1 INFOGR – Computer Graphics J. Bikker - April-July 2015 - Lecture 6: "Transformations" Welcome! 1
2 Today’s Agenda: Projection Pipeline Recap Rasterization 2
INFOGR – Lecture 6 – "Transformations" 3 Perspective Projection – Applying matrices, working our way backwards Goal: create 2D images of 3D scenes Standard approach: linear perspective (in contrast to e.g. fisheye views) Parallel projection: Perspective projection: 3
INFOGR – Lecture 6 – "Transformations" 4 Perspective Parallel projection: Maps 3D points to 2D by moving them along a projection direction until they hit an image plane . Perspective projection: Maps 3D points to 2D by projecting them along lines that pass through a single viewpoint until they hit an image plane. 4
INFOGR – Lecture 6 – "Transformations" 5 Perspective 5
INFOGR – Lecture 6 – "Transformations" 6 Perspective 6
INFOGR – Lecture 6 – "Transformations" 7 Perspective 7
INFOGR – Lecture 6 – "Transformations" 8 Perspective 8
INFOGR – Lecture 6 – "Transformations" 9 Perspective 9
INFOGR – Lecture 6 – "Transformations" 10 Perspective 10
INFOGR – Lecture 6 – "Transformations" 11 Perspective Perspective projection World space (3D) Screen space (2D) We get our 3D objects perspective correct on the 2D screen by applying a sequence of matrix operations. 11
INFOGR – Lecture 6 – "Transformations" 12 Perspective Perspective projection E y 𝑒 The camera is defined by: 𝑚 𝑜 Its position E 𝑊 The view direction 𝑊 The image plane (defined by its distance 𝑔 𝑒 and the field of view) 𝑠 FOV The view frustum is the volume visible from the camera. It is defined by: x z A near and a far plane 𝑜 and 𝑔 ; A left and a right plane 𝑚 and 𝑠 ; The world according to the camera: A top and a bottom plane 𝑢 and 𝑐 (in 3D). Camera space 12
INFOGR – Lecture 6 – "Transformations" 13 Perspective 𝑨 = 𝑔 x Perspective projection 𝑨 = 𝑜 Camera space: looking down negative 𝑨 . -z We can now map from (𝑦, 𝑧, 𝑨) to (𝑦 𝑡 , 𝑧 𝑡 ) y (but this mapping is not trivial) Projection (and later: clipping) becomes easier when we switch to an orthographic view volume. x This time the mapping is: 𝑨 = 𝑔 𝑨 = 𝑜 𝑦, 𝑧, 𝑨 → 𝑦, 𝑧 → 𝑦 𝑡 , 𝑧 𝑡 . Going from camera space to the orthographic -z view volume can be achieved using a matrix y multiplication. 13
