors. 4. Scaling Both uniform and non-uniform scaling are important in 3-dimensional Computer Graphics. Below we treat each of these transformations in turn. 4.1 Uniform Scaling. Uniform scaling is defined by specifying a point Q from which to scale distance and a scale factor s. The formulas for uniform scaling in 3-dimensions are identical to the formulas for uniform scaling in 2-dimensions (see Figure 6): If is a point and is a ector, then = s = Q+ s( Q) = s + (1 s)q. (6) 6

7 Q { Figure 6: Uniform scaling for points and ectors. 4.2 Non-Uniform Scaling. For non-uniform scaling, we need to designate a unit ector w specifying the scaling direction, as well as a point Q and a scale factor s (see Figure 7, left). Q w Q w { Figure 7: Non-uniform scaling is specified around a point Q in a direction w (left). The decomposition of a ector into components parallel and perpendicular to the direction ector (right). If is a point and is a ector, then the formulas for non-uniform scaling are = + (s 1)( w)w = + (s 1)(( Q)w)w. (7) As usual non-uniform scaling for points follows easily from non-uniform scaling for ectors, since = Q + ( Q) so = Q + ( Q) = Q + ( Q). Therefore once again it is enough to proe the non-uniform scaling formula for ectors. To derie the non-uniform scaling formula for ectors, we again split into two components: the component parallel to w and the component perpendicular to w (see Figure 7, right). Then, as usual, 7

8 so by linearity = +, = +. Since is perpendicular to the scaling direction w, =, and since is parallel to the scaling direction w, Therefore, = s. = + = + s. (***) But once again = = ( w)w. Substituting these identities into (***) yields = + (s 1)( w)w, which is the non-uniform scaling formula for ectors. Notice, by the way, that mirror image is just non-uniform scaling along the normal direction to the mirror plane, with scale factor s = rojections To display 3-dimensional geometry on a 2-dimensional screen, we need to project from 3- dimensions to 2-dimensions. The two most important types of projections are orthogonal projection and perspectie projection. 5.1 Orthogonal rojection. The simplest projection is orthogonal projection (see Figure 8). n Q S Figure 8: Orthogonal projection. 8

9 The screen is represented by a plane S defined by a point Q and a unit normal ector n. If is a point and is a ector, then the formulas for orthogonal projection are = ( n)n = (( Q)n)n. (8) These formulas are easy to derie. The formula for ectors is immediate from the obseration that is just the perpendicular component of relatie to the normal ector n, so = = = ( n)n. Moreoer, for points, we simply obsere, as usual, that orthogonal projection for points follows easily from orthogonal projection for ectors. Indeed, since = Q + ( Q). it follows that = Q + ( Q) = Q + ( Q) = (( Q) n)n. 5.2 erspectie. erspectie projection is required in Computer Graphics to simulate isual realism. erspectie projection is defined by specifying an eye point E and a plane S into which to project the image. The plane S is specified, as usual, by a point Q on S and a unit ector n normal to S. The image of a point under perspectie projection from the eye point E to the plane S is the intersection of the line E with the plane S (see Figure 9, left). Notice that unlike all of the other transformations that we hae deeloped in this lecture, perspectie projection is not an affine transformation because perspectie projection is not a well defined transformation at eery point nor is perspectie well defined for any ector (see Figure 9, right). n Q E E u u S S Figure 9: erspectie projection from the eye point E to the point is defined by the intersection of the line E with the iewing plane S (left). Notice that perspectie projection is not well defined on points where the line E is parallel to the plane S nor is perspectie defined on any ector, since u = but u (right). 9

