1. We ll look at: Types of geometrical transformation. Vector and matrix representations

Transcription

1 Tob Howard COMP272 Computer Graphics and Image Processing 3: Transformations Introduction We ll look at: Tpes of geometrical transformation Vector and matri representations Homogeneous coordinates Using transformations in OpenGL Some hand vector geometr 2

2 Tob Howard 3D Cartesian coordinates A coordinate represents a point in space, measured with respect to an origin and set of X, Y, Z aes Y P(,,z) René Descartes, , inventor of Cartesian coordinates o z X Z 3 Vectors A vector represents a direction in space, with respect to a set of X, Y, Z aes. It has a characteristic length. A vector of length is known as a unit vector Y o V (,,z) X Z 4

3 Tob Howard Coordinates and Vectors Both coordinates and vectors can be represented b a triple of,,z values. In computer graphics we usuall write these in column format: In the previous slides The melon is at Point P! " # $ # $ #% z $ & The spatial relationship between the origin and the melon is described b the vector V 5 Danger: 2 different representations We can write a vector as either a column or a row:! " # $ # $ #% z $ & OpenGL uses column vectors MATLAB uses row vectors Yes, this is confusing! or [ z ] The two representations are equivalent, but a transformation matri used with column vectors is the transpose of the equivalent matri used with row vectors 6

4 Tob Howard Geometrical Transformations We define the shape of a geometric primitive using points We can appl geometrical transformations to points to change them These include translation, scaling and rotation To transform an entire shape, we transform all its individual points 7 Translation Applies a 3D shift (t, t, tz) to all coordinates ' = + t ' = + t z' = z + tz Translate z z 8

5 Tob Howard Scaling Applies a scale factor (s, s, sz) to all coordinates (with respect to the origin) ' =. s ' =. s z' = z. sz Scale z z 9 Rotation (2D) Before we look at 3D, let s look at the 2D case. We want to rotate point P about the origin, b angle θ P (, ) P (,) θ 0

6 Tob Howard Rotation (2D) = RcosΦ = RsinΦ = Rcos(θ+Φ) = RcosΦcosθ - RsinΦsinθ = Rsin(θ+Φ) = RcosΦsinθ + RsinΦcosθ R R P (, ) P (,) Substituting for RcosΦ and RsinΦ gives: = cosθ sinθ = sinθ + cosθ θ Φ Rotation (2D) about a point What if we want to rotate about some other point than the origin? How do we rotate point P about point Q? We break the problem into 3 simpler steps: Translate b (-q,-q) to place Q at the origin Do the rotation b θ Translate back again b (+q, +q) P (, ) θ Q (q,q) This is alwas the wa we do comple transformations see later P (,) 2

7 Tob Howard Rotation (3D) Rotation of P about origin in 2D is the same in 3D, as rotation of P, in the XY plane, about the Z-azis So: = cosθ sinθ = sinθ + cosθ z = z (no change) θ P (, ) P (,) z 3 Rotation (3D) Rotations are relative to an ais, e.g. a rotation b angle θ about the Z ais ' =. cosθ. sinθ ' =. sinθ +. cosθ z' = z Rotate Around Z Ais z z 4

8 Tob Howard Rotation (3D) about a vector In 3D we often want to rotate (or scale) about an arbitrar ais vector This is analogous to the 2D case of rotating (or scaling) about an arbitrar point And we approach it in the same wa: we do it as sequence of steps (which we will describe later) z 5 Representing transformations We ve seen the following equations for representing some transformations: Translation Scale Rotation (about Z) ' = + t ' = + t z' = z + tz ' =. s ' =. s z' = z. sz ' =. cosθ. sinθ ' =. sinθ +. cosθ z' = z The re all different! It would be ver convenient if we could use a single homogeneous (== the same ) representation We use vectors and matrices 6

9 Tob Howard Using matrices: scaling A transformation changes a vector into another vector We can represent this change using a matri Eample, scale (,,z) b (2,3,5): ' ' = z' z Multipl elements row b column: = 2* + 0* + 0*z = 2 = 0* + 3* + 0*z = 3 z = 0* + 0* + 5*z = 5z 7 Using matrices: scaling A transformation changes a vector into another vector We can represent this change using a matri Eample, scale (,,z) b (2,3,5): ' ' = z' z Multipl elements row b column: = 2* + 0* + 0*z = 2 = 0* + 3* + 0*z = 3 z = 0* + 0* + 5*z = 5z 8

