Shadows in Computer Graphics
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1 Shadows in Computer Graphics Steven Janke November 2014 Steven Janke (Seminar) Shadows in Computer Graphics November / 49
2 Shadows (from Doom) Steven Janke (Seminar) Shadows in Computer Graphics November / 49
3 Simple Shadows Steven Janke (Seminar) Shadows in Computer Graphics November / 49
4 Shadows give position information Steven Janke (Seminar) Shadows in Computer Graphics November / 49
5 Shadow Geometry Steven Janke (Seminar) Shadows in Computer Graphics November / 49
6 Camera Model Steven Janke (Seminar) Shadows in Computer Graphics November / 49
7 View Frustum Near Far z Camera 0,0,0 View Plane Steven Janke (Seminar) Shadows in Computer Graphics November / 49
8 Ideal City (circa 1485) Steven Janke (Seminar) Shadows in Computer Graphics November / 49
9 Perspective Projection y C C z E D W E = (0, 0, e), C = (x, y, z), C = (x s, y s ) EW = e, CD = y, ED = e z EWC EDC y s = y e e z = y 1 z e Steven Janke (Seminar) Shadows in Computer Graphics November / 49
10 Albrecht Durer (Man drawing a Lute ) Steven Janke (Seminar) Shadows in Computer Graphics November / 49
11 Alberti Perspective Diagram 1. Draw lines from front tile corners to center point C. 2. Select point R on horizontal line so CR is distance to painting. 3. Connect R with front tile corners. 4. Draw horizontal lines through intersections of lines in 3 and vertical line through C. 5. Diagonals through tiles are projected into diagonals. Steven Janke (Seminar) Shadows in Computer Graphics November / 49
12 Pedro Berreuguete - Anunciation (circa 1500) Steven Janke (Seminar) Shadows in Computer Graphics November / 49
13 Masaccio - Trinity (1426) Steven Janke (Seminar) Shadows in Computer Graphics November / 49
14 Calculating Cube Perspective y s = y 1 z e x s = x 1 z e Example (Cube Vertices) Eye Coordinates: (0,0,4) World Coordinates: (1, 1, 1), (1, 1, 1), ( 1, 1, 1), ( 1, 1, 1) (1, 1, 1), (1, 1, 1), ( 1, 1, 1), ( 1, 1, 1) Screen Coordinates: (1.33, 1.33), (1.33, 1.33), ( 1.33, 1.33), ( 1.33, 1.33) (0.75, 0.75), (0.75, 0.75), ( 0.75, 0.75), ( 0.75, 0.75) Steven Janke (Seminar) Shadows in Computer Graphics November / 49
15 Cubes in Perspective Steven Janke (Seminar) Shadows in Computer Graphics November / 49
16 Projections Summary Perspective projections and Shadow projections are both projections from a point onto a plane. Next steps: For shadows, we generalize projection to arbitrary plane. Describe calculations compactly and efficiently. Steven Janke (Seminar) Shadows in Computer Graphics November / 49
17 Vectors v = (x 1, y 1, z 1 ) and w = (x 2, y 2, z 2 ) are displacements. Algebra: v + w = (x 1 + x 2, y 1 + y 2, z 1 + z 2 ) and a v = (ax 1, ay 1, az 1 ). Dot Product: v w = v w cos(θ) = x 1 x 2 + y 1 y 2 + z 1 z 2. (If v and w are perpendicular, then v w = 0.) Cross Product: i j k v w = x 1 y 1 z 1 x 2 y 2 z 2 = y 1 z 1 y 2 z 2 i x 1 z 1 x 2 z 2 j + x 1 y 1 x 2 y 2 k Steven Janke (Seminar) Shadows in Computer Graphics November / 49
18 Vector Products x 1,y 1 A B v v w B 0,0 Θ w Dot Product x 2,y 2 A Cross Product Steven Janke (Seminar) Shadows in Computer Graphics November / 49
19 Transformations a 11 a 12 a 13 x a 11 x + a 12 y + a 13 z T v = a 21 a 22 a 23 y = a 21 x + a 22 y + a 23 z a 31 a 32 a 33 z a 31 x + a 32 y + a 33 z Rotation around z-axis uses this matrix: cos θ sin θ 0 R z = sin θ cos θ Matrices give linear transformations: T ( v + w) = T v + T w and T (a v) = at v Cannot represent translations or projections. Steven Janke (Seminar) Shadows in Computer Graphics November / 49