10 by The image of a point under perspectie projection from the eye point E to the plane S is gien ((E Q)n) + (Q ) n = ( )E (E ) n. (9) To derie this formula, we simply compute the intersection of the line E with the plane S. Since the point lies on the line E, there is a constant c such that = + c( E ). Moreoer, since the point also lies on the plane S, the ector Q must be perpendicular to the normal ector n; therefore ( Q) n = 0. Substituting the first equation for into the second equation for gies ( Q) n + c(e ) n = 0. Soling for c, we find that (Q ) n c = (E ) n. Therefore = + c( E ) = + (Q ) n (E ) n ((E Q) n) + (Q ) n (E ) = ( )E (E ) n. Notice that if the line E is parallel to the plane S, then the denominator (E ) n = 0. In this case the expression for is not defined, so perspectie projection is not well-defined for any point for which the line E is parallel to the plane S. This problem arises because unlike any of our preious transformation, the expression for has in the denominator, so perspectie projection is a rational function. We shall hae more to say about rational transformations in the next lecture. 6. Summary Below is a summary of the highlights of this lecture. Also listed below for your conenience are the coordinate free formulas from ector algebra for all the transformations that we hae studied in this lecture. 6.1 Affine and rojectie Transformations without Matrices. One of the main themes of this lecture is that we do not need matrices to compute the effects of affine and projectie transformations on points and ectors. Instead we can use ector algebra to derie coordinate free 10

11 expressions for all of the important affine and projectie transformations in 3-dimensional Computer Graphics. Moreoer, except for rotation, these formulas from ector algebra are actually more efficient than the corresponding matrix formulations. Neertheless, formulas from ector algebra are not conenient for composing transformations, whereas matrix representations are easily composed by matrix multiplication. Therefore, in the next lecture we shall use these formulas from ector algebra to derie matrix representations for each of the corresponding affine and projectie transformations. Our main strategy, used oer and oer again, for deriing explicit formulas for transformations on ectors is to decompose an arbitrary ector into components parallel and perpendicular to a particular direction ector in which is it especially easy to compute the transformation. We then analyze the transformation on each of these components and sum the transformed components to find the effect of the transformation on the original ector. This strategy works because most of the transformations that we hae inestigated are specified in terms of either a fixed line or a fixed plane: rotations reole around an axis line; projections collapse into a projection plane. Both lines and planes can be specified by a point Q and a unit ector u. For lines L, the ector u is parallel to L; for planes S, the ector u is normal to S. Our strategy for computing transformations on a ector is to decompose into components and parallel and perpendicular to the unit ector u. Typically one of these components is unaffected by the transformation and the effect of the transformation on the other component is easy to compute. We then reassemble the transformation on these components to derie the effect of the transformation on the original ector. Formulas for transforming points follow easily from formulas for transforming ectors, since a point can typically be expressed as the sum of a ector Q and a fixed point Q of the transformation. This fixed point is typically the point Q that defines the line or plane that specifies the transformation. Thus once we know the effect of the transformation on arbitrary ectors, we can easily find the effect of the transformation on arbitrary points The general approach outlined aboe is important and powerful; you should incorporate this strategy into your analytic toolkit. We shall hae occasion to use this strategy again in other applications later in this course. 11

12 6.2 Formulas for Affine and rojectie Transformations. Translation -- by the ector w = = + w Rotation -- by the angle θ around the line L determined by the point Q and the unit direction ector w (Formula of Rodrigues) = (cosθ) + (1 cosθ)( w)w + (sinθ)w = Q+ (cosθ)( Q)+ (1 cosθ)(( Q) w)w + (sinθ)w ( Q) Mirror Image -- in the plane S determined by the point Q and the unit normal n = 2( n)n = + 2 (( Q) n)n Uniform Scaling -- around the point Q by the scale factor s = s = Q+ s( Q) = s + (1 s)q Non-Uniform Scaling -- around the point Q by the scale factor s in the direction of the unit ector w = + (s 1)( w)w = + (s 1)(( Q)w)w Orthogonal rojection -- into the plane S determined by the point Q and the unit normal n = ( n)n = (( Q)n)n erspectie rojection -- from the eye point E into the plane S determined by the point Q and the unit normal ector n ((E Q)n) + (Q ) n = ( )E (E ) n 12