10 Tob Howard Using matrices: scaling A transformation changes a vector into another vector We can represent this change using a matri Eample, scale (,,z) b (2,3,5): ' ' = z' z Multipl elements row b column: = 2* + 0* + 0*z = 2 = 0* + 3* + 0*z = 3 z = 0* + 0* + 5*z = 5z 9 Using matrices: scaling A transformation changes a vector into another vector We can represent this change using a matri Eample, scale (,,z) b (2,3,5): ' ' = z' z Multipl elements row b column: = 2* + 0* + 0*z = 2 = 0* + 3* + 0*z = 3 z = 0* + 0* + 5*z = 5z 20

11 Tob Howard Using matrices: rotation (about Z) Eample, rotate (,,z) about Z ais b θ: ' cosθ sinθ 0 ' = sinθ cosθ 0 z' 0 0 z Multipl elements row b column: = cosθ* - sinθ* + 0*z = cosθ - sinθ = sinθ* + cosθ* + 0*z = sinθ + cosθ z = 0* + 0* + *z = z 2 Using matrices: translation Eample, translate (,,z) b (t,t,tz) : '??? ' =??? z'??? z Q: What should we put in the matri? A: We can t do it!! What s the solution then? 22

12 Tob Howard Using matrices: translation To incorporate translation, we have to add an etra row and column to the matri, and an etra term to our coordinates: ʹ 0 0 t ʹ 0 0 t = zʹ 0 0 tz z w ʹ Multipl elements row b column: = * + 0* + 0*z + t*= + t = 0* + * + 0*z + t*= + t z = 0* + 0* + *z + tz*= z + tz w = 0* + 0* + 0*z + *= This ma seem like an arbitrar fi, but it goes deeper. And later we will see a use for the new bottom row of the matri, for doing projection from 3D to 2D. 23 Homogeneous coordinates In order to use a consistent matri representation for all kinds of linear transformations, we ve had to add an etra coordinate, w, to our usual 3D coordinate (,,z). This (,,z,w) form is called homogeneous coordinates But where is this w? Is it in a 4 th spatial dimension? Yes it is! Unfortunatel the mathematical details are beond the scope of this course Usuall, w=, and we just ignore it When it is not, we need to normalise see later, when we cover perspective August Möbius, , inventor of homogeneous coordinates 24

13 Tob Howard Scale matri P! = T P ʹ S ʹ 0 S 0 0 = zʹ 0 0 Sz 0 z z 25 Translation matri P! = T P! ( "! 0 0 T "! " # ( $ # 0 0 T $ # $ # $ = # $ # $ # z( $ # 0 0 Tz $ # z$ # $ # $ # $ & ' & ' & ' z 26

14 Tob Howard Rotation matri (around X ais) P! = T P " ) # " 0 0 0# " # $ ) % $ 0 cosθ sinθ 0 % $ % $ % = $ % $ % $ z) % $ 0 sinθ cosθ 0% $ z% $ % $ % $ % ' ( ' ( ' ( z 27 Rotation matri (around Y ais) P! = T P " ) # " cosθ 0 sinθ 0# " # $ ) % $ % $ % $ % = $ % $ % $ z) % $ sinθ 0 cosθ 0% $ z% $ % $ % $ % ' ( ' ( ' ( z 28

15 Tob Howard Rotation matri (around Z ais) P! = T P # " $ # cosθ sinθ 0 0$ # $ % " & % sinθ cosθ 0 0 & % & % & = % & % & % z" & % 0 0 0& % z& % & % & % & ( ) ( ) ( ) z 29 Matri transformations recap The transformation T changes point P to P Pʹ = T P ʹ a b c d ʹ e f g h = zʹ i j k l z m n o p Where the 6 values a... p in the matri determine what kind of transformation it is 30

16 Tob Howard Composing transformations What if we now appl a transformation T 2 to P? Pʹʹ = T 2 Pʹ Pʹ = T T P 2 We can appl this double transformation to P in one go, if we multipl the matrices T and T 2 together to obtain the composite transformation T C TC = T2 T Pʹʹ = P T C 3 Non-commutativit, or order matters Matri multiplications (and therefore transformations) are in general non-commutative Given two matrices M and M 2, M * M 2 M 2 * M In the left eample, P is first rotated b θ, and then shifted b (T,T). In the right eample, P is first shifted b (T,T) and then rotated b θ. The results are different. Performing transformations in the correct order is crucial. P P T T θ θ T T P P 32