INFOGR – Lecture 6 – "Transformations" 14 Perspective Perspective projection The final transform is the one that takes us from the orthographic view volume to the canonical view volume. Again, this is done using a matrix. 𝑨 = −1 𝑨 = 1 𝑦 = 1 -z 𝑦 = −1 14
INFOGR – Lecture 6 – "Transformations" 15 Perspective Perspective projection World space camera space orthographic view canonical view I × M camera × M ortho × M canonical These can be collapsed into a single 4 × 4 matrix. 15
INFOGR – Lecture 6 – "Transformations" 16 Perspective Perspective projection We need one last transform: From canonical view (-1..1) to 2D screen space ( 𝑂 𝑦 × 𝑂 𝑧 ). Screen space (2D) Canonical view screen 16
INFOGR – Lecture 6 – "Transformations" 17 Perspective Perspective projection STEP ONE: canonical view to screen space This is assuming we already threw away 𝑨 to get an orthographic projection. We will however Vertices in the canonical view are combine all matrices in the end, so we actually orthographically projected on an 𝑜 𝑦 × 𝑜 𝑧 image. need a 4 × 4 matrix: We need to map the square [-1,1] 2 onto a rectangle 0, 𝑜 𝑦 × [0, 𝑜 𝑧 ] . Matrix: 𝑜 𝑦 𝑜 𝑦 𝑜 𝑦 𝑜 𝑦 0 0 0 2 2 2 2 𝑜 𝑧 𝑜 𝑧 𝑜 𝑧 𝑜 𝑧 𝑁 𝑤𝑞 = 0 0 0 2 2 2 2 0 0 1 0 0 0 1 0 0 0 1 17
INFOGR – Lecture 6 – "Transformations" 18 Perspective Perspective projection STEP ONE: canonical view to screen space We now know the final transform for the vertices: 𝑦 𝑡𝑑𝑠𝑓𝑓𝑜 𝑦 𝑑𝑏𝑜𝑝𝑜𝑗𝑑𝑏𝑚 𝑧 𝑡𝑑𝑠𝑓𝑓𝑜 𝑧 𝑑𝑏𝑜𝑝𝑜𝑗𝑑𝑏𝑚 = 𝑁 𝑤𝑞 𝑨 𝑑𝑏𝑜𝑝𝑜𝑗𝑑𝑏𝑚 𝑨 𝑑𝑏𝑜𝑝𝑜𝑗𝑑𝑏𝑚 1 1 Next step: getting from the orthographic view volume to the canonical view volume. 18
INFOGR – Lecture 6 – "Transformations" 19 Perspective Perspective projection STEP TWO: orthographic view volume to canonical view volume The orthographic view volume is an axis aligned box 𝑚, 𝑠 × 𝑐, 𝑢 × [𝑜, 𝑔] . We want to scale this to a 2 × 2 × 2 box centered around the origin. Combined: Scaling to [-1,1]: Moving the center to the origin: 2 2 − 𝑚 + 𝑠 − 𝑚 + 𝑠 0 0 0 1 0 0 0 0 𝑠 − 𝑚 2 𝑠 − 𝑚 𝑠 − 𝑚 2 2 − 𝑐 + 𝑢 − 𝑐 + 𝑢 0 0 0 0 1 0 0 0 × = 𝑢 − 𝑐 2 𝑢 − 𝑐 𝑢 − 𝑐 2 2 − 𝑜 + 𝑔 − 𝑜 + 𝑔 0 0 0 0 0 1 0 0 𝑜 − 𝑔 2 𝑜 − 𝑔 𝑜 − 𝑔 0 0 0 1 0 0 0 1 0 0 0 1 19