13 Exercises: 1. Let A be an affine transformation. Show that A( k λ k k ) = k λ k A( k ) wheneer k λ k 0,1. 2. A shear transformation is defined in terms of a shearing plane S, a unit ector u in the plane S, and an angle φ in the following fashion. Gien any point, project orthogonally onto a point ' in the shearing plane S. Now slide parallel to u to a point so that = φ. The point is the result of applying the shearing transformation to the point (see Figure 10). n S Q u φ Let Figure 10: Shear S = Shearing plane n = Unit ector perpendicular to S Q = oint on S u = Unit ector in S (i.e. unit ector perpendicular to n) φ = Shear angle a. Show that for any point = + tan(φ) (( Q)n)u b. For each ector = R, define = R. Show that = + tan(φ)( n)u c. Conclude that i, is independent of the choices of the points and R that represent the ector. ii. = n d. Interpret the result of part b geometrically. 13

14 3. arallel projection is projection onto a plane S along the direction of a unit ector u. As usual the plane S is defined by a point Q and a unit normal ector n (see Figure 11). Show that: a. = n u n u b. = ( Q) n u n u n c u Q S Figure 11: arallel projection along a direction u into a plane S. 4. Let 1, 2 be unit ectors, and let u = 1 2. a. Show that the transformation u R() = ( 1 2 ) + u + u, rotates the ector 1 into the ector 2. b. Interpret this result geometrically. 5. Recall from Lecture 4 that the image of one point and two linearly independent ectors determines a unique affine transformation in the plane. a. Show that the image of one point and three linearly independent ectors determines a unique affine transformation in 3-space. b. Fix 2 sets of 3 ectors 1, 2, 3 and w 1,w 2,w 3, and define A(u) = Det ( u 2 3)w 1 + Det( 1 u 3 )w 2 + Det( 1 2 u)w 3. Show that A( i ) = w i i =1,2,3. ( ) Det

15 6. Recall from Lecture 4 that the image of three non-collinear points determines a unique affine transformation in the plane. a. Show that the image of four non-coplanar points determines a unique affine transformation in 3-space. b Fix 2 sets of 4 points 1, 2, 3, 4 and Q 1,Q 2,Q 3,Q 4, and define A(R) = Det( 2 R 3 R 4 R)Q 1 Det( 1 R 3 R 4 R)Q 2 Det( ) + Det( 1 R 2 R 4 R)Q 3 Det( 1 R 2 R 3 R)Q 4. Show that A( i ) = Q i i =1,2,3,4. ( ) Det Recall from Lecture 4, Exercise 19 that eery nonsingular affine transformation in the plane is the composition of 1 translation, 1 rotation, 1 shear, and 2 non-uniform scalings. Show that eery nonsingular affine transformation in 3-dimensions is the composition of 1 translation, 2 rotations, 2 shears, and 3 non-uniform scalings. 8. Let u,w be unit ectors and define the map w u by setting (w u)() = ( w)u. Show that a. w u is a linear transformation b. (u u)() =, where is the parallel projection of on u. 9. Let u be a unit ector and define the map u _ by setting (u _)() = u. Show that u _ is a linear transformation. 10. Let R() denote the ector generated by rotating the ector by the angle θ around the ector w. Using the Formula of Rodrigues, show that a. R(u) R() = u b. R(u ) = R(u) R() 11. Let M() denote the mirror image of the ector in a plane normal to the ector n. Show that a. M(u) M() = u b. M(u ) = M(u) M() 12. Let S() denote the transformation that scales each ector by the scale factor s. Show that a. S(u) S() = s 2 (u ) b. S(u ) = ( S(u) S() ) / s 15

16 13. Derie the formula for spherical linear interpolation (SLER, see Lecture 11) in 3-dimensions ( ) sin(φ) ( ) sin (1 t)φ sin tφ (t) = 0 + sin(φ) 1, where φ is the angle between the ectors 0 and 1 by rotating the ector 0 by the angle t φ around the ector w = Let T 1,T 2 be transformations from R 3 to R 3, and let T(λ) = (1 λ)t 1 + λt 2. Show that: a. If T 1,T 2 are affine transformations, then T(λ) is also an affine transformation. b. If T 1,T 2 are rotations, then T(λ) is not necessarily a rotation. 16