17 Tob Howard Composite transformations, 2D eample How to scale an object about an arbitrar point P, in 2D? We split the operation into three simpler transformations: P P P M : Translate to origin M 2 : Scale about the origin M 3 : Translate back 33 Composite transformations, 2D eample We want to scale an object b (s, s) about an arbitrar 2D point P (p, p). We split this into simpler steps: Step : Construct the translation matri M which shifts the object to the origin, b ( p, p) Step 2: Construct the matri M 2 which scales the object b (s, s) with respect to the origin Step 3: Construct the translation matri M 3 which shifts the object back b (p, p) The composite transformation is M 3 M 2 M Note the ORDER: M first, then M 2 then M 3 The ke to this process is in Step 3, where matri M 3 undoes the effect of matri M 34

18 Tob Howard Undoing a transformation Matri A: shift b (T, T, Tz): Matri B: shift b (-T, -T, -Tz): Matri product of A and B: A and B are called inverses 0 0 T 0 0 T 0 0 Tz T 0 0 T 0 0 Tz So multipling a point P b A, then B, has no effect on P identit matri, I 35 Matri inverses Two matrices A and B are said to be inverses of each other if: A B = I, where I is the identit matri For a matri M, we write its inverse as M - So, M M - = I In other words, if a matri M does some transformation on a point P, M - undoes it, restoring P Given M, there are algorithms for computing M - BUT, not all matrices actuall have an inverse! Eample: how can ou undo a transformation that makes all -coordinates 0? The original information has been destroed 36

19 Tob Howard Non-invertible transformations Eample: this scale transformation will set all -coordinates to 0: This matri has no inverse it is singular. In general, a transformation is singular if it throws awa information 37 Composite transformations, 3D eample We want to rotate b θ about an arbitrar 3D vector A But we onl know how to rotate around the X, Y and Z aes We reduce the problem to a series of transformations Y θ A X Z 38

20 Tob Howard Composite transformations, 3D eample We want to rotate b θ about an arbitrar 3D vector A Y A But we onl know how to rotate around the X, Y and Z aes We reduce the problem to a series of transformations X (let s remove the cow) Z 39 Composite transformations, 3D eample Step : We shift vector A until it passes through the origin. This is transformation M Call this new vector A Y M A A X Z 40

21 Tob Howard Composite transformations, 3D eample So now A is passing through the origin. Step 2: Rotate A about the X-ais until it lies in the XY plane. This is transformation M 2, and it produces the new vector A 2 M 2 Y A X Z 4 Composite transformations, 3D eample So now A 2 is passing through the origin, and it lies in the XY plane. Step 3: Rotate A 2 about the Z ais until it it is coincident with the X-ais. This is transformation M 3, and it produces the new vector A 3 Y A 2 is in the XY plane M 3 A 2 X Z 42

22 Tob Howard Composite transformations, 3D eample So now A 3 is passing through the origin, and it is coincident with the X-ais. Step 4: Perform the desired rotation about the X-ais b θ (which we know how to do). This is transformation M 4 M 4 A 3 Y X Z 43 Composite transformations, 3D eample So that s the rotation done. All we have to do now is to undo the effect of all the other transformations, in reverse order Step 5: appl M 3 - Step 6: appl M 2 - Step 7: appl M - Y θ If we appl all these transformations, in order, we achieve what we want Z 44 X

23 Tob Howard Rotation about an arbitrar 3D ais Summar of the steps we ve gone through:. Construct the matri M which translates A so it passes through the origin. The new vector is A. 2. Construct M 2 which rotates A about the X-ais (although we could use a different ais), mapping it into the XY plane. The new vector is A Construct M 3, which rotates A 2 about the Z-ais, mapping it onto the X-ais. The new vector is A Construct M 4, which applies the required rotation b θ about the X-ais. 5. Construct the inverse matrices, to undo the effects of M 3, M 2 and M. The entire transformation is thus: P = M - M 2 - M 3 - M 4 M 3 M 2 M P 45 Transformations in OpenGL () OpenGL maintains two transformation matrices internall: the modelview matri, used for transforming the geometr ou draw, and specifing the camera the projection matri, used for controlling the wa the camera image is projected onto the screen (see later) Ever 3D point ou ask OpenGL to draw is automaticall transformed b these two matrices before it is drawn (and ou cannot prevent this happening) P drawn = ProjectionMatri ModelviewMatri P specifed For full details, see Chapter 5 of the OpenGL manual. 46

