Graphics 2009/2010, period 1. Lecture 6: perspective projection

Transcription

1 Graphics 2009/2010, period 1 Lecture 6 Perspective projection

2 Orthographic vs. perspective projection Introduction Projecting from arbitrary camera positions Orthographic projection and the canonical view volume Windowing transform Goal: Projecting the 3D model to a 2D viewing window 2 approaches: Orthographic vs. perspective projection

3 From 3D worlds to 2D screens Introduction Projecting from arbitrary camera positions Orthographic projection and the canonical view volume Windowing transform Given an arbitrary camera position, we want to display the objects in the model in an image

4 From 3D worlds to 2D screens Introduction Projecting from arbitrary camera positions Orthographic projection and the canonical view volume Windowing transform The projection should be perspective projection.

5 From 3D worlds to 2D screens Introduction Projecting from arbitrary camera positions Orthographic projection and the canonical view volume Windowing transform First, we define a view frustum that contains everything we want to project onto the image.

6 Introduction Projecting from arbitrary camera positions Orthographic projection and the canonical view volume Windowing transform We simplify by moving the camera viewpoint to the origin, such that we look into the direction of the negative z-axis. Z

7 Orthographic projection Introduction Projecting from arbitrary camera positions Orthographic projection and the canonical view volume Windowing transform Orthographic projection is a lot simpler than perspective projection, so we transform the clipped view frustum to an axis-parallel box Z

8 The canonical view volume Introduction Projecting from arbitrary camera positions Orthographic projection and the canonical view volume Windowing transform To simplify our calculations, we transform to the canonical view volume (1, 1) ( 1, 1)

9 Windowing transform Introduction Projecting from arbitrary camera positions Orthographic projection and the canonical view volume Windowing transform We apply a windowing transform to display the square [ 1, 1] [ 1, 1] onto an m n image.

10 The graphics pipeline (part I) Every step in the sequence can be represented by a matrix operation, so the whole process can be applied by performing a single matrix operation! (Well: almost.) We call this sequence a graphics pipeline. Introduction Projecting from arbitrary camera positions Orthographic projection and the canonical view volume Windowing transform

11 The graphics pipeline (part I) Introduction Projecting from arbitrary camera positions Orthographic projection and the canonical view volume Windowing transform Graphics pipeline = a special software or hardware subsystem that efficiently draws 3D primitives in perspective.

12 The canonical view volume The canonical view volume The orthographic view volume The orthographic projection matrix The canonical view volume is a box, centered at the origin. The clipped view frustum is transformed to this box (and the objects within the view frustum undergo the same transformation). y z x (1, 1, 1) (1, 1, 1) Vertices in the canonical view volume are orthographically projected onto an m n image. ( 1, 1, 1)

13 Mapping the canonical view volume The canonical view volume The orthographic view volume The orthographic projection matrix We need to map the square [ 1, 1] 2 onto a rectangle [ 1 2, m 1 2 ] [ 1 2, n 1 2 ]. (1, 1) The following matrix takes care of that: m m n n ( 1, 1) n m

14 Mapping the canonical view volume The canonical view volume The orthographic view volume The orthographic projection matrix We need to map the square [ 1, 1] 2 onto a rectangle [ 1 2, m 1 2 ] [ 1 2, n 1 2 ]. (1, 1) The following matrix takes care of that: m m n n ( 1, 1) n m

15 The orthographic view volume The canonical view volume The orthographic view volume The orthographic projection matrix The orthographic view volume is an axis-aligned box [l, r] [b, t] [n, f]. (r, t, f) (l, b, n) y (1, 1, 1) z x (1, 1, 1) ( 1, 1, 1)

16 The orthographic view volume The canonical view volume The orthographic view volume The orthographic projection matrix Transformation to canonical view volume is done by 2 r l l+r t b b+t n f n+f (l, b, n) y z x (r, t, f) (1, 1, 1) (1, 1, 1) ( 1, 1, 1)

17 The orthographic projection matrix The canonical view volume The orthographic view volume The orthographic projection matrix We can combine the matrices into one: m m n n M o = r l l+r t b b+t n f n+f Given a point p in the orthographic view volume, we map it to a pixel [i, j] as follows: i j z canonical 1 = M o x p y p z p 1

18 Transforming the view frustum Transforming the view frustum matrix Homogeneous coordinates (cf. book, fig. 7.12)

19 Transforming the view frustum Transforming the view frustum matrix Homogeneous coordinates (cf. book, fig. 7.10)

20 Transforming the view frustum Transforming the view frustum matrix Homogeneous coordinates We have to transform the view frustum into the orthographic view volume. The transformation needs to Map lines through the origin to lines parallel to the z axis Map points on the viewing plane to themselves. Map points on the far plane to (other) points on the far plane. Preserve the near-to-far order of points on a line.