INFOGR – Lecture 6 – "Transformations" 20 Perspective Perspective projection STEP TWO: orthographic view volume to canonical view volume The final transforms for the vertices are thus: 𝑦 𝑡𝑑𝑠𝑓𝑓𝑜 𝑦 𝑝𝑠𝑢ℎ𝑝 𝑧 𝑡𝑑𝑠𝑓𝑓𝑜 𝑧 𝑝𝑠𝑢ℎ𝑝 = 𝑁 𝑤𝑞 𝑁 𝑑𝑏𝑜𝑝𝑜𝑗𝑑𝑏𝑚 𝑨 𝑑𝑏𝑜𝑝𝑜𝑗𝑑𝑏𝑚 𝑨 𝑝𝑠𝑢ℎ𝑝 1 1 Next step: getting from camera space to the orthographic view volume. 20
INFOGR – Lecture 6 – "Transformations" 21 Perspective Perspective projection Translate: 1 0 0 −𝐹 𝑦 STEP THREE: camera space to orthographic view volume 0 1 0 −𝐹 𝑧 0 0 1 −𝐹 𝑨 0 0 0 1 E y x i.e., the inverse of the camera translation. Rotate: -z We will use the inverse y of the basis defined by the camera orientation. x z 21
INFOGR – Lecture 6 – "Transformations" 22 Perspective Perspective projection 𝑣𝑞 STEP THREE: camera space to orthographic view volume 𝑧 𝑨 Basis defined by the camera orientation: 𝑦 z-axis: −𝑊 (convention says we look down – z) x-axis: −𝑊 × 𝑣𝑞 y-axis: 𝑊 × 𝑦 𝑊 Matrix: Inverse: 𝑌 𝑦 𝑍 −𝑊 0 𝑌 𝑦 𝑌 𝑧 𝑌 𝑨 0 1 0 0 −𝐹 𝑦 𝑦 𝑦 𝑌 𝑧 𝑍 −𝑊 0 0 1 0 −𝐹 𝑧 𝑍 𝑍 𝑍 0 × = 𝑁 𝑑𝑏𝑛𝑓𝑠𝑏 𝑧 𝑧 𝑦 𝑧 𝑨 𝑌 𝑨 𝑍 −𝑊 0 0 0 1 −𝐹 𝑨 −𝑊 −𝑊 −𝑊 0 𝑨 𝑨 𝑦 𝑧 𝑨 0 0 0 1 0 0 0 1 0 0 0 1 22
INFOGR – Lecture 6 – "Transformations" 23 Perspective 𝑨 = 𝑔 x Perspective projection 𝑨 = 𝑜 STEP THREE: camera space to orthographic view volume -z The combined transform so far: y 𝑦 𝑡𝑑𝑠𝑓𝑓𝑜 𝑦 𝑥𝑝𝑠𝑚𝑒 𝑧 𝑡𝑑𝑠𝑓𝑓𝑜 𝑧 𝑥𝑝𝑠𝑚𝑒 = 𝑁 𝑤𝑞 𝑁 𝑑𝑏𝑜𝑝𝑜𝑗𝑑𝑏𝑚 𝑁 𝑑𝑏𝑛𝑓𝑠𝑏 𝑨 𝑑𝑏𝑜𝑝𝑜𝑗𝑑𝑏𝑚 𝑨 𝑥𝑝𝑠𝑚𝑒 1 1 x 𝑨 = 𝑔 One thing is still missing: perspective. 𝑨 = 𝑜 -z y 23
INFOGR – Lecture 6 – "Transformations" 24 Perspective Perspective projection Q: What is perspective? A: The size of an object on the screen is proportional to 1/𝑨 . More precisely: 𝑒 𝑒 𝑧 𝑡 = 𝑨 𝑧 (and 𝑦 𝑡 = 𝑨 𝑦 ) where 𝑒 is the distance of the view plane to the camera. Q: How do we capture scaling based on distance in a matrix? A: … Dividing by z can’t be done using linear nor affine transforms. 24
INFOGR – Lecture 6 – "Transformations" 25 Perspective Perspective projection Let’s have a look at homogeneous coordinates again. Recall: 𝑏 1 𝑦 + 𝑐 1 𝑧 + 𝑑 1 𝑨 𝑦 𝑏 1 𝑐 1 𝑑 1 𝑧 𝑏 2 𝑦 + 𝑐 2 𝑧 + 𝑑 2 𝑨 = 𝑏 2 𝑐 2 𝑑 2 𝑨 𝑏 3 𝑦 + 𝑐 3 𝑧 + 𝑑 3 𝑨 𝑏 3 𝑐 3 𝑑 3 With homogeneous coordinates, we get: (𝑏 1 𝑦 + 𝑐 1 𝑧 + 𝑑 1 𝑨 + 𝑈 𝑦 )/1 𝑏 1 𝑐 1 𝑑 1 𝑈 𝑏 1 𝑦 + 𝑐 1 𝑧 + 𝑑 1 𝑨 + 𝑈 𝑦 𝑦 𝑦 (𝑏 2 𝑦 + 𝑐 2 𝑧 + 𝑑 2 𝑨 + 𝑈 𝑧 )/1 𝑏 2 𝑐 2 𝑑 2 𝑈 𝑧 𝑏 2 𝑦 + 𝑐 2 𝑧 + 𝑑 2 𝑨 + 𝑈 𝑧 𝑧 = = 𝑨 (𝑏 3 𝑦 + 𝑐 3 𝑧 + 𝑑 3 𝑨 + 𝑈 𝑨 )/1 𝑏 3 𝑐 3 𝑑 3 𝑈 𝑏 3 𝑦 + 𝑐 3 𝑧 + 𝑑 3 𝑨 + 𝑈 𝑨 𝑨 1 0 0 0 1 1 1 25
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