24 Tob Howard Transformations in OpenGL (2) OpenGL provides functions for easil dealing with transformations. Here are some: gltranslatef(t, t, tz) glscalef(s, s, sz) glrotatef(theta, r, r, rz) OpenGL When we call one of these functions, OpenGL creates a corresponding temporar matri TMP, and then multiples the modelview matri b TMP, and then throws awa TMP 47 Transformations in OpenGL (3) An eample. We want to first rotate and then shift the teapot. glmatrimode(gl_modelview); glloadidentit(); // M= identit matri (I) gltranslatef(t, t, tz); // OpenGL computes temp translation matri T, // then sets M= M T, so now M is T glrotatef(theta, 0.0,.0, 0.0); // OpenGL computes temp rotation matri R, // then sets M= M R, so M is now T R glutwireteapot(.0); Notice the order we call the functions in it s the reverse of how we would write it down logicall. 48

25 Tob Howard Transformations in OpenGL (4) What if we want a series of steps, as we saw earlier? Sometimes there are OpenGL functions which come to our rescue. For eample, glrotatef() will convenientl compute a matri for rotation of angle θ about the vector (,, z) which passes through the origin. glrotatef(glfloat theta, GLfloat, GLfloat, GLfloat z); OpenGL Therefore, if we call this transformation R, we can epress our previous rotation-about-arbitrar-vector eample as M - R M There are OpenGL functions for loading our own matri from the modelview or projection matrices, and for multipling them together. But in practice, it s not necessar to use these much. 49 Model and world coordinate sstems Often an object is defined in a local modelling coordinate sstem. E.g. modelling a car wheel with an origin at the wheel centre. Modelling transformations are used to instance multiple copies of an object in the scene, e.g. translate and rotate the wheels onto a car bod The entire car ma then have further transformations applied, like translation to simulate its movement. A global world coordinate sstem is used to specif the position of objects in the entire scene. We ll cover this later in the course 50

26 Tob Howard Reference: Useful vector geometr Vectors provide a ver convenient wa of thinking about man of the manipulations we might want to perform on an object in 3D space In fact, it s the onl sensible wa to work (and essential for rendering, as we shall see later) Understanding a small amount of vector maths goes a long wa in 3D graphics Addition and Subtraction Scalar multiplication Vector normalization Dot product Cross product 5 Vector addition To add two vectors of the same order, add the components = 2 2 z z z + z Wh is this useful? moves a point through space in a known direction 52

27 Tob Howard Vector subtraction To subtract two vectors of the same order, subtract the components = 2 2 z z z z 2 2 z 2 P2 V P z Wh is this useful? represents a line between two points V= P - P2 53 Multiplication b a scalar Multipl the individual components b a scalar C V C C C = z z C Wh is this useful? moves a point along a vector b a given amount V 3 54

28 Tob Howard Vector magnitude Gives the length or size of a vector V = V = + + z z P2 V P If we have a line joining two points, the magnitude of the vector between them represents their distance in 3D space V= P - P2 Distance from P to P2 is V 55 Vector normalization Normalization is the process of taking an arbitrar (but non-zero) vector V, and converting it into a vector ˆV (V-hat) of length, which points in the same direction Calculate the length L of V, and divide its, and z components b this value V = z L= + + z Essential operation in rendering / L ˆ / L V = z/ L 56

29 Tob Howard Vector multiplication There are two was of multipling vectors One results in a scalar value, and is called the dot product (aka inner product ) The other results in a vector, and is called the cross product (aka outer product ) Both are essential operations in 3D graphics 57 The Dot Product is the scalar product of the individual components 2 2 = ( 2) + ( 2) + ( z z2) z z 2 θ V V 2 For normalized vectors, their dot product is the cosine of the angle between them Essential for rendering cosθ = V iv2 58

30 Tob Howard The Cross Product is a vector, defined as follows: V 3 V 2 z2 z 2 2 z2 z 2 = z z V 2 For two normalized vectors, their cross product is a third vector perpendicular to them both (forming a right handed sstem) V = 3 V V 2 59 Vector geometr is essential All these properties of vectors are essential in 3D graphics: for defining and manipulating geometr for specifing and evaluating rendering ˆL ˆN ˆR θ φ ˆV There are man vector manipulation libraries available that hide the underling maths and make vector manipulation eas 60