To Do To Do Foundations of Computer Graphics Foundations of Computer Graphics Questions/concerns about assignment 1? (Spring 2012) (Spring 2012) Remember it is due next Thu. Ask me or TAs re problems CS 184, Lecture 5: Viewing http://inst.eecs.berkeley.edu/~cs184 Motivation Demo (Projection Tutorial) Motivation Demo (Projection Tutorial) Nate Robbins OpenGL We have seen transforms (between coord systems) tutors Projection.exe But all that is in 3D Download others We still need to make a 2D picture Project 3D to 2D. How do we do this? This lecture is about viewing transformations What we’ What we ’ve seen so far ve seen so far Outline Outline Transforms (translation, rotation, scale) as 4x4 homogeneous matrices Orthographic projection (simpler) Last row always 0 0 0 1. Last w component always 1 Perspective projection, basic idea Derivation of gluPerspective (handout: glFrustum) In new OpenGL, glm macro glm::lookAt glm::Perspective For viewing (perspective), we will use that last row Brief discussion of nonlinear mapping in z and w component no longer 1 (must divide by it) Not well covered in textbook chapter 7. We follow section 3.5 of real-time rendering most closely. Handouts on this will be given out. 1
Projections Orthographic Projection Projections Orthographic Projection To lower dimensional space (here 3D -> 2D) Characteristic: Parallel lines remain parallel Preserve straight lines Useful for technical drawings etc. Trivial example: Drop one coordinate (Orthographic) Fig 7.1 in text Perspective Orthographic Example In general Example In general We have a cuboid that we want to map to the Simply project onto xy plane, drop z coordinate normalized or square cube from [-1, +1] in all axes We have parameters of cuboid (l,r ; t,b; n,f) Orthographic Matrix Orthographic Matrix Transformation Matrix Transformation Matrix First center cuboid by translating Scale Translation (centering) Then scale into unit cube 2 l r 0 0 0 1 0 0 r l 2 2 t b 0 0 0 0 1 0 M t b 2 2 f n 0 0 0 0 0 1 f n 2 0 0 0 1 0 0 0 1 2
Caveats Final Result Caveats Final Result Looking down –z, f and n are negative (n > f) OpenGL convention: positive n, f, negate internally 2 r l 2 r l 0 0 0 0 r l r l r l r l 2 t b 2 t b 0 0 0 0 M glm :: ortho t b t b t b t b 2 f n 2 f n 0 0 0 0 f n f n f n f n 0 0 0 1 0 0 0 1 Outline Perspective Projection Outline Perspective Projection Most common computer graphics, art, visual system Orthographic projection (simpler) Further objects are smaller (size, inverse distance) Perspective projection, basic idea Parallel lines not parallel; converge to single point Derivation of gluPerspective (handout: glFrustum) A In modern OpenGL, glm::perspective Plane of Projection Brief discussion of nonlinear mapping in z A’ B B’ Center of projection (camera/eye location) Pinhole Camera Pinhole Camera Perspective Projection Perspective Projection Center of Projection (one point) Foreshortening: Distant objects appear smaller Very common model in graphics (but real cameras use lenses; a bit more complicated) 3
Overhead View of Our Screen In Matrices Overhead View of Our Screen In Matrices Note negation of z coord (focal plane –d) x y z , , x y d , , (Only) last row affected (no longer 0 0 0 1) 0,0,0 w coord will no longer = 1. Must divide at end d 1 0 0 0 0 1 0 0 P 0 0 1 0 Looks like we’ve got some nice similar triangles here? x x d x y y d * y 1 x y 0 0 0 z d z z d z d Verify Vanishing Points Verify Vanishing Points Parallel lines “meet” at vanishing point d * x (x,y) ~ (d x , d y ) [directions] 1 0 0 0 x x z Every pixel vanishing pt for 0 1 0 0 y y d * y some dirn (lines parallel to ? z 0 0 1 0 z image plane vanish infinity) z 1 z d Horizon 0 0 0 1 d d 1 Perspective Distortions Perspective Distortions Outline Outline Perspective can distort; artists often correct Orthographic projection (simpler) Computers can too (Zorin and Barr 95) Perspective projection, basic idea Derivation of gluPerspective (handout: glFrustum) Brief discussion of nonlinear mapping in z 4
Remember projection tutorial Viewing Frustum Remember projection tutorial Viewing Frustum Far plane Near plane Screen (Projection Plane) gluPerspective Screen (Projection Plane) gluPerspective gluPerspective(fovy, aspect, zNear > 0, zFar > 0) width Fovy, aspect control fov in x, y directions zNear, zFar control viewing frustum Field of view (fovy) height Aspect ratio = width / height Overhead View of Our Screen Overhead View of Our Screen In Matrices In Matrices 1 Simplest form: 0 0 0 x y z , , aspect x y d , , 0,0,0 1 0 1 0 0 P d 0 0 1 0 1 ? d ? 0 0 0 d Aspect ratio taken into account fovy d cot Homogeneous, simpler to multiply through by d 2 Must map z vals based on near, far planes (not yet) 5
In Matrices Z mapping derivation In Matrices Z mapping derivation 1 0 0 0 d 0 0 0 aspect A B z Az B B aspec t A ? 0 1 0 0 z z 1 0 1 0 d 0 0 P 0 0 1 0 0 0 A B Simultaneous equations? 1 0 0 0 0 0 1 0 d B f n A A 1 A and B selected to map n and f to -1, +1 respectively f n n 2 fn B A 1 B f f n Outline Mapping of Z is nonlinear Outline Mapping of Z is nonlinear Az B B A Orthographic projection (simpler) z z Perspective projection, basic idea Many mappings proposed: all have nonlinearities Derivation of gluPerspective (handout: glFrustum) Advantage: handles range of depths (10cm – 100m) Brief discussion of nonlinear mapping in z Disadvantage: depth resolution not uniform More close to near plane, less further away Common mistake: set near = 0, far = infty. Don’t do this. Can’t set near = 0; lose depth resolution. We discuss this more in review session Summary: The Whole Viewing Pipeline Summary: The Whole Viewing Pipeline Eye coordinates Model coordinates Perspective Transformation (glm::perspective) Model transformation Screen coordinates World coordinates Viewport transformation Camera Transformation Window coordinates (glm::lookAt) Raster Device coordinates transformation Slide courtesy Greg Humphreys 6
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