20 Homogeneous Coordinates The point P 0 = ( 1, 5) is on the 2D line 3x + 2y = 7. The vector equation of the line: (3, 2) (P P 0 ) = (3, 2) (x + 1, y 5) = 0 Let P = (x, y) = ( x h w h, y h w h ) be any point on the line 3x + 2y = 7. = 3x h + 2y h 7w h = (3, 2, 7) (x h, y h, w h ) = 0 (x h, y h, w h ) are homogeneous coordinates for the point P. Since w h is arbitrary, there are infinitely many sets of homogeneous coordinates representing P. For example, P 0 = ( 1, 5, 1) = ( 2, 10, 2) = ( 0.5, 2.5, 0.5) Two-dimensional Homogeneous Line equation: n P = 0 Steven Janke (Seminar) Shadows in Computer Graphics November / 49
21 Homogeneous Coordinates for Lines Example (2D Line coordinates) P 1 = (3, 2, 1) and P 2 = (5, 7, 3) determine a two-dimensional line. n (3, 2, 1) = 0 and n (5, 7, 3) = 0. n = (3, 2, 1) (5, 7, 3) = ( 1, 4, 11) The homogeneous coordinates ( 1, 4, 11) represent the line. Both points and lines in two dimensions can be represented by homogeneous coordinates (x, y, w). Steven Janke (Seminar) Shadows in Computer Graphics November / 49
22 Calculating with Homogeneous Coordinates Example (2D Intersection Point) Consider two lines: (2, 2, 1) and (6, 5, 2) (They represent the lines 2x + 2y 1 = 0 and 6x 5y + 2 = 0) P is the point of intersection. (2, 2, 1) P = 0 and (6, 5, 2) P = 0 P must be a vector perpendicular to the two homogeneous line vectors. P = (2, 2, 1) (6, 5, 2) = ( 1, 10, 22) is the cross product. P = ( 1, 10, 22) represents the Cartesian point P = ( 1 22, ) Steven Janke (Seminar) Shadows in Computer Graphics November / 49
23 Points at Infinity Lines 2x + 4y 8 = 0 and 2x + 4y 10 = 0 are parallel. The homogeneous point (4, 2, 0) is on both lines. Points of the form (x h, y h, 0) are points at infinity. Notice that (4, 2, 0) and (5, 3, 0) are distinct points at infinity. Points in 2D Points at Infinity Steven Janke (Seminar) Shadows in Computer Graphics November / 49
24 Homogeneous Coordinates in Three Dimensions Homogeneous coordinates for three dimensional points add a fourth coordinate: Cartesian (x, y, z) = Homogeneous (x, y, z, 1) or (tx, ty, tz, t) Since planes are determined by a normal and a point, (tx, ty, tz, t) also represents a plane. Homogeneous plane equation: n P = 0 Lines have homogeneous coordinates called Plücker coordinates. Steven Janke (Seminar) Shadows in Computer Graphics November / 49
25 Perspective Matrix Now we can express the perspective transformation as a matrix multiplication: x x T (P) = MP = y z = y e z e In the space of homogeneous coordinates (Projective Space), the perspective transformation is a linear function. Steven Janke (Seminar) Shadows in Computer Graphics November / 49
26 Perspective Drawing Steven Janke (Seminar) Shadows in Computer Graphics November / 49
27 Two Point Perspective Steven Janke (Seminar) Shadows in Computer Graphics November / 49
28 Projective Geometry Girard Desargues ( ) was the founding father. Parallel lines intersect in a point at infinity: (x h, y h, 0) Points at infinity fall on a line. Duality: In 2D, for any theorem about points there is a theorem about lines. No concept of length or angle. Projective transformations are linear transformations. In 2D, there is a projective transformation that sends a given four points to another specified four points. Steven Janke (Seminar) Shadows in Computer Graphics November / 49