21 Transforming the view frustum Transforming the view frustum matrix Homogeneous coordinates How do we calculate this? (cf. book, fig. 7.9)

22 Transforming the view frustum Transforming the view frustum matrix Homogeneous coordinates So we need a matrix that gives us x = nx z y = ny z and a z-value that stays the same for all points on the near and fare planes does not change the order along the z-axis for all other points Problem: we can t do division with matrix multiplication

23 matrix Transforming the view frustum matrix Homogeneous coordinates But the following matrix M p does the trick: x y n+f 0 0 n f z = n 0 z n+f n z n Hmmmm... is that what we want...? x y f

24 Homogeneous coordinates Transforming the view frustum matrix Homogeneous coordinates We have seen homogeneous coordinates before, but so far, the fourth coordinate of a 3D point has alway been 1. In general, however, the homogeneous representation of a point (x, y, z) in 3D is (hx, hy, hz, h). Choosing h = 1 has just been a convenience. So we have: x M p y z = 1 z n+f n z n nx x z y f homogenize ny z n + f fn z 1

25 Transforming the view frustum matrix Homogeneous coordinates Homogeneous coordinates and perspective transformation x y n+f 0 0 n f z = n 0 z n+f n z n nx x z y f homogenize ny z n + f fn z 1

26 Transforming the view frustum matrix Homogeneous coordinates Homogeneous coordinates and perspective transformation x y n+f 0 0 n f z = n 0 z n+f n z n nx x z y f homogenize ny z n + f fn z 1

27 Homogeneous coordinates Transforming the view frustum matrix Homogeneous coordinates Note: this fits well in our existing framework For example: translation t x hx hx + ht x x + t x t y hy t z hz = hy + ht y hz + ht z homogenize y + t y z + t z h h 1

28 Aligning coordinate systems Aligning coordinate systems Transformation matrix The only thing left: Given a camera coordinate system with origin e and orthonormal base (u, v, w), we need to tranform objects in world space coordinates into objects in camera coordinates. This can alternatively be considered as aligning the canonical coordinate system with the camera coordinate system.

29 Aligning coordinate systems Transformation matrix The required transformation is taken care of by x u y u z u x e M v = x v y v z v y e x w y w z w z e

30 The graphics pipeline: part I If we combine all steps, we get: compute M o compute M v compute M p M = M o M p M v for each line segment (a i, b i ) do p = Ma i q = Mb i drawline(x p /h p, y p /h p, x q /h q, y q /h q )

31 Some announcements... I was asked to draw your attention to the Master-voorlichtingsavond that takes place tomorrow at 18h00. More info on the website (which is linked from the news section of this course s website).

32 Some announcements... When will the results from the midterm exam be ready? Exact date unpredictable due to illness and interesting answers from some participants which need to be checked carefully. But we are working on it... A general hint for future exams (not only in graphics): Most exercises are easier to solve if you read them first!!! For example, if it starts with Assume v and w are two unit vectors..., it is rather unlikely that One of them must be the nullvector is a correct answer.

33 Some announcements... Remember the deadline for the programming assignment P1: Friday, October 2, 12h00 Submit online (via link on website) Read the instructions (and follow them) If you have problems finishing, make sure you upload what you have and give a clear description of problems, gaps, etc. in the README file

34 Some announcements... Finally, there s a change of the schedule: Lectures: I have to go to a PhD defense on October 19, therefore: No lecture on October 19 [old schedule: lecture 11] Lecture 11 on October 21 [old schedule: lecture 12] Lecture 12 on October 26 [old schedule: no lecture] No lecture on October 28 [as before] You find the updated schedule online.

35 Some announcements... Tutorials: We have the impression that covering the content of a lecture in the tutorial directly afterwards is not optimal. Therefore, tutorials move one step, i.e. No tutorial on Mon, October 5 Lecture 6 and 7 will be covered in the tutorials on Thu, Oct 7 and Mon, Oct 12 Lecture 8 and 9 will be covered in the tutorials on Thu, Oct 14 and Mon, Oct 19 and so on You find the updated schedule online.