Shadows in Computer Graphics

Steven Janke

November 2014

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 1 / 49 Shadows (from Doom)

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 2 / 49 Simple Shadows

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 3 / 49 Shadows give position information

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 4 / 49 Shadow Geometry

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 5 / 49 Camera Model

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 6 / 49 View Frustum

Far Near -z

Camera

H0,0,0L View Plane

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 7 / 49 Ideal City (circa 1485)

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 8 / 49 Perspective Projection

y

* C C

E W z D

∗ E = (0, 0, e), C = (x, y, z), C = (xs , ys )

EW = e, CD = y, ED = e − z

∗ y · e y 4EWC ∼ 4EDC ys = = z e − z 1 − e

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 9 / 49 Albrecht Durer (Man drawing a Lute - 1525)

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 10 / 49 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 2014 11 / 49 Pedro Berreuguete - Anunciation (circa 1500)

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 12 / 49 Masaccio - Trinity (1426)

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 13 / 49 Calculating Cube Perspective

y x ys = z xs = z 1 − e 1 − 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 2014 14 / 49 Cubes in Perspective

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 15 / 49 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 2014 16 / 49 Vectors

~v = (x1, y1, z1) and w~ = (x2, y2, z2) are displacements.

Algebra: ~v + w~ = (x1 + x2, y1 + y2, z1 + z2) and a~v = (ax1, ay1, az1).

Dot Product: ~v · w~ = |~v||w~ |cos(θ) = x1x2 + y1y2 + z1z2. (If ~v and w~ are perpendicular, then ~v · w~ = 0.) Cross Product:

~i ~j ~k y1 z1 ~ x1 z1 ~ x1 y1 ~ ~v × w~ = x1 y1 z1 = i − j + k y2 z2 x2 z2 x2 y2 x2 y2 z2

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 17 / 49 Vector Products

Hx1,y1L A ´ B

v v - w B

Θ w H0,0L Hx2,y2L A Dot Product

Cross Product

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 18 / 49 Transformations

      a11 a12 a13 x a11x + a12y + a13z T~v = a21 a22 a23 y = a21x + a22y + a23z a31 a32 a33 z a31x + a32y + a33z Rotation around z-axis uses this matrix: cos θ − sin θ 0 Rz = sin θ cos θ 0 0 0 1

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 2014 19 / 49 Homogeneous Coordinates

The point P0 = (−1, 5) is on the 2D line 3x + 2y = 7. The vector equation of the line:

(3, 2) · (P − P0) = (3, 2) · (x + 1, y − 5) = 0

Let P = (x, y) = ( xh , yh ) be any point on the line 3x + 2y = 7. wh wh

=⇒ 3xh + 2yh − 7wh = (3, 2, −7) · (xh, yh, wh) = 0

(xh, yh, wh) are homogeneous coordinates for the point P. Since wh is arbitrary, there are infinitely many sets of homogeneous coordinates representing P. For example,

P0 = (−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 2014 20 / 49 Homogeneous Coordinates for Lines

Example (2D Line coordinates)

P1 = (3, 2, 1) and P2 = (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 2014 21 / 49 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. 1 10 P = (−1, −10, −22) represents the Cartesian point P = ( 22 , 22 )

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 22 / 49 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 (xh, yh, 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 2014 23 / 49 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 Pl¨ucker coordinates.

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 24 / 49 Perspective Matrix

Now we can express the perspective transformation as a matrix multiplication:

1 0 0 0 x  x  0 1 0 0 y  y  T (P) = MP =     =   0 0 0 0 z  0  1 z 0 0 − e 1 1 1 − e

In the space of homogeneous coordinates (Projective Space), the perspective transformation is a linear function.

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 25 / 49 Perspective Drawing

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 26 / 49 Two Point Perspective

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 27 / 49 Projective Geometry

Girard Desargues (1591 - 1661) was the founding father.

Parallel lines intersect in a point at infinity: (xh, yh, 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 2014 28 / 49 Desargues Theorem

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 29 / 49 Additional Vector Algebra

Tensor Product     vx vx wx vx wy vx wz T   ~v ⊗ w~ = ~vw~ = vy  wx wy wz = vy wx vy wy vy wz  vz vz wx vz wy vz wz

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 2014 30 / 49 2D Projection

L

E P’=THPL P

Line through E and P is E × P.

T (P) = P0 = ~L × (E × P) = (~L · P)E − (~L · E)P = ((E ⊗ ~L) − (~L · E)I )P = MP

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 31 / 49 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 48 8 −40 45 0 0   3 8 −40 = 12 2 −10 −  0 45 0  = 12 −43 −10 6 1 −5 0 0 45 6 1 −50

 3 8 −40 3 −23 T (P) = MP = 12 −43 −10 1 = −17 6 1 −50 1 −31 Cartesian coordinates for P are (23/31, 17/31).

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 32 / 49 3D Projection

E

P

n 0 P’ P = αP + βE.

−β(~n · E) ~n · (αP + βE) = 0 =⇒ α = (~n · P) −β(~n · E) P0 = P + βE ~n · P

P0 = 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 2014 33 / 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)

14 −7 14 28 28 0 0 0   4 −2 4 8   0 28 0 0  M = (E ⊗ ~n) − (~n · E)I =   −   12 −6 12 24  0 0 28 0  2 −1 2 4 0 0 0 28 −14 −7 14 28   4 −30 4 8  0 =   =⇒ P = (−63, −126, 42, −21)  12 −6 −16 24  2 −1 2 −24

This gives Cartesian coordinates (3, 6, −2) Steven Janke (Seminar) Shadows in Computer Graphics November 2014 34 / 49 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).

−e 0 0 0  1 0 0 0  0 −e 0 0  0 1 0 0 M =   = −e    0 0 0 0  0 0 0 0 1 0 0 1 −e 0 0 − e 1

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 35 / 49 Shadow Geometry

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 36 / 49 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 2014 37 / 49 Multiple Shadows

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 38 / 49 Ray Tracing

To Light Source Camera v

View Window

Reflected Ray

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 39 / 49 Shadow Ray

Light

Shadow P

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 40 / 49 Soft Shadows

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 41 / 49 Penumbra & Umbra

Light

Object

Umbra

Penumbra Penumbra

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 42 / 49 More Soft Shadows

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 43 / 49 Multiple Lights

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 44 / 49 Shadow Volume

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 45 / 49 More Multiple Lights

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 46 / 49 Further Refinements

Shadow Maps

Curves (NURBS)

Complex Lighting Models (Radiosity)

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 47 / 49 Complex Shadows

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 48 / 49 Reference (2015

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 49 / 49