29 Desargues Theorem Steven Janke (Seminar) Shadows in Computer Graphics November / 49
30 Additional Vector Algebra Tensor Product v x v w = v w T = v y [ ] v x w x v x w y v x w z w x w y w z = v y w x v y w y v y w z v z v z w x v z w y v z w z Vector Triple Product A ( B C) = ( A C) B ( A B) C Dot to Tensor ( A C) B = ( B A) C Steven Janke (Seminar) Shadows in Computer Graphics November / 49
31 2D Projection L E P P T P Line through E and P is E P. T (P) = P = L (E P) = ( L P)E ( L E)P = ((E L) ( L E)I )P = MP Steven Janke (Seminar) Shadows in Computer Graphics November / 49
32 2D Projection Example Project P = (3, 1) onto the line 6x + y 5 = 0 from the point (8, 2). L = (6, 1, 5) E = (8, 2, 1) P = (3, 1, 1) M = (E L) ( L E)I = = T (P) = MP = = Cartesian coordinates for P are (23/31, 17/31). Steven Janke (Seminar) Shadows in Computer Graphics November / 49
33 3D Projection E P n P P = αp + βe. n (αp + βe) = 0 = α = β( n E) ( n P) P = β( n E) n P P + βe P = T (P) = ( n P)E ( n E)P = (E n)p ( n E)P = M = (E n) ( n E)I Steven Janke (Seminar) Shadows in Computer Graphics November / 49
34 This gives Cartesian coordinates (3, 6, 2) Steven Janke (Seminar) Shadows in Computer Graphics November / 49 3D Projection Example Eye point: E = (7, 2, 6) Vertex: P = (4, 5, 0) Plane: 2x y + 2z = 4 n = (2, 1, 2, 4) E = (7, 2, 6, 1) P = (4, 5, 0, 1) M = (E n) ( n E)I = = = P = ( 63, 126, 42, 21)
35 Compare to earlier 3D matrix M = (E n) ( n E)I Perspective projection: Viewing plane is the xy plane with homogeneous representation n = (0, 0, 1, 0). The eye point is E = (0, 0, e, 1) and P = (x, y, z, 1). M = e e e = e e 1 Steven Janke (Seminar) Shadows in Computer Graphics November / 49
36 Shadow Geometry Steven Janke (Seminar) Shadows in Computer Graphics November / 49
37 Points in Shadow Shadow: Fill in the polygon determined by projected vertices. For convex objects, shadows are convex. Vertices that are not incident on both visible and hidden faces are inside the shadow. Steven Janke (Seminar) Shadows in Computer Graphics November / 49
38 Multiple Shadows Steven Janke (Seminar) Shadows in Computer Graphics November / 49
39 Ray Tracing Camera v To Light Source View Window Reflected Ray Steven Janke (Seminar) Shadows in Computer Graphics November / 49
40 Shadow Ray Light P Shadow Steven Janke (Seminar) Shadows in Computer Graphics November / 49
41 Soft Shadows Steven Janke (Seminar) Shadows in Computer Graphics November / 49
42 Penumbra & Umbra Light Object Umbra Penumbra Penumbra Steven Janke (Seminar) Shadows in Computer Graphics November / 49
43 More Soft Shadows Steven Janke (Seminar) Shadows in Computer Graphics November / 49
44 Multiple Lights Steven Janke (Seminar) Shadows in Computer Graphics November / 49
45 Shadow Volume Steven Janke (Seminar) Shadows in Computer Graphics November / 49
46 More Multiple Lights Steven Janke (Seminar) Shadows in Computer Graphics November / 49
47 Further Refinements Shadow Maps Curves (NURBS) Complex Lighting Models (Radiosity) Steven Janke (Seminar) Shadows in Computer Graphics November / 49
48 Complex Shadows Steven Janke (Seminar) Shadows in Computer Graphics November / 49
49 Reference (2015 Steven Janke (Seminar) Shadows in Computer Graphics November / 49
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